Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture
aa r X i v : . [ m a t h . QA ] F e b Q-W–algebras, Zhelobenko operators and a proof ofDe Concini–Kac–Procesi conjecture
A. SevostyanovFebruary 8, 2021 ntroduction
In this monograph we consider a phenomenon which occurs in the study of certain classes and categories ofrepresentations of semisimple Lie algebras, groups of Lie type, and the related quantum groups. This phenomenonis similar to the classical Schur–Weyl duality. However, the relevant classes of representations are quite differentfrom the finite-dimensional irreducible representations of the general linear group or, more generally, of complexsemisimple Lie groups which appear in the Schur–Weyl setting.All examples of the above mentioned type are realizations of the following quite general construction a homo-logical version of which was suggested in [94] (in fact, in the quantum group case the construction presented belowrequires some technical modifications; we shall not discuss them in the introduction). Let A be an associativealgebra over a unital ring k , B ⊂ A a subalgebra with a character χ : B → k . Denote by k χ the correspondingrank one representation of B . Let Q χ = A ⊗ B k χ be the induced representation of A .Let Hk ( A, B, χ ) = End A ( Q χ ) opp be the algebra of A –endomorphisms of Q χ with the opposite multiplication.One says that the algebra Hk ( A, B, χ ) is obtained from A by a quantum constrained reduction with respect to thesubalgebra B . Hk ( A, B, χ ) is an algebra of Hecke type. Indeed, if A is the group algebra of a Chevalley groupover a finite field, B the group algebra of a Borel subgroup in it, and χ is the trivial complex representation of theBorel subgroup one obtains the Iwahori–Hecke algebra this way (see [49]).For any representation V of A the algebra Hk ( A, B, χ ) naturally acts in the space V χ = Hom A ( Q χ , V ) = Hom B ( k χ , V )by compositions of homomorphisms and for any Hk ( A, B, χ )–module
W Q χ ⊗ Hk ( A,B,χ ) W is a left A –module. Let Hk ( A, B, χ ) − mod be the category of left Hk ( A, B, χ )–modules and A − mod χB the category of left A –modulesof the form Q χ ⊗ Hk ( A,B,χ ) W , where W ∈ Hk ( A, B, χ ) − mod, with morphisms induces by morphisms of left Hk ( A, B, χ )–modules. Then we have the following theorem.
Theorem 1. A − mod χB is a full subcategory in the category of left A –modules, and the functors Hom A ( Q χ , · ) and Q χ ⊗ Hk ( A,B,χ ) · yield mutually inverse equivalences of the categories, A − mod χB ≃ Hk ( A, B, χ ) − mod . (1) Proof.
Let
W, W ′ ∈ Hk ( A, B, χ ) − mod. Then by the Frobenius reciprocity and by the definition of the algebra Hk ( A, B, χ )Hom A ( Q χ , Q χ ⊗ Hk ( A,B,χ ) W ) = Hom B ( k χ , Q χ ⊗ Hk ( A,B,χ ) W ) = Hom B ( k χ , Q χ ) ⊗ Hk ( A,B,χ ) W == Hom A ( Q χ , Q χ ) ⊗ Hk ( A,B,χ ) W = Hk ( A, B, χ ) opp ⊗ Hk ( A,B,χ ) W = W. This implies the second claim of this theorem.By the formula above we also haveHom A ( Q χ ⊗ Hk ( A,B,χ ) W ′ , Q χ ⊗ Hk ( A,B,χ ) W ) = Hom Hk ( A,B,χ ) ( W ′ , Hom A ( Q χ , Q χ ⊗ Hk ( A,B,χ ) W )) == Hom Hk ( A,B,χ ) ( W ′ , W ) , and hence A − mod χB is a full subcategory in the category of left A –modules.At first sight the category A − mod χB looks a bit exotic. But it turns out that in many situations it has alternativedescriptions in terms of the algebra A , its subalgebra B and the character χ only, and actually such categories and3the related algebras of Hecke type played a very important, if not central, role in representation theory for at leastlast sixty years.An important example of equivalences of the type mentioned in Theorem 1 was considered in [60]. In thispaper Kostant showed that algebraic analogues of the principal series representations, called irreducible Whittakermodules, for a complex semisimple Lie algebra g are in one–to–one correspondence with the one–dimensionalrepresentations of the center Z ( U ( g )) of the enveloping algebra U ( g )In the situation considered in [60] the algebra Z ( U ( g )) is isomorphic to Hk ( A, B, χ ) with A = U ( g ), B = U ( n − ),where n − is a nilradical of g , and χ being a non–singular character of U ( n − ) which does not vanish on all simpleroot vectors in n − . The category A − mod χB in this case can be described as the category of g –modules on which x − χ ( x ) acts locally nilpotently for any x ∈ n − .This correspondence was generalized in [76] and in the Appendix to [82] to a more general categorical settingand the categorical equivalence established in the Appendix to [82] is called the Skryabin equivalence.Similar equivalences were obtained in [82] in the case of semisimple Lie algebras over fields of prime characteristicand in [101] in case of quantum groups associated to complex semisimple Lie algebras for generic values of thedeformation parameter. Various approaches to the proofs of the above mentioned statements have been developedin [36, 105].An analogous construction appears also in the case of finite groups of Lie type (see [17], Chapter 10).Note that the problem of classification of Hk ( A, B, χ )–modules is usually very difficult (see e.g. [38] for the caseof the Iwahori–Hecke algebra). Sometimes it is easier to classify irreducible objects in the category A − mod χB andthen to translate the result to the category Hk ( A, B, χ ) − mod (see [64, 65] for the case of algebras Hk ( A, B, χ )considered in [82]).In all cases considered in [60, 76, 82, 101] the algebras A and B , the characters χ and the appropriate categories A − mod χB and Hk ( A, B, χ ) − mod of representations of A and of Hk ( A, B, χ ) are relatively easy to define. It ismuch more difficult to obtain alternative descriptions of the category A − mod χB . However, one should note thatthe approach to this problem in all papers mentioned above is slightly different: all those papers start with thedescription of a category of A –modules in intrinsic terms using the algebra A , its subalgebra B and the character χ . And then one proves that this category is equivalent to the category Hk ( A, B, χ ) − mod, the equivalence beingestablished using the functors Hom A ( Q χ , · ) and Q χ ⊗ Hk ( A,B,χ ) · . Finally one deduces that this category actuallycoincides with A − mod χB .In the Lie algebra case the most simple proofs of statements of this kind were proposed in the Appendix to[82] in the zero characteristic case and in [105] in the prime characteristic case. But the phenomenon behindthese proofs is already manifest in [60]. Namely, in the case of Lie algebras over fields of zero characteristic onealways has A = U ( g ) and B = U ( m ) for some reductive Lie algebra g and a nilpotent Lie subalgebra m ⊂ g , andthe above mentioned phenomenon amounts to introducing a second U ( m )–module structure on Q χ by tensoringwith the one–dimensional representation k χ and to demonstrating that for k = C a certain “classical limit” of the U ( m )–module Q χ ⊗ k χ is isomorphic to the algebra of regular functions C [ C ] on a closed algebraic variety C , andthe “classical limit” of the U ( m )–action on Q χ ⊗ k χ is induced by a free action of the complex unipotent algebraicgroup M corresponding to the Lie algebra m on C . The “classical limits” here are understood in the sense of takingassociate graded objects with respect to suitable filtrations.The action M × C → C has a global cross-section Σ ⊂ C , called a Slodowy slice, so that the action map M × Σ → C (2)is an isomorphism of varieties, and C [ C ] ≃ C [Σ] ⊗ C [ M ] . (3)The space W = C [Σ] ≃ C [ C ] M can be regarded as a “classical limit” of Hk ( A, B, χ ) which is called a W–algebrain this case. We can also write C [ C ] ≃ W [ M ], where W [ M ] is the algebra of regular functions on M with valuesin W . In fact W carries the natural structure of a Poisson algebra. It is called a Poisson W–algebra.Let A − mod χB be the category of left A –modules V for which the U ( m )–action on V ⊗ k χ is locally nilpotent.In the Appendix to [82] it is shown that if one equips V ∈ A − mod χB with a second U ( m )–module structure bytensoring with k χ then, as a U ( m )–module, V ⊗ k χ is isomorphic to hom k ( U ( m ) , V χ ), V ⊗ k χ ≃ hom k ( U ( m ) , V χ ) ≃ V χ [ M ] , (4)where hom k stands for the space of homomorphisms vanishing on some power of the augmentation ideal of U ( m ), V χ = Hom U ( m ) ( k χ , V ) is called the space of Whittaker vectors in V , and, as above, the latter isomorphism holdsif k = C . In the Appendix to [82] it is shown that isomorphisms (4) directly imply an isomorphism between thecategory of left A –modules V for which the U ( m )–action on V ⊗ k χ is locally nilpotent and the correspondingcategory A − mod χB introduced before Theorem 1.Isomorphisms of type (2) occur in the quantum group setting as well (see [98, 99, 101]), and the same idea isapplied in [101] to establish similar categorical equivalences in the quantum group case for generic values of thedeformation parameter.In [100, 104] it was observed that an isomorphism of type (2) gives rise to a natural projection operatorΠ : C [Σ] → C [ C ] M ≃ C [Σ] = W . Namely, according to (2) any x ∈ C can be uniquely represented in the form x = n ( x ) ◦ σ ( x ) , n ( x ) ∈ M, σ ( x ) ∈ Σ . (5)If for f ∈ C [ C ] we define Π f ∈ C [ C ] by (Π f )( x ) = f ( n − ( x ) ◦ x ) = f ( σ ( x )) (6)then Π f is an M –invariant function, and any M –invariant regular function on C can be obtained this way. Moreover,by the definition Π = Π, i.e. Π is a projection onto C [ C ] M .In the quantum group setting considered in [98, 99, 101] the “classical limiting” variety C is always a closedsubvariety in a complex semisimple algebraic Lie group G , Σ is an analogue of a Slodowy slice for G introduced in[98, 103], and M is a unipotent subgroup of G , where the “classical limit” simply corresponds now to the q = 1specialization of the deformation parameter q . The peculiarity of the quantum group case is that every elementof M can be uniquely represented as an ordered product of elements of some one–parameter subgroups M i ⊂ G , i = 1 , . . . , c corresponding to roots, i.e. M = M . . . M c . If we denote by t i the parameter in M i and by X i ( t i ) theelement of M i corresponding to the value t i ∈ C of the parameter then factorizing n ( x ) in (5) as follows n ( x ) = X ( t ( x )) . . . X c ( t c ( x )) (7)one can express the operator Π as a composition of operators Π i ,(Π i f )( x ) = f ( X i ( − t i ( x )) ◦ x ) , (8)Π f = Π . . . Π c f. (9) t i ( x ) here can be regarded as regular functions on C ⊂ G .The first miracle of the quantum group case is that there are explicit formulas for the functions t i ( x ) in (7)expressing them in terms of matrix elements of finite-dimensional irreducible representations of G . These formulaswere obtained in [104].The main objective of this book is to obtain quantum group counterparts of these formulas and to define properquantum group analogues P i and Π q of the operators Π i and Π. This provides a description of quantum groupanalogues of W–algebras, called q-W–algebras, as images of operators Π q . This description implies that q-W–algebras belong to the class of the so-called Mickelsson algebras (see e.g. [115, 118, 119, 120, 122] and [123], Ch.4). Magically, the classical formulas for t i ( x ) and formulas (8) can be directly extrapolated to the quantum case,so the operator Π q is given in a factorized form similar to (9). Note that no operators similar to P i and Π q can bedefined in the Lie algebra setting discussed above.Using the quantum group analogues B i of the functions t i ( x ) one can also construct natural bases in modules V from the corresponding category A − mod χB and establish isomorphisms similar to (4) in the case when thedeformation parameter is not a root of unity. Recall that in the Lie algebra case with k = C for any V ∈ A − mod χB the Skryabin equivalence provides an isomorphism V ≃ Q χ ⊗ Hk ( A,B,χ ) V χ . If we denote by V χ the “classical limit”of V χ then recalling that the “classical limit” of Q χ is C [ C ] and the “classical limit” of Hk ( A, B, χ ) is C [Σ] we inferfrom (3) that the “classical limit” of Q χ ⊗ Hk ( A,B,χ ) V χ ≃ V is C [ C ] ⊗ C [Σ] V χ ≃ ( C [Σ] ⊗ C [ M ]) ⊗ C [Σ] V χ ≃ C [ M ] ⊗ V χ . These isomorphisms together with (5) and (7) give a hint how to construct natural bases in modules from thecategory V ∈ A − mod χB in the quantum group case. Namely, if V is such a module it is natural to expect thatif one picks up a linear basis v j , j = 1 , . . . in the space of Whittaker vectors V χ then the elements of V given byproperly defined ordered monomials in B i applied to v j , j = 1 , . . . form a linear basis in V . We show that thisis indeed the case. These bases are key ingredients for an alternative proof of the Skryabin correspondence forquantum groups.Operators conceptually similar to Π q appeared in the literature a long time ago as the projection operators ontosubspaces of singular vectors in some modules over a complex finite-dimensional semisimple Lie algebra g the actionof a nilradical n − ⊂ g on which is locally nilpotent. The first example of such operators, called extremal projectionoperators, for g = sl was explicitly constructed in [66]. In papers [2, 3, 4] the results of [66] were generalized tothe case of arbitrary complex semisimple Lie algebras, and explicit formulas for extremal projection operators wereobtained. A summary of these results can be found in [111]. Later, using a certain completion of an extensionof the universal enveloping algebra of g , Zhelobenko observed in [115] that the existence of extremal projectionoperators is an almost trivial fact. In [115] he also introduced a family of operators which are analogues to ouroperators P i . These operators are called now Zhelobenko operators. Properties of extremal projection operatorsand of the Zhelobenko operators have been extensively studied in [115]–[122], and the results obtained in thesepapers were summarized in book [123].In out terminology the situation considered in these works corresponds to the case when A = U ( g ), B = U ( n − )and χ is the trivial character of U ( n − ). As observed in [100], in this case C = b − , the Borel subalgebra b − ⊂ g containing n − , and the action of the unipotent group N − corresponding to n − on C is induced by the adjoint actionof a Lie group G with the Lie algebra g on g . This action is not free but is gives rise to a birational equivalence N − × h → b − , where h = b − / n − is a Cartan subalgebra. In [100] it is shown that using this birational equivalence one can stilldefine operators similar to Π i and Π acting on a certain localization of the algebra of regular functions C [ b − ] andthese operators are “classical limits” of the Zhelobenko and of the extremal projection operators, respectively.Remarkably, as observed in [105], the arguments from the Appendix to [82] are applicable to obtain alternativedescriptions of the corresponding categories A − mod χB from [82] for Lie algebras over fields of prime characteristic.Although no “classical” geometric group action picture is available in this case.Along the same line, formulas for the quantum group analogues B i of the functions t i ( x ) and for the operators P i and Π q can be specialized to the case when q is a primitive odd m -th root of unity ε subject to a few otherconditions depending on the Cartan matrix of the corresponding semisimple Lie algebra g . This provides technicaltools for the proof of a root of unity version of the Skryabin correspondence for quantum groups. Similarly to thecase of generic q one can construct bases in modules V from the corresponding category A − mod χB . In case when q is a root of unity all such modules are finite-dimensional, and if one picks up a linear basis v j , j = 1 , . . . , n in thespace of Whittaker vectors V χ then the elements of V given by applied to v j , j = 1 , . . . , n properly defined orderedmonomials in B i , i = 1 , . . . , c powers of which are truncated at the degree m form a linear basis in V . In particular,the dimension of V is divisible by m c . It turns out that any finite-dimensional module over the standard quantumgroup U ε ( g ), where ε is a primitive odd m -th root of unity subject to the extra conditions mentioned above, belongsto one of the categories A − mod χB with appropriate A , B and χ , so its dimension is divisible by b = m c . Moreover,the number b is equal to the number from the De Concini–Kac–Procesi conjecture on dimensions of irreduciblemodules over quantum groups at roots of unity suggested in [22]. Thus our result confirms this conjecture. Due toits importance we are going to discuss the De Concini–Kac–Procesi conjecture in more detail.It is very well known that the number of simple modules for a finite-dimensional algebra over an algebraicallyclosed field is finite. However, often it is very difficult to classify such representations. In some important particularexamples even dimensions of simple modules over finite-dimensional algebras are not known.One of the important examples of that kind is representation theory of semisimple Lie algebras over algebraicallyclosed fields of prime characteristic. Let g ′ be the Lie algebra of a semisimple algebraic group G ′ over an algebraicallyclosed field k of characteristic p >
0. Let x x [ p ] be the p -th power map of g ′ into itself. The structure of theenveloping algebra of g ′ is quite different from the zero characteristic case. Namely, the elements x p − x [ p ] , x ∈ g ′ are central. For any linear form θ on g ′ , let U θ be the quotient of the enveloping algebra of g ′ by the ideal generatedby the central elements x p − x [ p ] − θ ( x ) p with x ∈ g ′ . Then U θ is a finite-dimensional algebra. Kac and Weisfeilerproved that any simple g ′ -module can be regarded as a module over U θ for a unique θ as above (this explainswhy all simple g ′ –modules are finite-dimensional). The Kac–Weisfeiler conjecture formulated in [54] and provedin [83] says that if the G ′ –coadjoint orbit of θ has dimension dim O θ then p dim O θ divides the dimension of everyfinite-dimensional U θ –module.One can identify θ with an element of g ′ via the Killing form and reduce the proof of the Kac–Weisfeilerconjecture to the case of nilpotent θ . In that case Premet defines in [83] a subalgebra U θ ( m θ ) ⊂ U θ generatedby a Lie subalgebra m θ ⊂ g ′ such that U θ ( m θ ) has dimension p dim O θ and every finite-dimensional U θ –module is U θ ( m θ )–free. Verification of the latter fact uses the theory of support varieties (see [33, 34, 35, 84]). Namely,according to the theory of support varieties, in order to prove that a U θ –module is U θ ( m θ )–free one should checkthat it is free over every subalgebra U θ ( x ) generated in U θ ( m θ ) by a single element x ∈ m θ .There is a more elementary and straightforward proof of the Kac–Weisfeiler conjecture given in [81]. The mostsimple proof of this conjecture follows from the results of [105] on a prime characteristic version of the Skryabinequivalence which we already discussed above. A proof of the conjecture for p > h , where h is the Coxeter numberof the corresponding root system, using localization of D –modules was presented in [6].Another important example of finite-dimensional algebras is related to the theory of quantum groups at rootsof unity. Let g be a complex finite-dimensional semisimple Lie algebra. A remarkable property of the standardDrinfeld-Jimbo quantum group U ε ( g ) associated to g , where ε is a primitive m -th root of unity, is that its centercontains a huge commutative subalgebra isomorphic to the algebra Z G of regular functions on (a finite coveringof a big cell in) a complex algebraic group G with Lie algebra g . In this book we consider the simply connectedversion of U ε ( g ) and the case when m is odd. In that case G is the connected, simply connected algebraic groupcorresponding to g .Consider finite-dimensional representations of U ε ( g ), on which Z G acts according to non–trivial characters η g given by evaluation of regular functions at various points g ∈ G . Note that all irreducible representations of U ε ( g )are of that kind, and every such representation is a representation of the algebra U η g = U ε ( g ) /U ε ( g )Ker η g for some η g . In [22] De Concini, Kac and Procesi showed that if g and g are two conjugate elements of G then the algebras U η g and U η g are isomorphic. Moreover in [22] De Concini, Kac and Procesi formulated the following conjecture. De Concini–Kac–Procesi conjecture.
The dimension of any finite-dimensional representation of the algebra U η g is divisible by b = m dim O g , where O g is the conjugacy class of g . This conjecture is the quantum group counterpart of the Kac–Weisfeiler conjecture for semisimple Lie algebrasover fields of prime characteristic.As it is shown in [21] it suffices to verify the De Concini–Kac–Procesi conjecture in case of exceptional elements g ∈ G (an element g ∈ G is called exceptional if the centralizer in G of its semisimple part has a finite center).However, the De Concini–Kac–Procesi conjecture is related to the geometry of the group G which is much morecomplicated than the geometry of the linear space g ′ in case of the Kac–Weisfeiler conjecture.The De Concini–Kac–Procesi conjecture is known to be true for the conjugacy classes of regular elements (see[23]), for the subregular unipotent conjugacy classes in type A n when m is a power of a prime number (see [12]),for all conjugacy classes in A n when m is a prime number (see [14]), for the conjugacy classes O g of g ∈ SL n whenthe conjugacy class of the unipotent part of g is spherical (see [13]), and for spherical conjugacy classes (see [11]).In [61] a proof of the De Concini–Kac–Procesi conjecture using localization of quantum D –modules was outlinedin case of unipotent conjugacy classes. In contract to many papers quoted above the strategy of the proof of theDe Concini–Kac–Procesi conjecture developed in this book does not use the reduction to the case of exceptionalelements, and all conjugacy classes are treated uniformly.Namely, following Premet’s philosophy we use certain subalgebras U η g ( m − ) ⊂ U η g introduced in [102]. Thesesubalgebras have non–trivial characters χ : U η g ( m − ) → C . In terms of the previously introduced notation, weshow that for A = U η g , B = U η g ( m − ) and an appropriate χ the category A − mod χB can be identified with thecategory of finite-dimensional representations of U η g and equivalence (1) holds if Hk ( A, B, χ ) − mod is the categoryof finite-dimensional representations of the corresponding algebra Hk ( A, B, χ ).As observed in [102] every finite-dimensional U η g –module is also equipped with an action of the algebra U η ( m − )corresponding to the trivial character η of Z G given by the evaluation at the identity element of G . In the settingof quantum groups at roots of unity this action is a counterpart of the second U ( m )–module structure on objects V of the category A − mod χB which appeared in (4) in the case of Lie algebras over fields of zero characteristic.Since the De Concini–Kac–Procesi conjecture is related to the structure of the set of conjugacy classes in G it is natural to look at transversal slices to the set of conjugacy classes. It turns out that the definition of thesubalgebras U η g ( m − ) is related to the existence of some special transversal slices Σ s to the set of conjugacy classesin G . These slices Σ s associated to (conjugacy classes of) elements s in the Weyl group of g were introduced bythe author in [98]. The slices Σ s play the role of Slodowy slices in algebraic group theory. In the particular caseof elliptic Weyl group elements these slices were also introduced later by He and Lusztig in paper [46] within adifferent framework.A remarkable property of a slice Σ s observed in [102] is that if g is conjugate to an element in Σ s then thedimension of the corresponding subalgebra U η g ( m − ) ⊂ U η g is equal to m codim Σ s . The dimension of the algebra U η ( m − ) is also equal to m codim Σ s . If g ∈ Σ s (in fact g may belong to a larger variety) then the correspondingsubalgebras U η g ( m − ) and U η ( m − ) can be explicitly described in terms of quantum group analogues of root vectors.Note that one can also define analogues U sh ( m − ) of subalgebras U η g ( m − ) in the standard Drinfeld–Jimbo quantumgroup U h ( g ) over the ring of formal power series C [[ h ]] (see [99]).In [99], Theorem 5.2 it is shown that for every conjugacy class O in G one can find a transversal slice Σ s suchthat O intersects Σ s and dim O = codim Σ s . Using this result we showed in [102] that for every element g ∈ G one can find a a subalgebra U η g ( m − ) in U η g of dimension m dim O g with a non–trivial character χ . The dimensionof the corresponding algebra U η ( m − ) is also equal to m dim O g .Following the strategy outlined in the first part of the introduction we show that if m satisfies a certain conditionthen every finite-dimensional U η g –module is free over U η ( m − ). Thus the dimension of every such module is divisibleby m dim O g . This establishes the De Concini–Kac–Procesi conjecture.Note that in the case of restricted representations of a small quantum group similar results were obtained in[29]. The situation in [29] is rather similar to the case of the trivial character η = η in our setting.We also show that the rank of every finite-dimensional U η g –module V over U η ( m − ) is equal to the dimensionof the space V χ and that U η g is the algebra of matrices of size m dim O g over the corresponding q-W–algebra Hk ( A, B, χ ) = Hk ( U η g , U η g ( m − ) , χ ) which has dimension m dim Σ s . In case of Lie algebras over fields of primecharacteristic similar results were obtained in [82].Note that the support variety technique used in [83] to prove the Kac–Weisfeiler conjecture can not be transferredto the case of quantum groups straightforwardly. The notion of the support variety is still available in case ofquantum groups (see [29, 41, 78]). But in practical applications it is much less efficient since in the case of quantumgroups there is no any underlying linear space.In conclusion we would like to make a few remarks on the structure of the book. It consists of six chapters. Inthis introduction we have given a very superficial and incomplete review of the content of the book which ratheraims to provide the reader with a general guide outlining the main ideas and the strategy of the main proofs. Moretechnical comments are given in the beginning of each chapter.In Chapters 1 and 2 we summarize results from [98, 99, 101, 103, 104] on the algebraic group analogues Σ s ofthe Slodowy slices and the related results on quantum groups and on the subalgebras U sh ( m − ) ⊂ U h ( g ). Chapter1 also contains some results on combinatorics of Weyl groups and on root systems required for the definition ofthe slices Σ s , and Chapter 2 contains some advanced results on quantum groups required later for the study ofq-W–algebras.In Chapter 3, following [99, 101], we recall the definition of q-W–algebras and the description of their classicalPoisson counterparts given in [104] in terms of the Zhelobenko type operators Π i and Π. The main purpose of thischapter is to bring this description to a form suitable for quantization. Formulas (3.5.10), (3.5.11) and (3.5.12)obtained in this chapter for Π i and Π have direct quantum analogues (4.5.5), (4.6.1) and (4.7.3) obtained in Chapter4 for P i and Π q . The main result of Chapter 4 (Theorem 4.7.2) is the description of the q-W–algebra as the imageof the operator Π q .In Chapter 5 we prove a version of the Skryabin equivalence of type (1) for equivariant modules over quantumgroups established in [101]. The new proof of this equivalence in Theorem 5.2.1 is based on Proposition 4.6.7 whichallows to construct some nice bases in modules from the category A − mod χB (see the discussion in the introductionabove). Theorem 5.2.1 also gives precise values of ε of the deformation parameter q for which the categoricalequivalence holds while in [101] it was established for generic ε only.Finally in Chapter 6 we apply the results of Chapter 4 to the study of representations of quantum groups atroots of unity and prove the De Concini–Kac–Procesi conjecture. The strategy of this proof has already beendiscussed above.Citations in the main text are reduced to a minimum. References to proofs which are omitted in the body ofthe text and some historic remarks are given in the bibliographic comments after each chapter. Acknowledgements
The author is greatly indebted to Giovanna Carnovale, Iulian Ion Simion, Lewis Topley and to the members ofthe representation theory seminars at the Universities of Bologna and Padua for careful reading of some parts ofthe manuscript.The results presented in this book have been partially obtained during several research stays at Institut desHaut ´Etudes Scientifiques, Paris, Max–Planck–Instut f¨ur Mathematik, Bonn and at the Representation Theorythematic program held at Institut Henri Poincar´e, Paris, January 6–April 3, 2020. The author is grateful to theseinstitutions for hospitality.The research on this project received funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368) during the visit of theauthor to Institut des Haut ´Etudes Scientifiques, Paris.0 ontents
Introduction 31 Algebraic group analogues of Slodowy slices 13
CONTENTS
Appendix 167
Appendix 1. Normal orderings of root systems compatible with involutions in Weyl groups . . . . . . . . 167Appendix 2. Transversal slices for simple exceptional algebraic groups . . . . . . . . . . . . . . . . . . . . 171Appendix 3. Irreducible root systems of exceptional types . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Bibliography 189 hapter 1
Algebraic group analogues of Slodowyslices
The q-W–algebras are non-commutative deformations of algebras of regular functions on certain algebraic varietiesin algebraic groups transversal to conjugacy classes. In this book these varieties play a role similar to that of theSlodowy slices in the theory of W–algebras and of generalized Gelfand-Geraev representations of semisimple Liealgebras. In this chapter we define these varieties and study their properties. We also develop the relevant Weylgroup combinatorics.
Fix the notation used throughout the book. Let G be a connected finite-dimensional complex semisimple Lie group, g its Lie algebra. Fix a Cartan subalgebra h ⊂ g and let ∆ = ∆( g , h ) be the set of roots of the pair ( g , h ), Q thecorresponding root lattice, and P the weight lattice. Let α i , i = 1 , . . . , l, l = rank ( g ) be a system of simple roots,∆ + = { β , . . . , β D } the set of positive roots, P + the set of the corresponding integral dominant weights, ω , . . . , ω l the fundamental weights. Let H , . . . , H l be the set of simple root generators of h .Denote by a ij the corresponding Cartan matrix, and let d , . . . , d l , d i ∈ { , , } , i = 1 , . . . , l be coprime positiveintegers such that the matrix b ij = d i a ij is symmetric. There exists a unique non–degenerate invariant symmetricbilinear form ( , ) on g such that ( H i , H j ) = d − j a ij . It induces an isomorphism of vector spaces h ≃ h ∗ underwhich α i ∈ h ∗ corresponds to d i H i ∈ h . We denote by h ∨ the element of h that corresponds to h ∈ h ∗ under thisisomorphism. The induced bilinear form on h ∗ is given by ( α i , α j ) = b ij .Let W be the Weyl group of the root system ∆. W is the subgroup of GL ( h ) generated by the fundamentalreflections s , . . . , s l , s i ( h ) = h − α i ( h ) H i , h ∈ h . The action of W preserves the bilinear form ( , ) on h . We denote a representative of w ∈ W in G by the sameletter. For w ∈ W, g ∈ G we write w ( g ) = wgw − . For any root α ∈ ∆ we also denote by s α the correspondingreflection.For every element w ∈ W one can introduce the set ∆ w = { α ∈ ∆ + : w ( α ) ∈ − ∆ + } , and the number of theelements in the set ∆ w is equal to the length l ( w ) of the element w with respect to the system Γ of simple roots in∆ + . We also write ∆ − = − ∆ + .Let b + be the positive Borel subalgebra corresponding to ∆ + and b − the opposite Borel subalgebra; let n + =[ b + , b + ] and n − = [ b − , b − ] be their nilradicals. Let H, N + = exp n + , N − = exp n − , B + = HN + , B − = HN − bethe maximal torus, the maximal unipotent subgroups and the Borel subgroups of G which correspond to the Liesubalgebras h , n + , n − , b + and b − , respectively. Note that ∆ can also be defined as the root system of the pair( G, H ), ∆ = ∆(
G, H ).We identify g and its dual by means of the canonical bilinear form. Then the coadjoint action of G on g ∗ isnaturally identified with the adjoint one. Using the canonical bilinear form we shall also identify n + ∗ ≃ n − , b + ∗ ≃ b − , h ≃ h ∗ .Let g β be the root subspace corresponding to a root β ∈ ∆, g β = { x ∈ g | [ h, x ] = β ( h ) x for every h ∈ h } . g β ⊂ g is a one–dimensional subspace. It is well known that for α = − β the root subspaces g α and g β are orthogonal withrespect to the canonical invariant bilinear form. Moreover g α and g − α are non–degenerately paired by this form.134 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Let X α ∈ g be a non–zero root vector corresponding to a root α ∈ ∆. Root vectors X α ∈ g α satisfy the followingrelations: [ X α , X − α ] = ( X α , X − α ) α ∨ . Note also that in this book we denote by N the set of non-negative integer numbers, N = { , , . . . } . Algebraic group analogues of the Slodowy slices are associated to (conjugacy classes) in the Weyl group. In thissection we recall the relevant combinatorics of the Weyl group and of root systems. We start by defining systemsof positive roots associated to Weyl group elements which play the key role in the definition of the algebraic groupanalogues of the Slodowy slices.Assume from now on that the group G is simply–connected. Let s be an element of the Weyl group W anddenote by h ′ the orthogonal complement in h , with respect to the canonical bilinear form on g , to the subspace h ′⊥ of h fixed by the natural action of s on h . Let h ′∗ be the image of h ′ in h ∗ under the identification h ∗ ≃ h inducedby the canonical bilinear form on g . By Theorem C in [16] s can be represented as a product of two involutions, s = s s , (1.2.1)where s = s γ . . . s γ n , s = s γ n +1 . . . s γ l ′ , the roots in each of the sets γ , . . . , γ n and γ n +1 , . . . , γ l ′ are positive andmutually orthogonal, and the roots γ , . . . , γ l ′ form a linear basis of h ′∗ .Let h R be the real form of h , the real linear span of simple coroots in h . The set of roots ∆ is a subset of thedual space h ∗ R .The Weyl group element s naturally acts on h R as an orthogonal transformation with respect to the scalarproduct induced by the symmetric bilinear form of g .Let f , . . . , f l ′ be the vectors of unit length in the directions of γ , . . . γ l ′ , and b f , . . . , b f l ′ the basis of h ′ R dual to f , . . . , f l ′ . Let M be the l ′ × l ′ symmetric matrix with real entries M ij = ( f i , f j ). I − M is also a symmetric realmatrix, and hence it is diagonalizable and has real eigenvalues.The following proposition gives a recipe for constructing a spectral decomposition for the action of the orthogonaltransformation s on h R . Proposition 1.2.1.
Let λ be a (real) eigenvalue of the symmetric matrix I − M , and u ∈ R l ′ a correspondingnon–zero real eigenvector with components u i , i = 1 , . . . , l ′ . Let a u , b u ∈ h R be defined by a u = n X i =1 u i b f i , b u = l ′ X i = n +1 u i b f i . (1.2.2) (i) If λ = 0 then the angle θ between a u and b u satisfies cos θ = λ , the plane h λ ⊂ h R spanned by a u and b u isinvariant with respect to the involutions s , , s acts on h λ as the reflection in the line spanned by b u , and s actson h λ as the reflection in the line spanned by a u . If λ > the orthogonal transformation s = s s acts on h λ as arotation through the angle θ .(ii) If λ = 0 , ± is an eigenvalue of I − M then − λ is also an eigenvalue of I − M , and if λ = µ are two positiveeigenvalues of I − M , λ, µ = 1 then the planes h λ and h µ are mutually orthogonal.(iii) Let λ = 0 , ± be an eigenvalue of I − M of multiplicity greater than , and u k ∈ R l ′ , k = 1 , . . . , mult λ abasis of the eigenspace corresponding to λ . If the basis u k is orthonormal with respect to the standard scalar producton R l ′ then the corresponding planes h kλ defined with the help of u k , k = 1 , . . . , mult λ are mutually orthogonal.(iv) λ = ± are not eigenvalues of I − M .(v) If λ = 0 is an eigenvalue of I − M , then there is a basis u k ∈ R l ′ , k = 1 , . . . , mult 0 of the eigenspacecorresponding to orthonormal with respect to the standard scalar product on R l ′ and such that the correspondingnon–zero elements a u k , b u k are all mutually orthogonal. Moreover, s a u k = − a u k , s a u k = a u k , s b u k = b u k , s b u k = − b u k for non–zero elements a u k , b u k . In particular, for non–zero elements a u k , b u k we have sa u k = − a u k , sb u k = − b u k , and non–zero elements a u k , b u k is a basis of the subspace of h R on which s acts by multiplication by − .Proof. By definition the matrix M can be written in a block form, M = (cid:18) I n AA ⊤ I l ′ − n (cid:19) , (1.2.3) .2. SYSTEMS OF POSITIVE ROOTS ASSOCIATED TO WEYL GROUP ELEMENTS A is an n × ( l ′ − n ) matrix, A ⊤ is the transpose to A , I n and I l ′ − n are the unit matrices of sizes n and l ′ − n . M − is also symmetric and has a similar block form, M − = (cid:18) B CC ⊤ D (cid:19) , B = B ⊤ , D = D ⊤ , (1.2.4)with the entries M − ij = ( b f i , b f j ).For any vector u ∈ R l ′ we introduce its R n and R l ′ − n components e u and ee u in a similar way, u = (cid:18) e u ee u (cid:19) . (1.2.5)We shall consider both e u and ee u as elements of R l ′ using natural embeddings R n , R l ′ − n ⊂ R l ′ associated to decom-position (1.2.5).If u is a non–zero eigenvector of I − M corresponding to an eigenvalue λ then the equation ( I − M ) u = λu gives − A ee u = λ e u, − A ⊤ e u = λ ee u. (1.2.6)From these equations we deduce that (cid:18) − e u ee u (cid:19) is a non–zero eigenvector of I − M corresponding to the eigenvalue − λ .Since M − M = I one has BA + C = 0 , C ⊤ + DA ⊤ = 0 . (1.2.7)Multiplying the first and the second equations in (1.2.6) from the left by B and D , respectively, and using (1.2.7)we obtain that C ee u = λB e u, C ⊤ e u = λD ee u. (1.2.8)Now if u , are two non–zero eigenvectors of I − M corresponding to an eigenvalue λ then by (1.2.4) we have( a u , a u ) = n X i,j =1 u i u j ( b f i , b f j ) = n X i,j =1 u i u j B ij = e u · B e u , (1.2.9)where · stands for the standard scalar product in R l ′ .Similarly, ( b u , b u ) = D ee u · ee u , ( a u , b u ) = e u · C ee u (1.2.10)From (1.2.8), (1.2.9) and the first identity in (1.2.10) we also obtain that if λ = 0 then( a u , a u ) = e u · B e u = 1 λ e u · C ee u = 1 λ C ⊤ e u · ee u = D ee u · ee u = ( b u , b u ) . (1.2.11)Similarly, for any real eigenvalue λ we have( a u , b u ) = e u · C ee u = λ ( a u , a u ) , ( b u , a u ) = ee u · C ⊤ e u = λ ( a u , a u ) . (1.2.12)Therefore if λ = 0, taking into account (1.2.11), we obtain λ = ( a u , b u )( a u , a u ) = ( a u , b u ) p ( b u , b u ) p ( a u , a u ) = cos θ. Now if λ = 0 , ± M − u = − λ u yield( a u + b u , a u + b u ) = 2( a u , a u )( λ + 1) = u · M − u = 11 − λ u · u . Thus if λ = 0 , ± u , are mutually orthogonal a u , a u are also mutually orthogonal, and from (1.2.11) and(1.2.12) we obtain that b u and b u , a u and b u , a u and b u are mutually orthogonal. Therefore the planes spannedby a u , b u and by a u , b u are mutually orthogonal.6 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Let again u be a non–zero eigenvector of I − M corresponding to an eigenvalue λ For i = 1 , . . . , n we have( b f i , λa u − b u ) = l ′ X j =1 u j ( b f i , b f j ) = λ ( B e u ) i − ( C ee u ) i = 0 , where at the last step we used the first identity in (1.2.8). From the last identity we deduce that λa u − b u is alinear combination of f n +1 , . . . , f l ′ , and hence s ( λa u − b u ) = − ( λa u − b u ) . However, by the definition of a u s a u = a u . Therefore s b u = 2 λa u − b u . (1.2.13)Let λ = 0. Then recalling that by (1.2.11) ( a u , a u ) = ( b u , b u ) we conclude that λa u = cos( θ ) a u is the orthogonalprojection of b u onto the line spanned by a u and that s b u is obtained from b u by the reflection in the line spannedby a u as shown at Figure 1. = = ③③③③③③③③③③③③③③③③③ b u / / a u ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ s b u ✤✤✤✤✤✤✤✤✤✤✤ λa u / / Fig. 1Similarly, s b u = b u , s a u is obtained from a u by the reflection in the line spanned by b u .Thus the plane h λ ⊂ h R spanned by a u and b u is invariant with respect to the involutions s , , s acts on h λ asthe reflection in the line spanned by b u , and s acts on h λ as the reflection in the line spanned by a u . Since theangle between a u and b u is θ , for λ > s = s s acts on h λ as a rotation throughthe angle 2 θ . λ = 1 is not an eigenvalue of I − M since the matrix M is invertible. λ = − I − M since otherwise the corresponding elements a u , b u would span a non–trivial fixed point subspace for the action of s in h ′ R which is impossible as s acts on h ′ R without non–trivial fixed points.From the general theory of orthogonal transformations it follows that if λ = µ are two positive eigenvalues of I − M , λ, µ = 1 then the planes h λ and h µ are mutually orthogonal.If λ = 0 is an eigenvalue of I − M then e u and ee u are the components of an eigenvector u of I − M with eigenvalue0 if and only if A ee u = 0 and A ⊤ e u = 0. Therefore using the usual orthogonalization procedure one can construct abasis u k ∈ R l ′ , k = 1 , . . . , mult 0 of the eigenspace corresponding to 0 orthonormal with respect to the standardscalar product on R l ′ and such that the components e u k and ee u k k = 1 , . . . , mult 0 are all mutually orthogonal.By (1.2.13) s b u k = − b u k . Also by the definition of a u k s a u k = a u k . Similarly, s a u k = − a u k and s b u k = b u k .Now using the definition of the eigenvector we deduce that for the basis u k the following relations hold: B e u k = e u k , D ee u k = ee u k . For k = l by (1.2.9) ( a u k , a u l ) = e u k · B e u l = e u k · e u l = 0and by (1.2.10) ( b u k , b u l ) = D ee u k · ee u l = ee u k · ee u l = 0 . .2. SYSTEMS OF POSITIVE ROOTS ASSOCIATED TO WEYL GROUP ELEMENTS a u k , b u l ) = λ ( a u k , a u l ) = 0 . This completes the proof.Using the previous proposition we can decompose h R into a direct orthogonal sum of s –invariant subspaces, h R = K M i =0 h i , (1.2.14)where each of the subspaces h i ⊂ h R , i = 1 , . . . , K is invariant with respect to both involutions s , in thedecomposition s = s s , and there are the following three possibilities for each h i : h i is two–dimensional ( h i = h kλ for an eigenvalue 0 < λ < I − M , and k = 1 , . . . , mult λ ) and the Weyl group element s acts onit as rotation with angle θ i , 0 < θ i < π or h i = h kλ , λ = 0, k = 1 , . . . , mult λ has dimension 1 and s acts on it bymultiplication by − h i coincides with the linear subspace of h R fixed by the action of s . Note that since s hasfinite order θ i = πn i m i , n i , m i ∈ { , , . . . } .Since the number of roots in the root system ∆ is finite one can always choose elements h i ∈ h i , i = 0 , . . . , K ,such that h i ( α ) = 0 for any root α ∈ ∆ which is not orthogonal to the s –invariant subspace h i with respect to thenatural pairing between h R and h ∗ R .Now we consider certain s –invariant subsets of roots ∆ i , i = 0 , . . . , K , defined as follows∆ i = { α ∈ ∆ : h j ( α ) = 0 , j > i, h i ( α ) = 0 } , (1.2.15)where we formally assume that h K +1 = 0. Note that for some indexes i the subsets ∆ i are empty, and that thedefinition of these subsets depends on the order of the terms in direct sum (1.2.14).Now consider the nonempty s –invariant subsets of roots ∆ i k , k = 0 , . . . , M . For convenience we assume thatindexes i k are labeled in such a way that i j < i k if and only if j < k .Observe also that the root system ∆ is the disjoint union of the subsets ∆ i k ,∆ = M [ k =0 ∆ i k . Now assume that | h i k ( α ) | > | X l ≤ j
We denote by (∆ i k ) s + the set of positive roots contained in ∆ i k , (∆ i k ) s + = ∆ s + ∩ ∆ i k .We also define other s –invariant subsets of roots ∆ i k , k = 0 , . . . , M ,∆ i k = [ i j ≤ i k ∆ i j . (1.2.20)According to this definition we have a chain of strict inclusions∆ i M ⊃ ∆ i M − ⊃ . . . ⊃ ∆ i , (1.2.21)such that ∆ i M = ∆, ∆ = ∆ , and ∆ i k \ ∆ i k − = ∆ i k .The following lemma shows that the subsets of roots ∆ i k ⊂ ∆ are root systems of some standard Levi subalgebrasin g . Lemma 1.2.2.
Let Γ s be the set of simple roots in ∆ s + . Then Γ s ∩ ∆ i k is a set of simple roots in ∆ i k .Proof. Indeed, let α ∈ ∆ i k ∩ ∆ s + , α = P li =1 n i α i , where n i ∈ { , , , . . . } and Γ s = { α , . . . , α l } . Assume that α does not belong to the linear span of roots from Γ s ∩ ∆ i k and t > i k is maximal possible such that for some α q ∈ ∆ t one has n q >
0. Then by (1.2.15) and (1.2.19) h t ( α ) = P li =1 n i h t ( α i ) = P α i ∈ ∆ t n i h t ( α i ) >
0, and by the choiceof t h r ( α ) = 0 for r > t . Therefore α ∈ ∆ t , and hence α ∆ i k . Thus we arrive at a contradiction. In this section we define analogues of the Slodowy slices for algebraic groups.Let s ∈ W be a Weyl group element, ∆ s + a system of positive roots associated to (the conjugacy class of) s in the previous section, Γ s the set of simple roots in ∆ s + . We shall assume that in sum (1.2.14) h is the linearsubspace of h R fixed by the action of s . According to this convention ∆ = { α ∈ ∆ : sα = α } is the set of rootsfixed by the action of s .We shall need the parabolic subalgebra p of g containing the Borel subalgebra corresponding to ∆ s − = − ∆ s + andassociated to the subset − Γ s of the set of simple roots − Γ s , where Γ s = Γ s ∩ ∆ . Denote by P the correspondingparabolic subgroup of G . Let n and l be the nilradical and the Levi factor of p , N and L the unipotent radical andthe Levi factor of P , respectively.Note that we have natural inclusions of Lie algebras p ⊃ n , and by Lemma 1.2.2 ∆ is the root system of thereductive Lie algebra l . We also denote by n the nilpotent subalgebra opposite to n and by N the subgroup in G corresponding to n .Denote a representative for the Weyl group element s in G by the same letter. Let Z be the connected subgroupof G generated by the semisimple part of the Levi subgroup L and by the identity component H of centralizer of s in H . H is the connected Lie subgroup of H corresponding to the Lie subalgebra h ′⊥ ⊂ h . We shall also denoteby z the Lie algebra of Z . Let H ′ ⊂ H be the subgroup corresponding to the Lie subalgebra h ′ ⊂ h . We obviouslyhave L = ZH ′ , and l = z + h ′ , where the sum is not direct.Recall that G is in fact an algebraic group and all its subgroups introduced above are algebraic subgroups (seee.g. see § Proposition 1.3.1.
Let N s = { v ∈ N | svs − ∈ N } . Then dim N s = l ( s ) , where l ( s ) is the length of the Weylgroup element s ∈ W with respect to the system of simple roots of ∆ s + , the subvarieties N ZsN and sZN s of G areclosed and the conjugation map N × sZN s → N ZsN (1.3.1) is an isomorphism of varieties. Moreover, the variety Σ s = sZN s is a transversal slice to the set of conjugacyclasses in G .Proof. dim N s = l ( s ) since N s ⊂ N is a closed subgroup generated by the one–parameter subgroups correspondingto the roots from the set { α ∈ − ∆ s + : sα ∈ ∆ s + } the cardinality of which is equal to l ( s ) (see e.g. [15], § N ZsN is closed in G . Using a decomposition of N as a product of one–dimensional subgroupscorresponding to roots one can write N = N ′ s N s , where N ′ s = N ∩ s − N s , and hence
N sZN = N sZN ′ s N s = N sZN s , (1.3.2) .3. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES Z normalizes N ′ s .Observe that an element g ∈ G belongs to N ZsN = N sZN s = N ZsN s if and only if gs − ∈ N ZsN s s − . Thevariety N ZsN s s − is a subvariety of N ZN . We prove that
N ZN is closed in G .Let h ′ R = h ′ ∩ h R . h ′ R and h are annihilators of each other with respect to the restriction of the bilinear formon g to h R . Let h ′∗ R and h ∗ be the images of h ′ R and h , respectively, under the isomorphism h R ≃ h ∗ R induced bythe bilinear form on g .Introduce the element ¯ h = P Mk =1 h i k ∈ h R . By the definition of ∆ s + for any x ∈ h ∗ one has ¯ h ( x ) = 0 and aroot α ∈ ∆ \ ∆ belongs to ∆ s + if and only if ¯ h ( α ) >
0. Let ¯ h ∗ ∈ h ∗ R be the image in h R of the element ¯ h ∈ h ′ R .Since ¯ h ∈ h ′ R we actually have ¯ h ∗ ∈ h ′∗ R .Let α , . . . , α p be the simple roots in Γ s which do not belong to ∆ , ω , . . . , ω p the corresponding fundamentalweights. h ′∗ R is a linear subspace in the real linear span Π of ω , . . . , ω p as Π is the annihilator of the subspace of h ∗ R spanned by the roots from ∆ which is contained in h ∗ . The subset Π + of Π which consists of x satisfying thecondition ( x, α ) > α ∈ ∆ s + \ ∆ is open in Π and by definition ¯ h ∗ ∈ Π + ∩ h ′∗ R . Therefore the intersection Π + ∩ h ′∗ R is not empty and open in h ′∗ R .The roots γ , . . . , γ l ′ form a linear basis of h ′∗ R . They also span a Z –sublattice Q ′ in the Z –lattice generated by ω , . . . , ω p as every root is a linear combination of fundamental weights with integer coefficients and γ , . . . , γ l ′ forma linear basis of h ′∗ R ⊂ Π. Linear combinations of elements of Q ′ with rational coefficients are dense in h ′∗ R , and, inparticular, in the open set Π + ∩ h ′∗ R . Since the subset Π + of Π consists of x satisfying the condition ( x, α ) > α ∈ ∆ s + \ ∆ there is a linear basis of h ′∗ R which consists linear combinations of ω , . . . , ω p with positive rationalcoefficients. Multiplying the elements of this basis by appropriate positive integer numbers we obtain a linear basisΩ i , i = 1 , . . . , l ′ of h ′∗ R which consists of integral dominant weights of the form Ω i = P pj =1 g ij ω j , g ij ∈ Z , g ij > h ′⊥ be the orthogonal compliment to h ′ in h with respect to the restriction of the symmetric bilinear formon g to h . h ′⊥ is the complexification of h , and hence we deduce that an element x ∈ h belongs to h ′⊥ if and onlyif Ω i ( x ) = 0, i = 1 , . . . , l ′ .Let B s + be the Borel subgroup of G corresponding to the system ∆ s + of positive roots, B s − the opposite Borelsubgroup , N s ± , b s ± their unipotent radicals and Lie algebras, respectively. Let V Ω i , i = 1 , . . . , l ′ be the irreduciblefinite–dimensional representation of g with highest weight Ω i with respect to the system ∆ s + of positive roots.Denote by v Ω i a nonzero highest weight vector in V Ω i and by < · , · > the contravariant bilinear form on V Ω i normalized in such a way that < v Ω i , v Ω i > = 1. The matrix element < v Ω i , · v Ω i > can be regarded as aregular function on G whose restriction to the big dense cell N s − HN s + is given by the character Ω i of H ,
N ZN is closed in G . The variety N ZsN s s − is a closed subvariety of N ZN as sN s s − is the closed algebraic subgroup in N generatedby the one–parameter subgroups corresponding to the roots from the set { α ∈ ∆ s + : s − α ∈ − ∆ s + } . So finally N ZsN s s − is closed in G , and hence N sZN = N ZsN s is also closed in G . sZN s ⊂ N sZN s = N sZN is a closed subvariety as N s ⊂ N is a closed subgroup generated by the one–parameter subgroups corresponding to the roots from the set { α ∈ − ∆ s + : sα ∈ ∆ s + } , and N is a closed algebraicsubgroup of G . Since N sZN is closed in G its closed subvariety sZN s is also closed in G .0 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Next we show that map (1.3.1) is an isomorphism of varieties. It suffices to show that this map is bijective.Then by Zariski’s main theorem this map is an isomorphism of varieties.Fix a system of root vectors X α ∈ g , α ∈ ∆ such that if [ X α , X β ] = N α,β X α + β for any pair α, β ∈ Γ s of simplepositive roots then [ X − α , X − β ] = N α,β X − α − β .Recall that by Theorem 5.4.2. in [43] one can uniquely choose a representative w ∈ G for any Weyl groupelement w ∈ W in such a way that the operator Ad w sends root vectors X ± α to X ± wα for any simple positive root α ∈ Γ s . We denote this representative by the same letter, w ∈ G . The representative w ∈ G is called the normalrepresentative of the Weyl group element w ∈ W . If the order of the Weyl group element w ∈ W is equal to p thenthe inner automorphism Ad w of the Lie algebra g has order at most 2 p , Ad w p = id.Since any representative of s ∈ W in the normalizer of h in G is H –conjugate to an element from Zs , where s ∈ G is the normal representative of s ∈ W , we can assume without loss of generality that s ∈ G is the normalrepresentative for the Weyl group element s ∈ W .Note that, since the operator Ad s sends root vectors X ± α to X ± sα for any simple positive root α ∈ Γ s and theroot system of the reductive Lie algebra l is fixed by the action of s , the semisimple part of the Levi subalgebra l is fixed by the action of Ad s . Therefore Z is the connected component of the centralizer of s in L containing theidentity element, Z = { z ∈ L | s − zs = z } ◦ , (1.3.3)Observe that map (1.3.1) is bijective if and only if for any given k s ∈ N s , u ∈ N and z ∈ Z the equation uszk s = nsz ′ n s n − (1.3.4)has a unique solution n ∈ N, n s ∈ N s , z ′ ∈ Z . First observe that any element uzsk s is uniquely conjugated by k s ∈ N to vsz, v = k s u , and hence we can assume that k s = 1 in (1.3.4), vsz = nsz ′ n s n − . (1.3.5)Now we show that for any given v ∈ N and z ∈ Z equation (1.3.5) has a unique solution n ∈ N, n s ∈ N s , z ′ ∈ Z .We shall use induction over certain s –invariant reductive subgroups in G that we are going to define. Considerthe reductive Lie subalgebras g i k , k = 0 , . . . , M defined by induction as follows: g i M = g , g i k − = z g ik ( h i k ), where z g ik ( h i k ) is the centralizer of h i k in g i k . We denote by G i k the corresponding subgroups in G .By construction ∆ i k is the root system of g i k , and we have chains of strict inclusions g = g i M ⊃ g i M − ⊃ . . . ⊃ g = l , (1.3.6) G = G i M ⊃ G i M − ⊃ . . . ⊃ G = L (1.3.7)corresponding to inclusions (1.2.21). Note that by Lemma 1.2.2 all subalgebras g i k are standard levi subalgebras in g and g i k − is the Levi factor of the parabolic subalgebra p i k − ⊂ g i k containing the Borel subalgebra b s − ∩ g i k ⊂ g i k and associated to the set of simple roots − Γ s ∩ ∆ i k − . Let n i k − be the nilradical of p i k − . We also denote by n i k − the nilradical of the opposite parabolic subalgebra. Let P i k − , N i k − , and N i k − be the corresponding subgroupsof G i k . Below we shall need the following direct vector space decompositions of linear spaces g i k = p i k − + n i k − = n i k − + g i k − + n i k − , (1.3.8) n = M − X k =0 n i k (1.3.9)following straightforwardly from the definitions of the subalgebras p i k − , n i k and n i k − . Decompositions (1.3.8)imply decompositions of dense subsets G i k ⊂ G i k , G i k = P i k − N i k − = N i k − G i k − N i k − , (1.3.10)and decomposition (1.3.9) implies two decompositions N = N i M − N i M − . . . N i , N = N i N i . . . N i M − . (1.3.11)Note that, since the subsets of roots ∆ i k are s –invariant, the subalgebras g i k are Ad s –invariant and the subgroups G i k are invariant with respect to the action of s on G by conjugations. .3. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES G k ⊂ G , G k = N k G i k N k , N k = N i M − N i M − . . . N i k , N k = N i M − N i M − . . . N i k . (1.3.12)Due to the inclusions[ g i k − , n i k − ] ⊂ n i k − , [ g i k − , n i k − ] ⊂ n i k − , n i k − ⊂ g i k − , n i k − ⊂ g i k − (1.3.13) N k and N k are Lie subgroups in G with Lie algebras P M − m = k n i m and P M − m = k n i m , respectively.We shall prove that equation (1.3.5) has a unique solution by induction over the reductive subgroups G i k startingwith k = 0. First we rewrite equation (1.3.5) in a slightly different form, vzsns − = nz ′ sn s s − . (1.3.14)To establish the base of induction we first observe that both the l.h.s. and the r.h.s. of equation (1.3.14) belongto the dense subset G ⊂ G and that the G = L –component of equation (1.3.14) with respect to decomposition(1.3.12) for k = 0 is reduced to z = z ′ . (1.3.15)Indeed, using a decomposition of N as a product of one–dimensional subgroups corresponding to roots one canwrite N = N ′ s N s , and hence sN s − = sN ′ s s − sN s s − ⊂ N N. (1.3.16)If n = mm s is the decomposition of n corresponding to the decomposition N = N ′ s N s then recalling that Z normalizes both N and N we deduce that the decompositions of the r.h.s. and of the l.h.s. of equation (1.3.14)corresponding to the decomposition G = N LN take the form vzsms − z − zsm s s − = nz ′ sn s s − , where vzsms − z − , n ∈ N and sm s s − , sn s s − ∈ N , z, z ′ ∈ Z ⊂ L . This implies (1.3.15) and establishes the baseof induction.Now let n = n i . . . n i M − n i M − , v = v i M − . . . v i v i , n s = n si . . . n si M − n si M − (1.3.17)be the decompositions of the elements n, v, n s corresponding to decompositions (1.3.11) and assume that n i j and n si j have already been uniquely defined for j < k −
1. We shall show that using equation (1.3.14) one can find n i k − and n si k − in a unique way.Observe that both the l.h.s. and the r.h.s. of equation (1.3.14) belong to the dense subset G k ⊂ G and that the G i k –component of equation (1.3.14) with respect to decomposition (1.3.12) is reduced to v i k − ( v ) k − zs ( n ) k − n i k − s − = ( n ) k − n i k − zs ( n s ) k − n si k − s − , (1.3.18)where ( n s ) k − = n si . . . n si k − ∈ G i k − , ( n ) k − = n i . . . n i k − ∈ G i k − , ( v ) k − = v i k − . . . v i ∈ G i k − and( n ) k − , ( n s ) k − , ( u ) k − , v i k − , z are already known. This follows, similarly to the case k = 0, from decompositions(1.3.17), inclusions (1.3.13), which also imply that G i k − normalizes both N i k − and N i k − , and the fact that thesubgroups G i k are invariant with respect to the action of s on G by conjugations.The same properties imply that after multiplying by ( n ) − k − from the left, equation (1.3.18) takes the form wz sn i k − s − = n i k − z ′ sn si k − s − , (1.3.19)with some known w ∈ N i k − , z , z ′ ∈ G i k − , and the compatibility of the equation of type (1.3.18) with k replacedby k − z = z ′ . Therefore (1.3.19) takes the form wz s ¯ ns − = ¯ nz s ¯ n s s − (1.3.20)where we renamed the unknowns ¯ n = n i k − , ¯ n s = n si k − to simplify the notation.Let ¯ n = ¯ m ¯ m s be the decomposition of the element ¯ n corresponding to the factorization N i k − = ( N i k − ∩ N ′ s )( N i k − ∩ N s ) . CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
In terms of this factorization equation (1.3.20) can be rewritten as follows wz s ¯ ms − s ¯ m s s − = ¯ nz s ¯ n s s − , (1.3.21)and the N i k − –component of the last equation with respect to factorization (1.3.10) is s ¯ m s s − = s ¯ n s s − . From this relation we obtain that ¯ m s = ¯ n s , (1.3.22)and hence (1.3.21) yields wz s ¯ ms − z − = ¯ n. (1.3.23)Now we show that the last equation defines ¯ n in a unique way.First observe that ¯ n ∈ N i k − , and N i k − is generated by one–parametric subgroups corresponding to roots fromthe set (∆ i k ) + = ∆ i k ∩ ∆ + .By the definition of the set ∆ ss each s –orbit in the s –invariant set ∆ i k contains a unique element from ∆ ss ∩ ∆ i k . This observation implies that the set (∆ i k ) + is the disjoint union of the subsets ∆ pi k = { α ∈ − (∆ i k ) + : s − α, . . . , s − ( p − α ∈ − (∆ i k ) + , s − p α ∈ (∆ i k ) + } , p = 1 , . . . , D k + 1. Here D k is chosen in such a way that∆ D k +1 i k ⊂ − ∆ ss ∩ ∆ i k , and ∆ D k i k = ∆ ′ D k i k ∪ ∆ ′′ D k i k (disjoint union), where ∆ ′ D k i k = ∆ D k i k ∩− ∆ ss , and ∆ ′′ D k i k = ∆ D k i k \ ∆ ′ D k i k .The set ∆ D k +1 i k may be empty.The orthogonal projections of the roots from the subsets ∆ pi k onto h i k are contained in the interior of the sectorslabeled ∆ pi k at Figure 2. All those sectors belong to the lower half plane and have the same central angles equal to θ i k , except for the last sector labeled by ∆ D k +1 i k , which can possibly have a smaller angle. h i k ∆ i k ∆ i k ∆ i k ∆ D k +1 i k ∆ D k i k ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ ✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✛✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣④④④④④④④④④④④④④④④④④④④④④④④④✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ ★★★★★★★★★★★★★★★★★★★★★★★★★ ✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ O O θ i k Fig. 2The vector h i k is directed upwards at the picture, and the orthogonal projections of elements from − (∆ i k ) + onto h i k are contained in the lower half plane. The element s ∈ W acts on the plane h i k by clockwise rotation by theangle θ i k .Now consider the unipotent subgroups N pi k − , p = 1 , . . . , D k + 1, N ′ i k − , N ′′ i k − generated by the one–dimensional subgroups corresponding to the roots from the sets ∆ pi k , ∆ ′ D k i k , ∆ ′′ D k i k , respectively.Obviously we have decompositions N i k − = N i k − N i k − . . . N D k +1 i k − , N D k i k − = N ′′ i k − N ′ i k − , N i k − ∩ N s = N ′ i k − N D k +1 i k − . (1.3.24) .3. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES n = ¯ n . . . ¯ n D k +1 , ¯ n D k = ¯ n ′′ ¯ n ′ , ¯ m = ¯ n . . . ¯ n D k − ¯ n ′′ , w = w w . . . w D k +1 , ¯ m s = ¯ n ′ ¯ n D k +1 (1.3.25)be the corresponding decomposition of elements ¯ n , ¯ m , w and ¯ m s , respectively.We claim that the components ¯ n p , p = 1 , . . . , D k + 1 can be uniquely calculated by induction starting with ¯ n .Indeed, substituting decompositions (1.3.25) into (1.3.23) we obtain w w . . . w D k +1 z s ¯ n . . . ¯ n D k − ¯ n ′′ s − z − = ¯ n . . . ¯ n D k +1 . Now comparing the N pi k − –components of the last equation, with respect to the first factorization in (1.3.24), andusing the fact that sN pi k − s − ⊂ N p +1 i k − , p = 1 , . . . , D k − sN ′′ i k − s − ⊂ N D k +1 i k − , and that z normalizes thesubgroups N pi k − , p = 1 , . . . , D k + 1, N ′ i k − , N ′′ i k − , we obtain¯ n = w , ¯ n p = ( w p . . . w D k +1 z s ¯ n . . . ¯ n p − s − z − ) p , p = 2 , . . . , D k , ¯ n D k +1 = ( w D k +1 z s ¯ n . . . ¯ n D k − ¯ n ′′ s − z − ) D k +1 , where ¯ n ′′ is defined from the factorization ¯ n D k = ¯ n ′′ ¯ n ′ , and the subscript ( . . . ) p stands for the N pi k − –componentwith respect to the first factorization in (1.3.24). From the formulas above one can recursively find the components¯ n p starting from ¯ n = w , and finally one can find ¯ n s using (1.3.22). This proves the induction step and establishesisomorphism (1.3.1).Finally we have to show that the variety sZN s ⊂ G is a transversal slice to the set of conjugacy classes in G ,i.e. that the differential of the conjugation map γ : G × sZN s → G (1.3.26)is surjective.Note that the set of smooth points of map (1.3.26) is stable under the G –action by left translations on thefirst factor of G × sZN s . Therefore it suffices to show that the differential of map (1.3.26) is surjective at points(1 , szn s ), n s ∈ N s , z ∈ Z .In terms of the left trivialization of the tangent bundle T G and the induced trivialization of T ( sZN s ) thedifferential of map (1.3.26) at points (1 , szn s ) takes the form dγ (1 ,szn s ) : ( x, ( n, w )) → − ( Id − Ad( szn s ) − ) x + n + w, (1.3.27) x ∈ g ≃ T ( G ) , ( n, w ) ∈ n s + z ≃ T szn s ( sZN s ) , where n s ⊂ g is the Lie algebra of n s .In order to show that the image of map (1.3.27) coincides with T szn s G ≃ g we shall need a direct orthogonal,with respect to the bilinear form, vector space decomposition of the Lie algebra g , g = n + z + n + h ′ . (1.3.28)We shall use isomorphism (1.3.1), α : N × sZN s → N sZN s = N sZN . By definition α is the restriction of themap γ to the subset N × sZN s ⊂ G × sZN s . Observe that in terms of the left trivialization of the tangent bundle T G the differential of the map α at points (1 , szn s ) ∈ N × sZN s , n s ∈ N s , z ∈ Z is given by dα (1 ,szn s ) : ( x, ( n, w )) → − ( Id − Ad( szn s ) − ) x + n + w, (1.3.29) x ∈ n ≃ T ( N ) , ( n, w ) ∈ n s + z ≃ T szn s ( sZN s ) . Recall that the conjugation map α : N × sZN s → N sZN s is an isomorphism, and hence its differential isan isomorphism of the corresponding tangent spaces at all points. Using the left trivialization of the tangentbundle T G the tangent space T szn s ( N sZN s ) can be identified with n s + z + Ad( szn s ) − n , T szn s z ( N sZN s ) = n s + z + Ad( szn s ) − n . Therefore using (1.3.29) and the fcat that dα (1 ,szn s ) is a linear space isomorphism we deducethat ( Id − Ad( szn s ) − ) n + n s + z = Ad( szn s ) − n + n s + z . (1.3.30)Now observe that by definition the subset ( Id − Ad( szn s ) − ) n ⊂ g is contained in the image of dα (1 ,szn s ) , and by(1.3.30) the subset Ad( szn s ) − n ⊂ g is also contained in the image of dα (1 ,szn s ) . Since n = ( Id − Ad( szn s ) − ) n +Ad( szn s ) − n , we deduce that n is contained in the image of dα (1 ,szn s ) , and hence in the image of dγ (1 ,szn s ) , n ⊂ Im dγ (1 ,szn s ) . (1.3.31)4 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Next observe that similarly to (1.3.1) one can show that the conjugation map N × N s − Zs → N ZsN = N s − ZsN, N s − = { n ∈ N : s − ns ∈ N } (1.3.32)is an isomorphism of varieties.Interchanging the roles of N and N in (1.3.32) we immediately obtain that the conjugation map α : N × N s − Zs → N s − ZsN , N s − = sN s s − is an isomorphism. Observe also that by definition the map α is therestriction of γ to the subset N × N s − Zs = N × sZN s ⊂ G × sZN s .Using the differential of the map α we immediately infer, similarly to inclusion (1.3.31), that n ⊂ Im dγ (1 ,szn s ) . (1.3.33)Now observe that since h normalizes n , and Z is a subgroup of L the adjoint action of which fixes h ′ , we havefor any x ′ ∈ h ′ (cid:0) ( Id − Ad( szn s ) − ) x ′ (cid:1) h ′ = ( Id − Ad s − ) x ′ , (1.3.34)where (cid:0) ( Id − Ad( szn s ) − ) x ′ (cid:1) h ′ stands for the h ′ –component of ( Id − Ad( szn s ) − ) x ′ with respect to decomposition(1.3.28).Since by definition the operator Ad s − has no fixed points in the invariant subspace h ′ , from formula (1.3.34)it follows that h ′ is contained in the image of ( Id − Ad( szn s ) − ), and hence, by formula (1.3.27), in the imageof dγ (1 ,szn s ) . Recalling also inclusions (1.3.31) and (1.3.33) and taking into account the obvious inclusion z ⊂ Im dγ (1 ,szn s ) and decomposition (1.3.28) we deduce that the image of the map dγ (1 ,szn s ) coincides with g ≃ T szn s G .Therefore the differential of the map γ is surjective at all points. This completes the proof.In the course of the proof of the previous proposition we obtained that the subvariety N ZN ⊂ G is closed in G . Similarly one can show that N ZN ⊂ G is closed in G . For later references we state this result as a corollary Corollary 1.3.2.
The subvarieties
N ZN, N ZN ⊂ G are closed in G . The subvarieties Σ s ⊂ G are analogues of the Slodowy slices in algebraic group theory. Remark 1.3.3.
In the proof of isomorphism (1.3.1) we only used commutation relations between one–parametersubgroups of G . Therefore isomorphisms similar to (1.3.1) hold in case when G is replaced with an algebraic groupof the same type as G over an algebraically closed field or with a Chevalley group of the same type as G over anarbitrary field. In this section G k is a connected finite–dimensional semisimple algebraic group over an algebraically closed field k of characteristic good for G k and of the same type as G . Let G p be a connected finite–dimensional semisimplealgebraic group of the same type as G over an algebraically closed field of characteristic exponent p .In the next section we shall show that for every conjugacy class O in G k one can find a subvariety Σ s, k ⊂ G k defined similarly to Σ s ⊂ G such that O intersects Σ s, k and dim O = codim Σ s, k . It turns out that there is aremarkable partition of the group G k the strata of which are unions of conjugacy classes of the same dimension. Foreach stratum of this partition there is a Weyl group element s such that all conjugacy classes O from that stratumintersect Σ s, k , and dim O = codim Σ s, k . In this section, which is rather descriptive, we define this partition calledthe Lusztig partition. The exposition in this section mainly follows paper [70] to which we refer the reader fortechnical details.For any Weyl group W let c W be the set of isomorphism classes of irreducible representations of W over Q . Forany E ∈ c W let b E be the smallest nonnegative integer such that E appears with non–zero multiplicity in the b E -thsymmetric power of the reflection representation of W . If this multiplicity is equal to 1 then one says that E isgood. If W ′ ⊂ W are two Weyl groups, and E ∈ c W ′ is good then there is a unique e E ∈ c W such that e E appears inthe decomposition of the induced representation Ind WW ′ E , b e E = b E , and e E is good. The representation e E is calledj-induced from E , e E = j WW ′ E .Let g ∈ G p , and g = g s g u its decomposition as a product of the semisimple part g s and the unipotent part g u .Let C = Z G p ( g s ) be the identity component of the centralizer of g s in G p . C is a reductive subgroup of G p of the .4. THE LUSZTIG PARTITION G p . Let H p be a maximal torus of C . H p is also a maximal torus in G p , and hence one has a naturalimbedding W ′ = N C ( H p ) /H p → N G p ( H p ) /H p = W, where N C ( H p ) , N G p ( H p ) stand for the normalizers of H p in C and in G p , respectively, W ′ is the Weyl group of C and W is the Weyl group of G p .Let E be the irreducible representation of W ′ associated with the help of the Springer correspondence to theconjugacy class of g u and the trivial local system on it. Then E is good, and let e E be the j-induced representationof W . This gives a well-defined map φ G p : G p → c W . The fibers of this map are called the strata of G p . Bydefinition the map φ G p is constant on each conjugacy class in G p . Therefore the strata are unions of conjugacyclasses.Moreover, by 1.3 in [70] we have the following formula for the dimension of the centralizer Z G p ( g ) of any element g ∈ G p in G p : dim Z G p ( g ) = rank G p + 2 b φ Gp ( g ) , (1.4.1)where rank G p is the rank of G p .It turns out that the image R ( W ) of φ G p only depends on W . It can be described as follows. Let N ( G p ) be theunipotent variety of G p and N ( G p ) the set of unipotent conjugacy classes in G p . Let X p ( W ) be the set of irreduciblerepresentations of W associated by the Springer correspondence to unipotent classes in N ( G p ) and the trivial localsystems on them. We shall identify X p ( W ) and N ( G p ). Let f p : N ( G p ) → X p ( W ) be the corresponding bijectivemap. Proposition 1.4.1. ([70], Proposition 1.4)
We have R ( W ) = X ( W ) ∪ r prime X r ( W ) . If G p is of type A n , ( n ≥
1) or E then R ( W ) = X ( W ).If G p is of type B n ( n ≥ C n ( n ≥ D n ( n ≥ F or E then R ( W ) = X ( W ).If G p is of type G then R ( W ) = X ( W ).If G p is of type E then R ( W ) = X ( W ) ∪ X ( W ), and X ( W ) ∩ X ( W ) = X ( W ).The above description of the set R ( W ) and the bijections N ( G p ) → X p ( W ) yield certain maps between sets N ( G p ) which preserve dimensions of conjugacy classes by (1.4.1). For instance, one always has an inclusion X ( W ) ⊂ X r ( W ) for any r ≥
2. The corresponding inclusion N ( G ) ⊂ N ( G p ) coincides with the Spaltensteinmap π G k p : N ( G ) → N ( G p ) (see [107], Th´eor`eme III.5.2).Fix a system of positive roots in ∆. Note that ∆ can be regarded as the root system of the pair ( G p , H p ),∆ = ∆( G p , H p ). Let B p be the Borel subgroup in G p associated to the corresponding system of negative roots, H p ⊂ B p the maximal torus, and l the corresponding length function on W . Denote by W the set of conjugacyclasses in W . For each w ∈ W = N G p ( H p ) /H p one can pick up a representative ˙ w ∈ G p . If p is good for G p , wewrite G p = G k , B p = B k , N ( G p ) = N ( G k ).Let C be a conjugacy class in W . Pick up a representative w ∈ C of minimal possible length with respect to l . By Theorem 0.4 in [73] there is a unique conjugacy class O ∈ N ( G k ) of minimal possible dimension whichintersects the Bruhat cell B k ˙ wB k and does not depend on the choice of the minimal possible length representative w in C . We denote this class by Φ G k ( C ).As shown in Section 1.1 in [73], one can always find a representative w ∈ C of minimal possible length withrespect to l which is elliptic in a parabolic Weyl subgroup W ′ ⊂ W , i.e. w acts without fixed points in the reflectionrepresentation of W ′ . Indeed, by Theorem 3.2.12 in [38] there is a parabolic subgroup W ′ ⊂ W such that C ∩ W ′ isa cuspidal conjugacy class in W ′ , i.e. every element in it is elliptic in W ′ . By Lemma 3.1.14 in [38] if w ∈ C ∩ W ′ isof minimal possible length with respect to the restriction of l to W ′ then it is also of minimal possible length withrespect to l .Let P ′ k ⊂ G k be the parabolic subgroup which contains B k and corresponds to W ′ , and M ′ k the semi-simplepart of the Levi factor of P ′ k , so that W ′ is the Weyl group of M ′ k . Let Φ G k p ( C ) be the unipotent class in G p containing the class π M ′ k p Φ M ′ k ( C ). This class only depends on the conjugacy class C , and hence one has a mapΦ G k p : W → N ( G p ) which is in fact surjective by 4.5(a) in [73].Let C ∈ W , and m C the dimension of the fixed point space for the action of any w ∈ C in the reflectionrepresentation. Then by Theorem 0.2 in [72] for any γ ∈ N ( G p ) there is a unique C ∈ (Φ G k p ) − ( γ ) such that thefunction m C : (Φ G k p ) − ( γ ) → N reaches its minimum at C . We denote C by Ψ G k p ( γ ). Thus one obtains an injectivemap Ψ G k p : N ( G p ) → W .6 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Now recall that using identifications f p : N ( G p ) → X p ( W ) one can define a bijection F : b N ( G k ) = N ( G ) ∪ r prime N ( G r ) → X ( W ) ∪ r prime X r ( W ) = R ( W ) . Using maps Φ G k p one can also define a surjective map Φ W : W → b N ( G k ) as follows. If Φ Gr ( C ) ∈ N ( G ) for all r > G k r ( C ) is independent of r , and one puts Φ W ( C ) = Φ G k r ( C ) for any r >
1. If Φ G k r ( C )
6∈ N ( G ) for some r > r is unique, one defines Φ W ( C ) = Φ G k r ( C ).By definition there is a right-sided injective inverse Ψ W to Φ W such that if γ ∈ N ( G ) then Ψ W ( γ ) = Ψ G k ( γ ),and if γ
6∈ N ( G ), and γ ∈ N ( G r ) then Ψ W ( γ ) = Ψ G k r ( γ ).Denote by C ( W ) the image of b N ( G k ) in W under the map Ψ W , C ( W ) = Ψ W ( b N ( G k )). We shall identify C ( W ), b N ( G k ) and R ( W ).Now assume that p is not a bad prime for G p . In this case the strata of the Lusztig partition can be describedgeometrically as follows. Let C ∈ C ( W ). Pick up a representative w ∈ C of minimal possible length with respectto l . Denote by G p the set of conjugacy classes in G p , and by G ′C the set of all conjugacy classes in G p whichintersect the Bruhat cell B p ˙ wB p . This definition does not depend on the choice of the the minimal possible lengthrepresentative w . Let d C = min γ ∈ G ′C dim γ. Then the stratum G C = φ − G p ( F (Φ W ( C ))) can be described as follows (see Theorem 2.2, [70]), G C = [ γ ∈ G ′C , dim γ = d C γ. Thus we have a disjoint union G p = [ C∈ C ( W ) G C . Note that by the definition of the stratum , if
C ∈
Im(Ψ G k ) then G C contains a unique unipotent class, and if C 6∈
Im(Ψ G k ) then G C does not contain unipotent classes.For good p the maps introduced above are summarized in the following diagram X ( W ) f ←− N ( G k ) ↓ ι ↓ π G k G k φ G k −→ R ( W ) F ←− b N ( G k ) Φ W ←−−→ Ψ W W , (1.4.2)where ι is an inclusion, bijections f and F are induced by the Springer correspondence with the trivial local data,and the inclusion π G k is induced by the Spaltenstein map.For exceptional groups the maps f and F can be described explicitly using tables in [108], the maps Φ W andΨ W can be described using the tables in Section 2 in [72], and the maps ι and π G can be described explicitlyusing the tables of unipotent classes in [63], Chapter 22 or [108] (note that the labeling for unipotent classes inbad characteristics in [63] differs from that in [108]). The dimensions of the conjugacy classes in the strata in G k can be obtained using dimension tables of centralizers of unipotent elements in case when a stratum contains aunipotent class (see [17, 63]), the tables for dimensions of the centralizers of unipotent elements in bad characteristicwhen a stratum does not contain a unipotent class (see [63]) or formula (1.4.1) and the tables of the values of the b –invariant b E for representations of Weyl groups (see [17, 38]). Note that formula (1.4.1) implies that if O is anyconjugacy class in G C , O ∈ G C then dim O = dim Φ W ( C ) . (1.4.3)In case of classical groups all those maps and dimensions are described in terms of partitions (see [17, 39, 63,71, 72, 73, 107]). In case of classical matrix groups the strata can also be described explicitly (see [70]). We recallthis description below. By (1.4.3) the dimensions of the conjugacy classes in every stratum of G k are equal to thedimension of the corresponding conjugacy class in b N ( G k ). The dimensions of centralizers of unipotent elements inarbitrary characteristic can be found in [48, 63].If λ = ( λ ≥ λ ≥ . . . ≥ λ m ) is a partition we denote by λ ∗ = ( λ ∗ ≥ λ ∗ ≥ . . . ≥ λ ∗ m ) the corresponding dualpartition. It is defined by the property that λ ∗ = m and λ ∗ i − λ ∗ i +1 = l i ( λ ), where l i ( λ ) is the number of times i appears in the partition λ . We also denote by τ ( λ ) the length of λ , τ ( λ ) = m . If a partition µ is obtained from λ by adding a number of zeroes, we shall identify λ and µ . .4. THE LUSZTIG PARTITION A n G k is of type SL( V ) where V is a vector space of dimension n + 1 ≥ k ofcharacteristic exponent p ≥ W is the group of permutations of n + 1 elements. All sets in (1.4.2), except for G k ,are isomorphic to the set of partitions of n + 1, and all maps, except for φ G k , are the identity maps.To describe φ G k for G k = SL( V ) we choose a sufficiently large m ∈ N . Let g ∈ G k . For any x ∈ k ∗ let V x bethe generalized x –eigenspace of g : V → V and let λ x ≥ λ x ≥ . . . ≥ λ xm be the sequence in N whose terms are thesizes of the Jordan blocks of x − g : V x → V x . Then φ G k ( g ) is the partition λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m given by λ ( g ) j = P x ∈ k ∗ λ xj .If g is any element in the stratum G λ corresponding to a partition λ = ( λ ≥ λ ≥ . . . ≥ λ m ), λ m ≥
1, thendim Z G k ( g ) = n + 2 m X i =1 ( i − λ i . (1.4.4)The element of W which corresponds to λ is the Coxeter class in the Weyl subgroup of the type A λ − + A λ − + . . . + A λ m − . (1.4.5)The summands in the diagram above are called blocks. Blocks of type A are called trivial. C n G k is of type Sp( V ) where V is a symplectic space of dimension 2 n , n ≥ k ofcharacteristic exponent p = 2. W is the group of permutations of the set E = { ε , . . . , ε n , − ε , . . . , − ε n } whichalso commute with the involution ε i
7→ − ε i . Each element s ∈ W can be expressed as a product of disjoint cyclesof the form ε k → ± ε k → ± ε k → . . . → ± ε k r → ± ε k . The cycle above is of length r ; it is called positive if s r ( ε k ) = ε k and negative if s r ( ε k ) = − ε k . The lengths ofthe cycles together with their signs give a set of positive or negative integers called the signed cycle-type of s . Toeach positive cycle of s of length r there corresponds a pair of positive orbits X, − X , | X | = r , for the action ofthe group h s i generated by s on the set E = { ε , . . . , ε n , − ε , . . . , − ε n } , and to each negative cycle of s of length r there corresponds a negative orbit X , | X | = 2 r , for the action of h s i on E . A positive cycle of length 1 is calledtrivial. It corresponds to a pair of fixed points for the action of h s i on E .Elements of W are parametrized by pairs of partitions ( λ, µ ), where the parts of λ are even (for any w ∈ C ∈ W they are the numbers of elements in the negative orbits X , X = − X , in E for the action of the group h w i generatedby w ), µ consists of pairs of equal parts (they are the numbers of elements in the positive h w i –orbits X in E ; theseorbits appear in pairs X, − X , X = − X ), and P λ i + P µ j = 2 n . We denote this set of pairs of partitions by A n . Anelement of W which corresponds to a pair ( λ, µ ), λ = ( λ ≤ λ ≤ . . . ≤ λ m ) and µ = ( µ = µ ≤ . . . ≤ µ k − = µ k )is the Coxeter class in the Weyl subgroup of the type C λ + C λ + . . . + C λm + A µ − + A µ − + . . . + A µ k − − . (1.4.6)Elements of N ( G k ) are parametrized by partitions λ of 2 n for which l j ( λ ) is even for odd j . We denote thisset of partitions by T n . In case of G k = Sp( V ) the parts of λ are just the sizes of the Jordan blocks in V of theunipotent elements from the conjugacy class corresponding to λ .In this case b N ( G k ) = N ( G ), and G is of type Sp( V ) where V is a symplectic space of dimension 2 n over an algebraically closed field of characteristic 2. Elements of N ( G ) are parametrized by pairs ( λ, ε ), where λ = ( λ ≤ λ ≤ . . . ≤ λ m ) ∈ T n , and ε : { λ , λ , . . . , λ m } → { , , ω } is a function such that ε ( k ) = ω if k is odd;1 if k = 0;1 if k > l k ( λ ) is odd;0 or 1 if k > l k ( λ ) is even. (1.4.7)We denote the set of such pairs ( λ, ε ) by T n .Elements of c W are parametrized by pairs of partitions ( α, β ) written in non–decreasing order, α ≤ α ≤ . . . ≤ α τ ( α ) , β ≤ β ≤ . . . ≤ β τ ( β ) , and such that P α i + P β i = n . By adding zeroes we can assume that the length τ ( α ) of α is related to the length of β by τ ( α ) = τ ( β ) + 1. The set of such pairs is denoted by X n, .8 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
The maps f , F can be described as follows. Let λ = ( λ ≤ λ ≤ . . . ≤ λ m +1 ) ∈ T n , and assume that λ = 0.If f ( λ ) = (( c ′ , c ′ , . . . , c ′ m +1 ) , ( c ′ , c ′ , . . . , c ′ m )) then the parts c ′ i are defined by induction starting from c ′ = 0, c ′ i = λ i if λ i is even and c ′ i − is already defined; c ′ i = λ i +12 if λ i = λ i +1 is odd and c ′ i − is already defined; c ′ i +1 = λ i − if λ i = λ i +1 is odd and c ′ i is already defined.The image of f consists of all pairs (( c ′ , c ′ , . . . , c ′ m +1 ) , ( c ′ , c ′ , . . . , c ′ m )) ∈ X n, such that c ′ i ≤ c ′ i +1 + 1 for all i . If F ( λ, ε ) = (( c , c , . . . , c m +1 ) , ( c , c , . . . , c m )) then the parts c i are defined by induction starting from c = 0, c i = λ i if λ i is even, ε ( λ i ) = 1 and c i − is already defined; c i = λ i +12 if λ i = λ i +1 is odd and c i − is already defined; c i +1 = λ i − if λ i = λ i +1 is odd and c i is already defined; c i = λ i +22 if λ i = λ i +1 is even, ε ( λ i ) = ε ( λ i +1 ) = 0 and c i − is already defined; c i +1 = λ i − if λ i = λ i +1 is even, ε ( λ i ) = ε ( λ i +1 ) = 0 and c i is already defined.The image R ( W ) of F consists of all pairs (( c , c , . . . , c m +1 ) , ( c , c , . . . , c m )) ∈ X n, such that c i ≤ c i +1 + 2for all i .The map Φ W is defined by Φ W ( λ, µ ) = ( ν, ε ), where the set of parts of ν is just the union of the sets of partsof λ and µ , and ε ( k ) = k ∈ N is a part of λ ;0 if k ∈ N is not a part of λ ; ω if k is odd.The map Ψ W associates to each pair ( ν, ε ) a unique point ( λ, µ ) in the preimage (Φ W ) − ( ν, ε ) such that thenumber of parts of µ is minimal possible. This point is defined by the conditions l k ( λ ) = (cid:26) k is odd or k is even, l k ( ν ) ≥ ε ( k ) = 0; l k ( ν ) otherwise, l k ( µ ) = (cid:26) l k ( ν ) if k is odd or k is even, l k ( ν ) ≥ ε ( k ) = 0;0 otherwise.The map π G k is given by π G k ( λ ) = ( λ, ε ′ ), where ε ′ ( k ) = (cid:26) ω if k is odd;1 if k is even.The map π G k is injective and its image consists of pairs ( λ, ε ) ∈ T n , where ε satisfies the conditions above.To describe φ G k for G k = Sp( V ) we choose a sufficiently large m ∈ N . Let g ∈ G k . For any x ∈ k ∗ let V x bethe generalized x –eigenspace of g : V → V . For any x ∈ k ∗ such that x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m +1 be thesequence in N whose terms are the sizes of the Jordan blocks of x − g : V x → V x .For any x ∈ k ∗ with x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m +1 be the sequence in N , where (( λ x ≥ λ x ≥ . . . ≥ λ x m +1 ) , ( λ x ≥ λ x ≥ . . . ≥ λ x m )) is the pair of partitions such that the corresponding irreducible representation ofthe Weyl group of type B dim V x / is the Springer representation attached to the unipotent element x − g ∈ Sp( V x )and to the trivial local data.Let λ ( g ) be the partition λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m +1 given by λ ( g ) j = P x λ xj , where x runs overa set of representatives for the orbits of the involution a a − of k ∗ . Now φ G ( g ) is the pair of partitions(( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m +1 ) , ( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m )).If g is any element in the stratum G ( λ,ε ) corresponding to a pair ( λ, ε ) ∈ T n , λ = ( λ ≥ λ ≥ . . . ≥ λ m ) thendim Z G k ( g ) = n + m X i =1 ( i − λ i + 12 |{ i : λ i is odd }| + |{ i : λ i is even and ε ( λ i ) = 0 }| . (1.4.8) .4. THE LUSZTIG PARTITION B n G k is of type SO( V ) where V is a vector space of dimension 2 n + 1, n ≥ k ofcharacteristic exponent p = 2 equipped with a non–degenerate symmetric bilinear form. W is the same as in caseof C n .An element of W which corresponds to a pair ( λ, µ ), λ = ( λ ≥ λ ≥ . . . ≥ λ m ) and µ = ( µ = µ ≥ . . . ≥ µ k − = µ k ) is the class represented by the sum of the blocks in the following diagram (we use the notation of[16], Section 7) A µ − + A µ − + . . . + A µ k − − ++ D λ λ ( a λ − ) + D λ λ ( a λ − ) + . . . + D λm − λm − ( a λm − − ) + B λm ( m is odd) , (1.4.9) A µ − + A µ − + . . . + A µ k − − ++ D λ λ ( a λ − ) + D λ λ ( a λ − ) + . . . + D λm − λm ( a λm − ) ( m is even) , where it is assumed that D k ( a ) = D k .The elements of N ( G k ) are parametrized by partitions λ of 2 n + 1 for which l j ( λ ) is even for even j . We denotethis set of partitions by Q n +1 . In case of G k = SO( V ) the parts of λ are just the sizes of the Jordan blocks in V of the unipotent elements from the conjugacy class corresponding to λ .In this case b N ( G k ) = N ( G ), and G is of type SO( V ) where V is a vector space of dimension 2 n + 1 over analgebraically closed field of characteristic 2 equipped with a bilinear form ( · , · ) and a non–zero quadratic form Q such that ( x, y ) = Q ( x + y ) − Q ( x ) − Q ( y ) , x, y ∈ V , and the restriction of Q to the null space V ⊥ = { x ∈ V : ( x, y ) = 0 ∀ y ∈ V } of ( · , · ) has zero kernel. In fact G isisomorphic to a group of type Sp( V ), dim V = 2 n , and hence N ( G ) ≃ T n .We also have c W ≃ X n, , and the map F is the same as in case of C n .The map f can be described as follows. Let λ = ( λ ≤ λ ≤ . . . ≤ λ m +1 ) ∈ Q n +1 . If f ( λ ) = (( c ′ , c ′ , . . . , c ′ m +1 ) , ( c ′ , c ′ , . . . , c ′ m ))then the parts c ′ i are defined by induction starting from c ′ , c ′ i = λ i − + i − − (cid:2) i − (cid:3) if λ i is odd and c ′ i − is already defined; c ′ i = λ i if λ i = λ i +1 is even and c ′ i − is already defined; c ′ i +1 = λ i if λ i = λ i +1 is even and c ′ i is already defined.The image of f consists of all pairs (( c ′ , c ′ , . . . , c ′ m +1 ) , ( c ′ , c ′ , . . . , c ′ m )) ∈ X n, such that c ′ i ≤ c ′ i +1 for all odd i and c ′ i ≤ c ′ i +1 + 2 for all even i .The image R ( W ) of F consists of all pairs (( c , c , . . . , c m +1 ) , ( c , c , . . . , c m )) ∈ X n, such that c i ≤ c i +1 + 2for all i .The maps Φ W and Ψ W are the same as in case of C n .The map π G is given by π G ( λ ) = ( ν, ε ′ ), λ = ( λ ≤ λ ≤ . . . ≤ λ m +1 ) ∈ Q n +1 , where ν i = λ i − λ i and i are odd and λ i − < λ i ; λ i + 1 if λ i is odd, i is even and λ i < λ i +1 ; λ i otherwise , and ε ′ ( k ) = ω if k is odd;0 if k is even, there exists even λ i = k with even i such that λ i − < λ i ;1 otherwise.The map π G k is injective and its image consists of pairs ( ν, ε ) ∈ T n such that ε ( k ) = 0 if ν ∗ k is odd and for eacheven i such that ν ∗ i is even we have ν ∗ i − = ν ∗ i , i.e. i − ν . Here ν ∗ ≥ ν ∗ ≥ . . . ≥ ν ∗ m is the partition dual to ν .To describe φ G k for G k = SO( V ) we choose a sufficiently large m ∈ N . Let g ∈ G k . For any x ∈ k ∗ let V x bethe generalized x –eigenspace of g : V → V . For any x ∈ k ∗ such that x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m +1 be thesequence in N whose terms are the sizes of the Jordan blocks of x − g : V x → V x .0 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
For any x ∈ k ∗ with x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m +1 be the sequence in N , where (( λ x ≥ λ x ≥ . . . ≥ λ x m +1 ) , ( λ x ≥ λ x ≥ . . . ≥ λ x m )) is the pair of partitions such that the corresponding irreducible representation ofthe Weyl group of type B (dim V x − / (if x = −
1) or D dim V x / (if x = −
1) is the Springer representation attachedto the unipotent element x − g ∈ SO( V x ) and to the trivial local data.Let λ ( g ) be the partition λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m +1 given by λ ( g ) j = P x λ xj , where x runs overa set of representatives for the orbits of the involution a a − of k ∗ . Now φ G ( g ) is the pair of partitions(( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m +1 ) , ( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m )).If g is any element in the stratum G ( λ,ε ) corresponding to a pair ( λ, ε ) ∈ T n , λ = ( λ ≥ λ ≥ . . . ≥ λ m ) thenthe dimension of the centralizer of g in G k is given by formula (1.4.8),dim Z G k ( g ) = n + m X i =1 ( i − λ i + 12 |{ i : λ i is odd }| + |{ i : λ i is even and ε ( λ i ) = 0 }| . (1.4.10) D n G k is of type SO( V ) where V is a vector space of dimension 2 n , n ≥ k ofcharacteristic exponent p = 2 equipped with a non–degenerate symmetric bilinear form. W is the group of evenpermutations of the set E = { ε , . . . , ε n , − ε , . . . , − ε n } which also commute with the involution ε i
7→ − ε i . W canbe regarded as a subgroup in the Weyl group W ′ of type C n .Let f W be the set of W ′ –conjugacy classes in W . Elements of f W are parametrized by pairs of partitions ( λ, µ ),where the parts of λ are even (for any w ∈ C ∈ f W they are the numbers of elements in the negative orbits X , X = − X , in E for the action of the group h w i generated by w ), the number of parts of λ is even, µ consists ofpairs of equal parts (they are the numbers of elements in the positive h w i –orbits X in E ; these orbits appear inpairs X, − X , X = − X ), and P λ i + P µ j = 2 n . We denote this set of pairs of partitions by A n . To each pair( − , µ ), where all parts of µ are even, there correspond two conjugacy classes in W . To all other elements of A n there corresponds a unique conjugacy class in W .An element of f W which corresponds to a pair ( λ, µ ), λ = ( λ ≥ λ ≥ . . . ≥ λ m ) and µ = ( µ = µ ≥ . . . ≥ µ k − = µ k ) is the class represented by the sum of the blocks in the following diagram (we use the notation of[16], Section 7) A µ − + A µ − + . . . + A µ k − − + D λ λ ( a λ − ) + D λ λ ( a λ − ) + . . . + D λm − λm ( a λm − ) . (1.4.11)Let G ′ k be the extension of G k by the Dynkin graph automorphism of order 2. Then G ′ k is of type O( V ). Denoteby e N ( G k ) the set of unipotent classes of G ′ k . Note that they are all contained in G k . The elements of e N ( G k )are parametrized by partitions λ of 2 n for which l j ( λ ) is even for even j . Note that the number of parts of suchpartitions is even. We denote this set of partitions by Q n . In case when G k = SO( V ) the parts of λ are just thesizes of the Jordan blocks in V of the unipotent elements from the conjugacy class corresponding to λ . If λ hasonly even parts then λ corresponds to two unipotent classes in G k of the same dimension. In all other cases thereis a unique unipotent class in G which corresponds to λ .Let V be a vector space of dimension 2 n over an algebraically closed field of characteristic 2 equipped with anon–degenerate bilinear form ( · , · ) and a non–zero quadratic form Q such that( x, y ) = Q ( x + y ) − Q ( x ) − Q ( y ) , x, y ∈ V . We remind that SO( V ) is the connected component containing the identity of the group of linear automorphismsof V preserving the quadratic, and hence the bilinear, form.One has b N ( G k ) = N ( G ), and G is of type SO( V ).Let G ′ be the extension of G by the Dynkin graph automorphism of order 2. Then G ′ is of type O( V ).Denote by e N ( G ) the set of unipotent classes of G ′ contained in G . Since the bilinear form ( · , · ) is also alternatingin characteristic 2 there is a natural injective homomorphism from O( V ) to Sp( V ), dim V = 2 n , and hence e N ( G ) ≃ e T n , where e T n is the set of elements ( λ, ε ) ∈ T n such that λ has an even number of parts.Let cf W be the set of orbits of irreducible characters of W under the action of W ′ . Elements of cf W are parametrizedby unordered pairs of partitions ( α, β ) written in non–decreasing order, α ≤ α ≤ . . . ≤ α τ ( α ) , β ≤ β ≤ . . . ≤ β τ ( β ) , and such that P α i + P β i = n . By adding zeroes we can assume that the length of α is equal to the lengthof β . The set of such pairs is denoted by Y n, . .4. THE LUSZTIG PARTITION e X ( W ) e f ←− e N ( G k ) ↓ ι ↓ e π G k G k e φ G k −→ e R ( W ) e F ←− e N ( G ) e Φ W ←−−→ e Ψ W f W , (1.4.12)where e f and e F are induced by the restrictions of the maps f and F for G ′ k , G ′ to e N ( G k ), e N ( G ), respectively, e X ( W ) and e R ( W ) are their images, e φ G k , e Ψ W , e Φ W and e π G are also induced by the corresponding maps for G ′ k , G ′ and W ′ .The map e f is defined as in case of B n . The image of e f consists of all pairs (( c ′ , c ′ , . . . , c ′ m +1 ) , ( c ′ , c ′ , . . . , c ′ m )) ∈ Y n, such that c ′ i ≤ c ′ i +1 for all odd i and c ′ i ≤ c ′ i +1 + 2 for all even i .If ( λ, ε ) ∈ e T n , λ = ( λ ≤ λ ≤ . . . ≤ λ m ) and e F ( λ, ε ) = (( c , c , . . . , c m − ) , ( c , c , . . . , c m )) then the parts c i are defined by induction starting from c , c i = λ i − + 2( i − − (cid:2) i − (cid:3) if λ i is even, ε ( λ i ) = 1 and c i − is already defined; c i = λ i − + 2( i − − (cid:2) i − (cid:3) if λ i = λ i +1 is odd and c i − is already defined; c i +1 = λ i − + 2 i − (cid:2) i (cid:3) if λ i = λ i +1 is odd and c i is already defined; c i = λ i + 2( i − − (cid:2) i − (cid:3) if λ i = λ i +1 is even, ε ( λ i ) = 0 and c i − is already defined; c i +1 = λ i + 2( i − − (cid:2) i − (cid:3) if λ i = λ i +1 is even, ε ( λ i ) = 0 and c i is already defined.The image e R ( W ) of e F consists of all pairs (( c , c , . . . , c m +1 ) , ( c , c , . . . , c m )) ∈ Y n, such that c i ≤ c i +1 forall odd i and c i ≤ c i +1 + 4 for all even i .The maps e Φ W and e Ψ W are defined by the same formulas as in case of C n .The map e π G k is given by e π G k ( λ ) = ( ν, ε ′ ), λ = ( λ ≤ λ ≤ . . . ≤ λ m ) ∈ Q n , where ν i = λ i − λ i is odd, i is even and λ i − < λ i ; λ i + 1 if λ i and i are odd, and λ i < λ i +1 ; λ i otherwise , and ε ′ ( k ) = ω if k is odd;0 if k is even, there exists even λ i = k with odd i such that λ i − < λ i ;1 otherwise.The map e π G k is injective and its image consists of pairs ( ν, ε ) ∈ e T n such that ε ( k ) = 0 if ν ∗ k is odd and for eacheven i such that ν ∗ i is even we have ν ∗ i − = ν ∗ i , i.e. i − ν . Here ν ∗ ≥ ν ∗ ≥ . . . ≥ ν ∗ m is the partition dual to ν .To describe e φ G k for G k = SO( V ) we choose a sufficiently large m ∈ N . Let g ∈ G k . For any x ∈ k ∗ let V x be the generalized x –eigenspace of g : V → V . For any x ∈ k ∗ such that x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m be thesequence in N whose terms are the sizes of the Jordan blocks of x − g : V x → V x .For any x ∈ k ∗ with x = 1 let λ x ≥ λ x ≥ . . . ≥ λ x m be the sequence in N , where (( λ x ≥ λ x ≥ . . . ≥ λ x m − ) , ( λ x ≥ λ x ≥ . . . ≥ λ x m )) is the pair of partitions such that the corresponding irreducible representation ofthe Weyl group of type D dim V x / is the Springer representation attached to the unipotent element x − g ∈ SO( V x )and to the trivial local data.Let λ ( g ) be the partition λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m +1 given by λ ( g ) j = P x λ xj , where x runs overa set of representatives for the orbits of the involution a a − of k ∗ . Now e φ G ( g ) is the pair of partitions(( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m − ) , ( λ ( g ) ≥ λ ( g ) ≥ . . . ≥ λ ( g ) m )).The preimage e φ − G k ( λ, µ ) is a stratum in G k in all cases except for the one when the pair ( λ, µ ) is of the form(( λ ≥ λ ≥ . . . ≥ λ m − ) , ( λ ≥ λ ≥ . . . ≥ λ m − )). In that case e φ − G k ( λ, µ ) is a union of two strata, and theconjugacy classes in each of them have the same dimension.If g is any element in the stratum G ( λ,ε ) corresponding to a pair ( λ, ε ) ∈ e T n , λ = ( λ ≥ λ ≥ . . . ≥ λ m ) thenthe dimension of the centralizer of g in G k is given by the following formuladim Z G k ( g ) = n + m X i =1 ( i − λ i − |{ i : λ i is odd }| − |{ i : λ i is even and ε ( λ i ) = 1 }| . (1.4.13)2 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Let ∆ s + be a set of positive roots associated to s ∈ W = W ( G k , H k ) in Section 1.2, where we again assume thatthat in sum (1.2.14) h is the linear subspace of h R fixed by the action of s . Denote by P k the parabolic subgroupof G k containing the Borel subgroup corresponding to − ∆ s + and assocuated to the subset − Γ s of the set of simpleroots in − Γ s , where Γ s = Γ s ∩ ∆ . Let N k and L k the unipotent radical and the Levi factor of P k , respectively,and N k the opposite unipotent radical.Denote a representative for the Weyl group element s in G k by ˙ s . Let Z k be the connected subgroup of G k generated by the semisimple part of the Levi subgroup L k and by the identity component H k of centralizer of ˙ s in H k . Let N s, k = { v ∈ N k | ˙ sv ˙ s − ∈ N k } .Following the definition of transversal slices Σ s we defineΣ s, k = sZ k N s, k . Recall that the definition of ∆ s + , and hence of Σ s, k , depends on the choice of ordering of terms in decomposition(1.2.14). In this section for every conjugacy class C ∈ C ( W ) we define a variety Σ s, k , s ∈ C such that everyconjugacy class O ∈ G C intersects Σ s, k and dim O = codim Σ s, k . (1.5.1)It turns out that in order to fulfill condition (1.5.1) the subspaces h i in (1.2.14) should be ordered in such a waythat h ⊂ h R is the subspace fixed by the action of s , and if h i = h kλ , h j = h lµ and 0 ≤ λ < µ < i < j , where λ and µ are eigenvalues of the corresponding matrix I − M for s . In the case of exceptional root systems this is verifiedusing a computer program, and in the case of classical root systems this is confirmed by explicit computation basedon a technical lemma. In order to formulate this lemma we recall realizations of classical irreducible root systems.Let V be a real Euclidean n –dimensional vector space with an orthonormal basis ε , . . . , ε n . The root systemsof types A n − , B n , C n and D n can be realized in V as follows. A n The roots are ε i − ε j , 1 ≤ i, j ≤ n , i = j , h R is the hyperplane in V consisting of the points the sum of whosecoordinates is zero. B n The roots are ± ε i ± ε j , 1 ≤ i < j ≤ n , ± ε i , 1 ≤ i ≤ n , h R = V . C n The roots are ± ε i ± ε j , 1 ≤ i < j ≤ n , ± ε i , 1 ≤ i ≤ n , h R = V . D n The roots are ± ε i ± ε j , 1 ≤ i < j ≤ n , h R = V .In all cases listed above the corresponding Weyl group W is a subgroup of the Weyl group of type C n actingon the elements of the basis ε , . . . , ε n by permuting the basis vectors and changing the sign of an arbitrary subsetof them.Now we formulate the main lemma. Lemma 1.5.1.
Let s be an element of the Weyl group of type C n operating on the set E = { ε , . . . , ε n , − ε , . . . , − ε n } as indicated in Section 1.4, where ε , . . . , ε n is the basis of V introduced above. Assume that s has either only onenontrivial cycle of length k/ ( k is even), which is negative, or only one nontrivial cycle of length k , which ispositive, < k ≤ n . Let ∆ be a root system of type A n − , B n , C n or D n realized in V as above.(i) If s has only one nontrivial cycle of length k/ , which is negative, then k is even, the spectrum of s in thecomplexification V C of V is ǫ r = exp( πi ( k − r +1) k ) , r = 1 , . . . , k/ , and possibly ǫ = 1 , all eigenvalues are simpleexcept for possibly .(ii) If s only has one nontrivial cycle of length k , which is positive, then the spectrum of s in the complexificationof V is ǫ r = exp( πi ( k − r ) k ) , r = 1 , . . . , k − , and ǫ = 1 , all eigenvalues are simple except for possibly . .5. THE STRICT TRANSVERSALITY CONDITION In both cases we denote by V r the invariant subspace in V which corresponds to ǫ r = exp( πi ([ k/ − r ) k ) , r = 1 , . . . , (cid:2) k (cid:3) or ǫ = 1 in case of a positive nontrivial cycle and to ǫ r = exp( πi (2 [ k/ ] +1 − r ) k ) , r = 1 , . . . , h k/ i or ǫ = 1 in case of a negative cycle. For r = 0 the space V r is spanned by the real and the imaginary parts of anonzero eigenvector of s in V C corresponding to ǫ r , and V is the subspace of fixed points of s in V . V r is two–dimensional if ǫ r = ± , one–dimensional if ǫ r = − or may have arbitrary dimension if ǫ r = 1 .Let ∆ s + be a system of positive roots associated to s and defined as in Section 1.2, where we use the decomposition V = M i V i (1.5.2) as (1.2.14) in the definition of ∆ s + . Denote by ∆ i ⊂ ∆ the corresponding subsets of roots defined as in (1.2.15).Let ∆ s be the root subsystem fixed by the action of s and l ( s ) the number of positive roots which become negativeunder the action of s .(iii) If s has only one nontrivial cycle of length k , which is positive, we have1. if ∆ = A n − then ∆ s = A n − k − , l ( s ) = 2 n − k − ;2. if ∆ = B n ( C n ) then ∆ s = B n − k ( C n − k ) , l ( s ) = 4 n − k for odd k and l ( s ) = 4 n − k + 1 for even k ;3. if ∆ = D n then ∆ s = D n − k , l ( s ) = 4 n − k − for odd k and l ( s ) = 4 n − k − for even k .(iv) If s has only one nontrivial cycle of length k , which is negative, we have1. if ∆ = B n ( C n ) then ∆ s = B n − k/ ( C n − k/ ) , l ( s ) = 2 n − k/ ;2. if ∆ = D n then ∆ s = D n − k/ , l ( s ) = 2 n − k/ − .(v) If s has only one nontrivial cycle of length k , which is positive, ∆ is of type B n , C n or D n , and k is eventhen ∆ = ∆ k/ ∪ ∆ k/ − ∪ ∆ s (disjoint union), and all roots in ∆ k/ − are orthogonal to the fixed point subspacefor the action of s on V .(vi) In all other cases ∆ = ∆ i max ∪ ∆ s (disjoint union), where i max is the maximal possible index i which appearsin decomposition (1.5.2).Proof. The proof is similar in all cases. We only give details in the most complicated case when s has only onenontrivial cycle, which is positive, ∆ is of type B n ( C n ), and k is even. Without loss of generality one can assumethat s corresponds to the cycle of the form ε → ε → ε → ε → · · · → ε k − → ε k → ε k − → ε k − → · · · → ε → ε ( k > , ε → ε → ε ( k = 2) . From this definition one easily sees that ∆ s = B n − k ( C n − k ) = ∆ ∩ V ′ , where V ′ ⊂ V is the subspace generatedby ε k +1 , . . . , ε n . Computing the eigenvalues of s in V C is a standard exercise in linear algebra. The eigenvalues areexpressed in terms of the exponents of the root system of type A k − (see [15], Ch. 10).The invariant subspace V r is spanned by the real and the imaginary parts of a nonzero eigenvector of s in V C corresponding to the eigenvalue ǫ r . If ǫ r = ± V r is two–dimensional, and for ǫ r = − V r is one–dimensional.In the former case V r will be regarded as the real form of a complex plane with the orthonormal basis 1 , i . Underthis convention the orthogonal projection operator onto V r acts on the basic vectors ε j as follows ε j +1 cǫ jr , j = 0 , . . . , k − , ε j cǫ − jr , j = 1 , . . . , k , (1.5.3)where c = q k . Consider the case when k >
2; the case k = 2 can be analyzed in a similar way.To compute l ( s ) using the definition of ∆ s + given in Section 1.2 one should first look at all roots which havenonzero projections onto V k/ on which s acts by rotation with the angle πk .From (1.5.3) we deduce that the roots which are not fixed by s and have zero orthogonal projections onto V k/ are ± ( ε j + ε k − j +1 ), j = 1 , . . . k . The number of those roots is equal to k , and they all have nonzero orthogonalprojections onto V k/ − . From (1.5.3) we also obtain that all the other roots which are not fixed by s have nonzeroorthogonal projections onto V k/ , hence | ∆ k/ − | = k . The number of roots fixed by s is 2( n − k ) since it is equalto the number of roots in ∆ = ∆ s = B n − k ( C n − k ). Hence ∆ = ∆ k/ ∪ ∆ k/ − ∪ ∆ (disjoint union), the numberof roots in ∆ k/ is | ∆ | − | ∆ | − | ∆ k/ − | = 2 n − n − k ) − k = 4 nk − k − k , | ∆ k/ | = 4 nk − k − k .4 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Now using the symmetry of the root system ∆ as a subset of V and the fact that s acts as rotation by theangles πk and πk in V k/ and V k/ − , respectively, we deduce that the number of positive roots in ∆ k/ (∆ k/ − )which become negative under the action of s is equal to the number of roots in ∆ k/ (∆ k/ − ) divided by the orderof s in V k/ ( V k/ − ). Therefore l ( s ) = | ∆ k/ | k + | ∆ k/ − | k/ nk − k − kk + kk/ n − k + 1 . This completes the proof in the considered case.Now we are in a position to prove the main statement of this section.
Theorem 1.5.2.
Let G k be a connected semisimple algebraic group over an algebraically closed filed k of charac-teristic good for G k , and O ∈ b N ( G k ) . Let H k be a maximal torus of G k , W the Weyl group of the pair ( G k , H k ) ,and s ∈ W an element from the conjugacy class Ψ W ( O ) . Let ∆ be the root system of the pair ( G k , H k ) and ∆ s + a system of positive roots in ∆ associated to s and defined in Section 1.2 with the help of decomposition (1.2.14),where the subspaces h i are ordered in such a way that h is the linear subspace of h R fixed by the action of s , andif h i = h kλ , h j = h lµ and ≤ λ < µ < then i < j . In the case of exceptional root systems we assume, in addition,that ∆ s + is chosen as in the tables in Appendix 2, so that s = s s is defined by the data from columns threeand four in the tables in Appendix 2. Then all conjugacy classes in the stratum G O = φ − G ( F ( O )) intersect thecorresponding variety Σ s, k at some points of the subvariety ˙ sH , k N s, k , where H , k ⊂ H k is the identity componentof the centralizer of ˙ s in H k . Moreover, if O ∈ N ( G p ) ⊂ b N ( G k ) for some p , then for any g ∈ G O dim Z G k ( g ) = dim Σ s, k = codim G p O . (1.5.4) Proof.
We shall divide the proof into several lemmas. First we compute the dimension of the slice Σ s, k , s ∈ Ψ W ( O )and justify that for any g ∈ G O equality (1.5.4) holds. Lemma 1.5.3.
Assume that the conditions of Theorem 1.5.2 are satisfied. Then for any g ∈ G O , where O ∈N ( G p ) ⊂ b N ( G ) for some p , equality (1.5.4) holds, i.e. dim Z G k ( g ) = dim Σ s, k = codim G p O . Proof.
Observe that by the definition of the slice Σ s, k dim Σ s, k = l ( s ) + | ∆ | + dim h , where l ( s ) is the length of s with respect to the system of simple roots in ∆ s + . Hence to compute dim Σ s, k we haveto find all numbers in the right hand side of the last equality.Consider the case of classical Lie algebras when each Weyl group element is a product of cycles in a permutationgroup. In this case identity (1.5.4) is proved by a straightforward calculation using Lemma 1.5.1.If G is of type A n let s be a representative in the conjugacy class of the Weyl group which corresponds to apartition λ = ( λ ≥ λ ≥ . . . ≥ λ m ). The particular ordering of the invariant subspaces h i in the formulationof this theorem implies that the length l ( s ) should be computed by successive application of Lemma 1.5.1 to thecycles s i of s , which correspond to λ i placed in a non–increasing order.We should first apply Lemma 1.5.1 to the cycle s of s which corresponds to the maximal part λ . In this case l ( s ) = 2 n − λ + 1 and ∆ s = A n − λ = ∆ \ ∆ i M in the notation of Section 1.2. The remaining cycles s , . . . , s m of s corresponding to λ ≥ . . . ≥ λ m act on ∆ s , and we can apply Lemma 1.5.1 to s acting on ∆ s to get l ( s ) = 2( n − λ ) − λ + 1 and ∆ s = A n − λ − λ = ∆ \ (cid:0) ∆ i M ∪ ∆ i M − (cid:1) . Iterating this procedure and observing that l ( s ) is equal to the number of positive roots which become negative under the action of s we obtain l ( s ) = X l ( s k ) , l ( s k ) = 2( n − k − X i =1 λ i ) − λ k + 1 , (1.5.5)where the first sum in (1.5.5) is taken over k for which λ k > s can be represented in a similar form, | ∆ | = X l ( s k ) , l ( s k ) = 2( n − k − X i =1 λ i ) − λ k + 1 , (1.5.6) .5. THE STRICT TRANSVERSALITY CONDITION k for which λ k = 1.Finally the dimension of the fixed point space h of s in h is m −
1, dim h = m − s, k = l ( s ) + | ∆ | + dim h , (1.5.7)and hence dim Σ s, k = m X k =1 l ( s k ) + m − m X k =1 n − k − X i =1 λ i ) − λ k + 1 ! + m − . Exchanging the order of summation and simplifying this expression we obtain thatdim Σ s, k = n + 2 m X i =1 ( i − λ i which coincides with (1.4.4).The computations of dim Σ s, k in case of B n and of C n are similar. If ( ν, ε ) ∈ T n , ν = ( ν ≥ ν ≥ . . . ≥ ν m ),corresponds to O ∈ b N ( G k ) = N ( G ) then Ψ W ( ν, ε ) = ( λ, µ ) ∈ A n ≃ W is defined in Section 1.4, part C n . λ consists of even parts ν i of ν for which ε ( ν i ) = 1, and µ consists of all odd parts of ν and of even parts ν i of ν for which ε ( ν i ) = 0, the last two types of parts appear in pairs of equal parts. Let s be a representative inthe conjugacy class Ψ W ( ν, ε ). Then each part λ i corresponds to a negative cycle of s of length λ i , and each pair µ i = µ i +1 of equal parts of µ corresponds to a positive cycle of s of length µ i . We order the cycles s k of s associatedto the (pairs of equal) parts of the partition ν in a way compatible with a non–increasing ordering of the parts ofthe partition ν = ( ν ≥ ν ≥ . . . ≥ ν m ), i.e. if we denote by s k the cycle that corresponds to an even part ν k of ν for which ε ( ν k ) = 1 or to a pair ν k = ν k +1 of odd parts of ν or of even parts of ν for which ε ( ν k ) = 0 then s k ≥ s l if ν k ≥ ν l .Similarly to the case of A n , by the definition of ∆ s + and by Lemma 1.5.1 applied iteratively to the cycles s k inthe order defined above, the length l ( s ) of s is the sum of the following terms l ( s k ).To each even part ν k of ν for which ε ( ν k ) = 1 we associate the term l ( s k ) = 2( n − k − X i =1 ν i − ν k ν k = ν k +1 > l ( s k ) = 4( n − k − X i =1 ν i − ν k = n − k − X i =1 ν i − ν k ! + n − k X i =1 ν i − ν k +1 ! ;note that the sum of these terms over all pairs ν k = ν k +1 = 1 gives the number | ∆ | of the roots fixed by s ;to each pair of even parts ν k = ν k +1 for which ε ( ν k ) = 0 we associate the term l ( s k ) = 4( n − k − X i =1 ν i − ν k + 1 = n − k − X i =1 ν i − ν k ! + n − k X i =1 ν i − ν k +1 ! . The dimension of the fixed point space h of s in h R is equal to a half of the sum of the number of all even parts ν k for which ε ( ν k ) = 0 and of the number of all odd parts ν k ,dim h = 12 |{ i : ν i is odd }| + 12 |{ i : ν i is even and ε ( ν i ) = 0 }| . (1.5.8)Finally substituting all the computed contributions into formula (1.5.7) we obtaindim Σ s, k = m X k =1 n − k − X i =1 ν i − ν k ! + 12 |{ i : ν i is even and ε ( ν i ) = 0 }| ++ 12 |{ i : ν i is odd }| + 12 |{ i : ν i is even and ε ( ν i ) = 0 }| . CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Exchanging the order of summation and simplifying this expression we obtain thatdim Σ s, k = n + m X i =1 ( i − ν i + 12 |{ i : ν i is odd }| + |{ i : ν i is even and ε ( ν i ) = 0 }| (1.5.9)which coincides with (1.4.8) or (1.4.10).In case of D n the number dim Σ s, k can be easily obtained if we observe that the map e Ψ W is defined by thesame formula as Ψ W in case of C n . In case when e Ψ W ( ν, ε ) = ( − , µ ), where all parts of µ are even, there are twoconjugacy classes in W which correspond to e Ψ W ( ν, ε ). However, the numbers l ( s ), | ∆ | and dim h are the samein both cases. They only depend on e Ψ W ( ν, ε ) in all cases. Let s ∈ W be a representative from the conjugacy class e Ψ W ( ν, ε ), ν = ( ν ≥ ν ≥ . . . ≥ ν m ).From Lemma 1.5.1 we deduce that in the case of D n the contributions of the cycles s k of s to the formula fordim Σ s, k can be obtained from the corresponding contributions in case of C n in the following way: for each pair ofodd parts ν k = ν k +1 and for each pair of even parts ν k = ν k +1 with ε ( ν k ) = 0 the corresponding contribution l ( s k )to l ( s ) should be reduced by 2 and for each even part ν k of ν with ε ( ν k ) = 1 the corresponding contribution l ( s k )to l ( s ) should be reduced by 1. This observation and formula (1.5.9) yielddim Σ s, k = n + m X i =1 ( i − ν i + 12 |{ i : ν i is odd }| + |{ i : ν i is even and ε ( ν i ) = 0 }| −−|{ i : ν i is odd }| − |{ i : ν i is even }| == n + m X i =1 ( i − ν i − |{ i : ν i is odd }| − |{ i : ν i is even and ε ( ν i ) = 1 }| which coincides with (1.4.13).In case of root systems of exceptional types dim Σ s, k can be found in the tables in Appendix 2. According tothose tables equality (1.5.4) holds in all cases.Now we show that all conjugacy classes in the stratum G O = φ − G ( F ( O )) intersect the corresponding varietyΣ s, k , s ∈ Ψ W ( O ). Let ∆ s + be the system of the positive roots introduced in the statement of Theorem 1.5.2, h R = K M i =0 h i the corresponding decomposition of h R and h i ∈ h i the corresponding elements of the subspaces h i .Recall that h is the subspace of h R fixed by the action of s . If h = 0, let h ′ i = h i and h ′ i = h i , i = 0 , . . . , K .Otherwise let h ′ K = h , h ′ i = h i +1 , h ′ i = h i +1 , i = 0 , . . . , K − h ′ K ∈ h ′ K such that h ′ K ( α ) = 0for any root α ∈ ∆ which is not orthogonal to the s –invariant subspace h ′ K with respect to the natural pairingbetween h R and h ∗ R .By a suitable rescaling of h ′ K we can assume that conditions (1.2.16) are satisfied for the elements h ′ i and roots α from sets ∆ ′ i defined as in (1.2.20) with h j , h i replaced by h ′ j , h ′ i . Indeed, observe that∆ ′ i = { α ∈ ∆ : h ′ j ( α ) = 0 , j > i, h ′ i ( α ) = 0 } ⊂ { α ∈ ∆ : h j ( α ) = 0 , j > i + 1 , h i +1 ( α ) = 0 } = ∆ i +1 , i = 0 , . . . , K − h ′ i . Thus, since (1.2.16) is satisfied for h i , i = 0 , . . . , K , it is also satisfied for h ′ i = h i if i < K . By a suitable rescaling of h ′ K we can assume that (1.2.16) is satisfied for h ′ K as well.Let ∆ be a system of positive roots in ∆ = ∆( G k , H k ) which corresponds to the Weyl chamber containing theelement ¯ h ′ = P Ki =0 h ′ i . By Lemma 1.2.2 the set of roots ∆( L , H k ) with zero orthogonal projections onto h is theroot system of a standard Levi subgroup L ⊂ G k with respect to the system of simple roots in ∆ .Using formula (1.2.1) and recalling that the roots γ , . . . γ l ′ form a linear basis of h ′ , i.e. for i = 1 , . . . , l ′ γ i haszero orthogonal projection onto h ′⊥ , we deduce that for i = 1 , . . . , l ′ γ i ∈ ∆( L , H k ), and hence s belongs to theWeyl group W ⊂ W of the root system ∆( L , H k ). Note that, as L is a standard Levi subgroup in G k , W is aparabolic subgroup in W with respect to the system of simple roots in ∆ . Since γ , . . . γ l ′ form a linear basis of h ′ , the linear span of roots from ∆( L , H k ) coincides with h ′ , and hence the element s is elliptic in W as s actswithout fixed points on h ′ . .5. THE STRICT TRANSVERSALITY CONDITION w be a minimal length representative in the conjugacy class of s in W with respect to the system of simpleroots in ∆( L , H k ) + = ∆ ∩ ∆( L , H k ). By Lemma 3.1.14 in [38] if w ∈ Ψ W ( O ) ∩ W is of minimal possible lengthwith respect to the system of simple reflections in W then it is also of minimal possible length with respect to thesystem of simple reflections in W , where in both cases the simple reflections are the reflections with respect to thesimple roots in ∆ . Note that w is elliptic in W as well.Let B be the Borel subgroup in G corresponding to − ∆ , P ⊃ B the parabolic subgroup of G correspondingto W . Thus L is the Levi factor of P .Denote by B = B ∩ L the Borel subgroup in L . One can always find a representative ˙ w ∈ L of w . Lemma 1.5.4.
Any conjugacy class in G O intersects B ˙ wB ⊂ B ˙ wB .Proof. By the definition the stratum G O consists of all conjugacy classes of minimal possible dimension whichintersect the Bruhat cell B ˙ wB . Denote by U the unipotent radical of P . Then by the definition of parabolicsubgroups one can always find a one parameter subgroup ρ : k ∗ → Z G k ( L ) such thatlim t → ρ ( t ) nρ ( t − ) = 1 (1.5.10)for any n ∈ U .Let γ ∈ G O be a conjugacy class which intersects B ˙ wB at point b ˙ wb ′ , b, b ′ ∈ B such that b ˙ wb ′ B ˙ wB .Since by definitions of B and U we have B = B U there are unique factorizations b = un , b ′ = u ′ n ′ , u, u ′ ∈ B , n, n ′ ∈ U . By (1.5.10) we havelim t → ρ ( t ) b ˙ wb ′ ρ ( t − ) = lim t → uρ ( t ) nρ ( t − ) ˙ wu ′ ρ ( t ) n ′ ρ ( t − ) = u ˙ wu ′ ∈ B ˙ wB , and hence the closure of γ contains a conjugacy class γ ′ which intersects B ˙ wB at some point of B ˙ wB ⊂ B ˙ wB .In particular, dim γ > dim γ ′ . This is impossible by the definition of G O , and hence γ intersects B ˙ wB at somepoint of B ˙ wB ⊂ B ˙ wB . Lemma 1.5.5.
Let G k be a connected semisimple algebraic group over an algebraically closed field k of charac-teristic good for G k . Let H k be a maximal torus of G k , W the Weyl group of the pair ( G k , H k ) , and s ∈ W anelliptic element. Denote by O s the conjugacy class of s in W . Then Φ W ( O s ) ⊂ N ( G k ) .Proof. The statement of this lemma is a consequence of the fact that s is elliptic. Indeed, it suffices to consider thecase when G k is simple.In case when G k is of type A n this is obvious since b N ( G k ) contains only unipotent classes. In fact in this case O s is the Coxeter class, and Φ W ( O s ) is the class of regular unipotent elements.If G k is of type B n , C n or D n , formula (1.5.8) implies that if Φ W ( O s ) corresponds to ( ν, ε ) ∈ T n ( e T n ) then ν has no odd parts and no even parts ν i with ε ( ν i ) = 0. According to the description given in the previous sectionthe map π G k ( e π G k ) is injective and its image consists of pairs ( ν, ε ) ∈ T n ( e T n ) such that ε ( k ) = 0 if ν ∗ k is odd andfor each even i such that ν ∗ i is even we have ν ∗ i − = ν ∗ i , i.e. i − ν . We deducethat Φ W ( O s ) is contained in the image of π G k ( e π G k ), i.e. Φ W ( O s ) ∈ N ( G k ) is a unipotent class in G k .In case when G k is of exceptional type this can be checked by examining the tables in Appendix 2.Now we show that in fact one can always take w = s . Lemma 1.5.6.
The element s is of minimal length representative in its conjugacy class in W with respect to thesystem of simple roots in ∆( L , H k ) + = ∆ ∩ ∆( L , H k ) .Proof. Let M be the semisimple part of L and O n = Φ W ( O w ) ⊂ M , where O w is the conjugacy class of w inthe Weyl group W = W ( L , H ). By the previous lemma applied to the group M and the elliptic element w ∈ W we have O n ∈ N ( M ).Therefore O n is the unipotent class of minimal possible dimension which intersects B ˙ wB . By Theorem 0.7in [73] the codimension of O n in M is equal to l ( w ), where l is the length function in W with respect to thesystem of simple roots in ∆( L , H k ) + = ∆ ∩ ∆( L , H k ),codim M O n = l ( w ) . (1.5.11)8 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
Now we show that s has minimal length in the Weyl group W with respect to the system of simple roots inthe set of positive roots ∆( L , H k ) + .Indeed, let Σ ′ s, k be the variety in M associated to s ∈ W in the beginning of this section, where we use∆( L , H k ) + as the system of positive roots in the definition of Σ ′ s, k .Formula (1.5.4) confirmed in Lemma 1.5.3 is applicable to the slice Σ ′ s, k and yieldscodim M O n = dim Σ ′ s, k . Formula (1.5.7) and the fact that s is elliptic in W imply thatdim Σ ′ s, k = l ( s ) , From the last two formulas we infer codim M O n = dim Σ ′ s, k = l ( s ) . The last formula and (1.5.11) yield l ( w ) = l ( s ), and hence s has minimal possible length in its conjugacy classin W with respect to the system of simple roots in ∆( L , H k ) + .Now we can assume that s = w . Lemma 1.5.7.
Any conjugacy class γ ∈ G O intersects Σ s, k at some point of ˙ sH , k N s, k .Proof. Let α ∈ ∆ ∩ h ′ . Then α ∈ ∆ i ∩ h ′ for some i >
0. Observe that by the definition of the subspaces h ′ k andby the choice of the elements h k , k = 0 , . . . , K ∆ i ∩ h ′ = { β ∈ ∆ : h j ( β ) = 0 , j > i, h i ( β ) = 0 , h ( β ) = 0 } = { β ∈ ∆ : h ′ j ( β ) = 0 , j > i − , h ′ i − ( β ) = 0 , } = ∆ ′ i − , and hence by (1.2.19) α ∈ (∆ i ) + ∩ h ′ if and only if h ′ i − ( α ) = h i ( α ) >
0, i.e. α ∈ (∆ i ) + ∩ h ′ if and only if α ∈ ∆ ∩ ∆ ′ i . Therefore if we denote B = B k ∩ L , where B k is the Borel subgroup corresponding to − ∆ s + , then B = B ∩ L = B .By Lemma 1.5.4 applied to the element s ∈ W any conjugacy class γ ∈ G O intersects B ˙ sB , and hence it alsointersects B ˙ sB ⊂ B ˙ sB as B = B . But by the definition of L s acts on the root system of the pair ( L , H k )without fixed points. Since B = B ∩ L and s fixes all the roots of the pair ( Z k H k , H k ) we have an inclusion B ⊂ H k N k , where N k is the unipotent radical of the parabolic subgroup P k associated to s in the beginning ofthis section. Hence B ˙ sB ⊂ N k ˙ sH k N k .Let H , k be the identity component of the centralizer of ˙ s in H k . Let h k be the Lie algebra of H k and h ′⊥ k theLie algebra of H , k . Let h ′ k be the s –invariant complementary subspace to h ′⊥ k in h k . Since h k is abelian h ′ k ⊂ h k is a Lie algebra. Let H ′ k ⊂ H k be the subgroup which corresponds to h ′ k in H . If h, h ′ ∈ H ′ k , h = e x , h ′ = e y , x, y ∈ h ′ k then h ′ ˙ sh ( h ′ ) − = ˙ se ( s − − x + y , and for any y one can find a unique x = − s − y such that h ′ ˙ sh ( h ′ ) − = ˙ s . Note also that H k normalizes N k .Therefore the factorization H k = H , k H ′ k implies that any element of N k ˙ sH k N k can be conjugated by an elementof H ′ k to an element from N k ˙ sH , k N k .Finally observe that N k ˙ sH , k N k ⊂ N k ˙ sZ k N k , and hence any conjugacy class γ ∈ G O intersects N k ˙ sH , k N k ⊂ N k ˙ sZ k N k . By Remark 1.3.3 γ also intersects Σ s, k . The proof of isomorphism (1.3.1) in Proposition 1.3.1 impliesthat the Z k –component of any element from N k ˙ sZ k N k is equal to the Z k –component in Σ s, k = ˙ sZ k N s, k of itsimage under the isomorphism N k ˙ sZ k N k ≃ N k × ˙ sZ k N s, k . Therefore any conjugacy class γ ∈ G O intersects Σ s, k at some point of ˙ sH , k N s, k . This completes the proof.By the previous lemma the statement of this theorem holds. .5. NORMAL ORDERINGS OF POSITIVE ROOT SYSTEMS For the purpose of quantization we shall need a certain normal ordering on the root system ∆ s + .An ordering of a set of positive roots ∆ + is called normal if for any three roots α, β, γ such that γ = α + β we have either α < γ < β or β < γ < α . Let α , . . . , α l be the simple roots in ∆ + , s , . . . , s l the correspondingsimple reflections. Let w be the element of W of maximal length with respect to the system s , . . . , s l of simplereflections. For any reduced decomposition w = s i . . . s i D of w the ordering β = α i , β = s i α i , . . . , β D = s i . . . s i D − α i D is a normal ordering in ∆ + , and there is a one–to–one correspondence between normal orderings of ∆ + and reduceddecompositions of w .From this fact and from properties of Coxeter groups it follows that any two normal orderings in ∆ + can bereduced to each other by the so–called elementary transpositions. The elementary transpositions for rank 2 rootsystems are inversions of the following normal orderings (or the inverse normal orderings): α, β A + A α, α + β, β A α, α + β, α + 2 β, β B α, α + β, α + 3 β, α + 2 β, α + 3 β, β G (1.6.1)where it is assumed that ( α, α ) ≥ ( β, β ). Moreover, any normal ordering in a rank 2 root system is one of orderings(1.6.1) or one of the inverse orderings.In general an elementary inversion of a normal ordering in a set of positive roots ∆ + is the inversion of an orderedsegment of form (1.6.1) (or of a segment with the inverse ordering) in the ordered set ∆ + , where α − β ∆.From now on we shall always assume that in sum (1.2.14) h is the linear subspace of h R fixed by the action of s and that the one–dimensional subspaces h i on which s acts by multiplication by − h in (1.2.14). According to this convention ∆ = { α ∈ ∆ : sα = α } is the set of roots fixed by the action of s .Suppose that the direct sum L rk =0 ,i k > h i k of the subspaces h i k on which s acts by multiplication by − h i on which s acts by multiplication by − h in sum (1.2.14), the roots from the union S rk =0 ,i k > ∆ i k must be orthogonal to all subspaces h i k , i k > s does not act by multiplication by − γ n +1 , . . . γ l ′ as s acts trivially on L rk =0 h i k .Pick up a root γ ∈ S rk =0 ,i k > ∆ i k . Then γ is orthogonal to the roots γ n +1 , . . . γ l ′ . Therefore, by the choice of γ , s = s s γ is an involution the dimension of the fixed point space of which is equal to the dimension of the fixedpoint space of the involution s plus one, and s = s γ s is another involution the dimension of the fixed pointspace of which is equal to the dimension of the fixed point space of the involution s minus one. We also have adecomposition s = s s .Now we can apply the above construction of the system of positive roots to the new decomposition of s . Iteratingthis procedure we shall eventually arrive at the situation when the direct sum L rk =0 ,i k > h i k of the subspaces h i k on which s acts by multiplication by − s = s s which satisfy this property. This implies s α = α ⇒ α ∈ ∆ . (1.6.2) Proposition 1.6.1.
Let s ∈ W be an element of the Weyl group W of the pair ( g , h ) , ∆ the root system of thepair ( g , h ) , ∆ s + a system of positive roots associated to s . Then the following statements are true.(i) The decomposition s = s s is reduced in the sense that l ( s ) = l ( s ) + l ( s ) , where l ( · ) is the length functionin W with respect to the system of simple roots in ∆ s + , and ∆ ss = ∆ ss ∪ s (∆ ss ) , ∆ ss − = ∆ ss ∪ s (∆ ss ) (disjointunions), where ∆ ss = { α ∈ ∆ s + : sα ∈ − ∆ s + } , ∆ ss − = { α ∈ ∆ s + : s − α ∈ − ∆ s + } , ∆ ss , = { α ∈ ∆ s + : s , α ∈ − ∆ s + } .Here s , s are the involutions in decomposition (1.2.1), s = s γ . . . s γ n , s = s γ n +1 . . . s γ l ′ satisfying (1.6.2), theroots in each of the sets γ , . . . , γ n and γ n +1 , . . . , γ l ′ are positive and mutually orthogonal.(ii) For any root α ∈ ∆ ss one has s α ∈ ∆ s + \ (cid:0) ∆ ss ∪ ∆ ss ∪ ∆ (cid:1) . CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES (iii) There is a normal ordering of the root system ∆ s + of the following form β , . . . , β t , β t +1 , . . . , β t + p − n , γ , β t + p − n +2 , . . . , β t + p − n + n , γ ,β t + p − n + n +2 . . . , β t + p − n + n , γ , . . . , γ n , β t + p +1 , . . . , β l ( s ) , . . . , (1.6.3) β , . . . , β q , γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . ,γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) , β q +2 m l ( s − ( l ′ − n )+1 , . . . , β l ( s ) ,β , . . . , β D , where { β , . . . , β t , β t +1 , . . . , β t + p − n , γ , β t + p − n +2 , . . . , β t + p − n + n , γ ,β t + p − n + n +2 . . . , β t + p − n + n , γ , . . . , γ n , β t + p +1 , . . . , β l ( s ) } = ∆ ss , { β t +1 , . . . , β t + p − n , γ , β t + p − n +2 , . . . , β t + p − n + n , γ ,β t + p − n + n +2 . . . , β t + p − n + n , γ , . . . , γ n } = { α ∈ ∆ s + | s α = − α } = ∆ − s , { β , . . . , β q , γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . ,γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) , β q +2 m l ( s − ( l ′ − n )+1 , . . . , β l ( s ) } = ∆ ss , { γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . ,γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) } = { α ∈ ∆ s + | s α = − α } = ∆ − s , { β , . . . , β D } = { α ∈ ∆ s + | s ( α ) = α } . (iv) The length of the ordered segment ∆ m + ⊂ ∆ in normal ordering (1.6.3), ∆ m + = γ , β t + p − n +2 , . . . , β t + p − n + n , γ , β t + p − n + n +2 . . . , β t + p − n + n ,γ , . . . , γ n , β t + p +1 , . . . , β l ( s ) , . . . , β , . . . , β q , (1.6.4) γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . , γ l ′ , is equal to D − (cid:18) l ( s ) − l ′ D (cid:19) , (1.6.5) where D is the number of roots in ∆ s + , and D is the number of positive roots fixed by the action of s .(v) For any two roots α, β ∈ ∆ m + such that α < β the sum α + β cannot be represented as a linear combination P tk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i t < β .(vi) The roots from the set ∆ ss form a minimal segment in ∆ s + of the form γ, . . . , β l ( s ) which contains ∆ ss , andthe roots from the set ∆ ss − form a minimal segment in ∆ s + of the form β , . . . , δ which contains ∆ ss .(vii) For any α ∈ (∆ i k ) + , i k > such that sα ∈ (∆ i k ) + one has sα > α , and if β, γ ∈ ∆ i j ∪ { } , j < k and sα + β, α + γ ∈ ∆ then sα + β, α + γ ∈ ∆ s + and sα + β > α + γ .In particular, for any α ∈ ∆ s + , α ∆ and any α ∈ ∆ such that sα ∈ ∆ s + one has sα > α and if sα + α ∈ ∆ then sα + α > α .(viii) If α, β ∈ ∆ i k ∩ ∆ s + , i k > , α < β and sβ ∈ ∆ s + then the orthogonal projection of sβ onto h i k is obtained by aclockwise rotation with a non–zero angle and by a rescaling with a positive coefficient from the orthogonal projectionof α onto h i k . .5. NORMAL ORDERINGS OF POSITIVE ROOT SYSTEMS Proof.
Firstly we describe the set (∆ i k ) + = ∆ i k ∩ ∆ s + for i k >
0. Suppose that the corresponding s –invariantsubspace h i k is a two–dimensional plane. The case when h i k is an invariant line on which s acts by reflection and s acts trivially can be treated in a similar way. The plane h i k is shown at Figure 3. h i k O O ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ v k ∆ i k ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ψ k ψ k i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ v k ∆ i k ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ ϕ k ϕ k ❉❉❉❉❉❉❉❉❉❉❉❉❉❉ s ∆ i k ✑✑✑✑✑✑✑✑✑✑✑✑ s ∆ i k Fig. 3The vector h i k is directed upwards at the picture. By (1.2.19) a root α ∈ ∆ i k belongs to the set (∆ i k ) + if andonly if h i k ( α ) >
0. Identifying h R and h ∗ R with the help of the bilinear form one can deduce that α ∈ ∆ i k is in ∆ s + if and only if its orthogonal, with respect to the bilinear form, projection onto h i k is contained in the upper–halfplane shown at Figure 3.The involutions s and s act in h i k as reflections with respect to the lines orthogonal to the vectors labeled by v k and v k , respectively, at Figure 3, the angle between v k and v k being equal to π − θ i k /
2. The nonzero projectionsof the roots from the set { γ , . . . γ n } ∩ ∆ i k onto the plane h i k have the same (or the opposite) direction as thevector v k , and the nonzero projections of the roots from the set { γ n +1 , . . . , γ l ′ } ∩ ∆ i k onto the plane h i k have thesame (or the opposite) direction as the vector v k .The element s acts on h i k by clockwise rotation with the angle θ i k = 2( ϕ k + ψ k ). Therefore the set ∆ ss ∩ ∆ i k consists of the roots the orthogonal projections of which onto h i k belong to the union of the sectors labeled s ∆ i k and ∆ i k at Figure 3.Note that by Lemma 1.2.2 each ∆ i k is the root system of a standard Levi subalgebra in g .Let i k > ( r ) i k the subset of roots in (∆ i k ) + orthogonal projections of which onto h i k are directed along aray r ⊂ h i k starting at the origin. We call ∆ ( r ) i k the family corresponding to the ray r . Below we shall only considerrays which correspond to sets of the form ∆ ( r ) i k . Lemma 1.6.2.
Suppose that i k > . Then each ∆ ( r ) i k is an additively closed set of roots.Let ∆ ( r ) i k and ∆ ( r ) i k be two families corresponding to rays r and r , and δ ∈ ∆ ( r ) i k , δ ∈ ∆ ( r ) i k two roots suchthat δ + δ = δ ∈ ∆ . Then δ ∈ ∆ ( r ) i k , where ∆ ( r ) i k is the family corresponding to a ray r such that r lies insideof the angle formed by r and r .Proof. All statements are simple consequences of the fact that the sum of the orthogonal projections of any tworoots onto h i k is equal to the orthogonal projection of the sum.In the first case the orthogonal projections of any two roots α, β from ∆ ( r ) i k onto h i k have the same directiontherefore the orthogonal projection of the sum α + β onto h i k has the same direction as the orthogonal projectionsof α and β , and hence α + β ∈ ∆ ( r ) i k .2 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
In the second case it suffices to observe that the sum of the orthogonal projections of δ and δ onto h i k is equalto the orthogonal projection of the sum, and the sum of the orthogonal projections of δ and δ onto h i k lies insideof the angle formed by r and r .Now we construct an auxiliary normal ordering on ∆ s + by induction starting from the set (∆ i ) + as follows.If i = 0 or h i is one–dimensional then we fix an arbitrary normal order on (∆ i ) + .If h i is two–dimensional then we choose a normal ordering in (∆ i ) + in the following way. First fix an initialarbitrary normal ordering on (∆ i ) + . Since by Lemma 1.6.2 each set ∆ ( r ) i is additively closed we obtain an inducedordering for ∆ ( r ) i which satisfies the defining property for the normal ordering.Now using these induced orderings on the sets ∆ ( r ) i we define an auxiliary normal ordering on (∆ i ) + suchthat on the sets ∆ ( r ) i it coincides with the induced normal ordering defined above, and if ∆ ( r ) i and ∆ ( r ) i are twofamilies corresponding to rays r and r such that r lies on the right from r in h i then for any α ∈ ∆ ( r ) i and β ∈ ∆ ( r ) i one has α < β . By Lemma 1.6.2 the two conditions imposed on the auxiliary normal ordering in (∆ i ) + are compatible and define it in a unique way for the given initial normal ordering on (∆ i ) + . Since s acts by aclockwise rotation on h i we have s (∆ ( r ) i ) = ∆ ( s ( r )) i for s ( r ) in the upper–half plane, and hence the new normalordering satisfies the condition that for any α ∈ (∆ i ) + such that sα ∈ (∆ i ) + one has sα > α .Now assume that an auxiliary normal ordering has already been constructed for the set ∆ i k − and define it forthe set ∆ i k .By Lemma 1.2.2 ∆ i k − ⊂ ∆ i k is the root system of a standard Levi subalgebra g i k − inside of the standardLevi subalgebra g i k of g with the root system ∆ i k . In particular, ∆ i k − is generated by some subset of simpleroots of the set of simple roots of (∆ i k ) + . Therefore there exists an initial normal ordering on (∆ i k ) + in which theroots from the set (∆ i k ) + \ (∆ i k − ) + = (∆ i k ) + form an initial segment and the remaining roots from (∆ i k − ) + areordered according to the previously defined auxiliary normal ordering. As in case of the induction base this initialnormal ordering gives rise to an induced ordering on each set ∆ ( r ) i k .Now using these induced orderings on the sets ∆ ( r ) i k we define an auxiliary normal ordering on (∆ i k ) + . Weimpose the following conditions on it. Firstly we require that the roots from the set (∆ i k ) + form an initial segmentand the remaining roots from (∆ i k − ) + are ordered according to the previously defined auxiliary normal ordering.Secondly, on the sets ∆ ( r ) i k the auxiliary normal ordering coincides with the induced normal ordering defined above,and if ∆ ( r ) i k and ∆ ( r ) i k are two families corresponding to rays r and r such that r lies on the right from r in h i k then for any α ∈ ∆ ( r ) i k and β ∈ ∆ ( r ) i k one has α < β . By Lemma 1.6.2 the conditions imposed on the auxiliarynormal ordering in (∆ i k ) + are compatible and define it in a unique way. Since s acts by a clockwise rotation on h i k we have s (∆ ( r ) i k ) = ∆ ( s ( r )) i k for s ( r ) in the upper–half plane, and hence the new normal ordering satisfies thecondition that for any α ∈ (∆ i k ) + such that sα ∈ (∆ i k ) + one has sα > α .Note also that the roots from ∆ i k − have zero orthogonal projections onto h i k . Therefore if α ∈ (∆ i k ) + , β, γ ∈ ∆ i k − are such that sα ∈ (∆ i k ) + , sα + β, α + γ ∈ ∆ then by (1.2.15) and (1.2.19) sα + β, α + γ ∈ ∆ s + and sα + β > α + γ as s (∆ ( r ) i k ) = ∆ ( s ( r )) i k for s ( r ) in the upper–half plane.These properties of the new normal ordering are summarized in the following lemma. Lemma 1.6.3.
Suppose that i k > . Then for any α ∈ (∆ i k ) + such that sα ∈ (∆ i k ) + one has sα > α and if β, γ ∈ ∆ i k − ∪ { } , sα + β, α + γ ∈ ∆ then sα + β, α + γ ∈ ∆ s + and sα + β > α + γ . Now we proceed by induction and obtain an auxiliary normal ordering on ∆ s + .Observe that, according to the definition of the auxiliary normal ordering of ∆ s + constructed above we have thefollowing properties of this normal ordering. Lemma 1.6.4.
Let i k > . Then for any α ∈ ∆ i k and β ∈ ∆ i k +1 we have α > β , and if ∆ ( r ) i k and ∆ ( r ) i k are twofamilies corresponding to rays r and r such that r lies on the right from r in h i k then for any α ∈ ∆ ( r ) i k and β ∈ ∆ ( r ) i k one has α < β . Moreover, the roots from the sets ∆ ( r ) i k form minimal segments, and the roots from theset (∆ ) + form a final minimal segment. For each of the involutions s and s we obviously have decompositions ∆ ss , = S Mk =0 ∆ , i k , ∆ ss = S Mk =0 ∆ si k ,where ∆ , i k = ∆ i k ∩ ∆ ss , , ∆ ss , = { α ∈ ∆ s + : s , α ∈ − ∆ s + } , ∆ si k = ∆ i k ∩ ∆ ss , ∆ si k = ∆ i k ∪ s ∆ i k . In the plane .5. NORMAL ORDERINGS OF POSITIVE ROOT SYSTEMS h i k , the elements from the sets ∆ , i k are projected onto the interiors of the sectors labeled by ∆ , i k and the elementsfrom the set ∆ si k are projected onto the interior of the union of the sectors labeled by ∆ i k and s ∆ i k . Thereforethe sets ∆ i k and ∆ i k have empty intersection and are the unions of the sets ∆ ( r ) i k with r belonging to the sector∆ , i k , and the sets ∆ si k have empty intersection and are the unions of the sets ∆ ( r ) i k with r belonging to the unionof the sectors labeled by ∆ i k and s ∆ i k .Let α ∈ ∆ ss . By Theorem C in [16] the roots γ , . . . , γ l ′ form a linear basis in the annihilator h ′ R ∗ of h withrespect to the pairing between h R and h ∗ R . Therefore s = s γ . . . s γ n fixes all roots from ∆ , and hence α ∆ .Also one obviously has α ∈ ∆ i k , where h i k is a two–dimensional plane, as by the assumption imposed before (1.6.2)there are no one–dimensional subspaces h i k on which s acts by multiplication by −
1. Thus in case if h i k is aninvariant line on which s acts by multiplication by − i k is empty and hence ∆ si k = ∆ i k . This set is theset ∆ ( r ) i k = (∆ i k ) + , where r is the positive semi-axis in h i k . From the observations made in the last two paragraphswe deduce that the sets ∆ ss and ∆ ss have always empty intersection. In particular, by the results of § s = s s is reduced in the sense that l ( s ) = l ( s ) + l ( s ), and ∆ ss = ∆ ss ∪ s (∆ ss ) (disjoint union).Similarly, ∆ ss − = ∆ ss ∪ s (∆ ss ) (disjoint union). This proves (i).Note also that for any root α fixed by the action of s one has α ∈ ∆ by (1.6.2). As we observed above for anyroot α ∈ ∆ ss one obviously has α ∈ ∆ i k , where h i k is a two–dimensional plane. Hence using Figure 3 we deducethat s α ∈ ∆ s + \ (cid:0) ∆ ss ∪ ∆ ss ∪ ∆ (cid:1) . This proves (ii). Lemma 1.6.5.
Assume that ∆ s + is equipped with an arbitrary normal ordering such that for some ray r ⊂ h i k theroots from the set ∆ ( r ) i k = { δ , . . . , δ a } form a minimal segment δ , . . . , δ a . Suppose also that for some natural p such that ≤ p < k , i p = 0 and some ray t ⊂ h i p the roots from the set ∆ ( t ) i p = { ξ , . . . , ξ b } form a minimal segment ξ , . . . , ξ b and that the segment δ , . . . , δ a , ξ , . . . , ξ b is also minimal. Then applying elementary transpositions onecan reduce the last segment to the form ξ i , . . . , ξ i b , δ j , . . . , δ j a .Proof. The proof is by induction. First consider the minimal segment δ , . . . , δ a , ξ .Since the orthogonal projection of the roots from the set ∆ i p onto h i k are equal to zero, for any α ∈ ∆ ( t ) i p and β ∈ ∆ ( r ) i k such that α + β ∈ ∆ we have α + β ∈ ∆ ( r ) i k . Assume now that α and β are contained in an ordered segmentof form (1.6.1) or in a segment with the inverse ordering. By the above observation this segment contains no otherroots from ∆ ( t ) i p , and α is the first or the last element in that segment. For the same reason the other roots in thatsegment must also belong to ∆ ( r ) i k . Therefore applying an elementary transposition, if necessarily, one can move α to the first position in that segment.Applying this procedure iteratively to the segment δ , . . . , δ a , ξ we can reduce it to the form ξ , δ k , . . . , δ k a .Now we can apply the same procedure to the segment δ k , . . . , δ k a , ξ to reduce the segment ξ , δ k , . . . , δ k a , ξ to the form ξ , ξ , δ l , . . . , δ l a .Iterating this procedure we obtain the statement of the lemma.Now observe that according to Lemma 1.6.4 the roots from each of the sets (∆ i k ) + form a minimal segment inthe auxiliary normal ordering of ∆ s + , and the roots from the sets ∆ ( r ) i k form minimal segments inside (∆ i k ) + . Aswe observed above the sets ∆ , i k are the unions of the sets ∆ ( r ) i k with r belonging to the sectors ∆ , i k and hence byLemma 1.6.4 the roots from the sets ∆ , i k form an initial and a final segment inside (∆ i k ) + .Therefore we can apply Lemmas 1.6.4 and 1.6.5 to move all roots from the segments ∆ i k , k = 0 , . . . , M to theleft and to move all roots from the segments ∆ i k , k = 0 , . . . , M to the right to positions preceding the final segmentformed by the roots from (∆ ) + .Now using similar arguments the roots from the sets s ∆ i k , k = 0 , . . . , M forming minimal segments by Lemma1.6.4 as well can be moved to the right to positions preceding the final segment formed by the roots from the set∆ ss ∪ (∆ ) + . After this modification the roots from the set ∆ ss = ∆ ss ∩ s (∆ ss ) will form a minimal segment in∆ s + of the form γ, . . . , β l ( s ) which contains ∆ ss .Similarly, the roots from the sets s ∆ i k , k = 0 , . . . , M forming minimal segments by Lemma 1.6.4 can be movedto the left to positions after the initial segment formed by the roots from the set ∆ ss . After this modification theroots from the set ∆ ss − = ∆ ss ∩ s (∆ ss ) will form a minimal segment in ∆ s + of the form β , . . . , δ which contains∆ ss . This proves (vi).Note that according to the algorithm given in Lemma 1.6.5 for each fixed k the order of the minimal segmentsformed by the roots from the sets ∆ ( r ) i k is preserved after applying that lemma. Therefore the new normal ordering4 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES obtained this way still satisfies the second property mentioned in Lemma 1.6.4, i.e. if ∆ ( r ) i k and ∆ ( r ) i k are twofamilies corresponding to rays r and r such that r lies on the right from r in h i k then for any α ∈ ∆ ( r ) i k and β ∈ ∆ ( r ) i k one has α < β .Now we can apply elementary transpositions to bring the initial segment formed by the roots from ∆ ss = { β , . . . , β l ( s ) } and the segment formed by the roots from ∆ ss = { β , . . . , β l ( s ) } and preceding the final segment(∆ ) + to the form described in (1.6.3).Recall that by Theorem A in [86] every involution w in the Weyl group W is the longest element of the Weylgroup of a Levi subalgebra in g , and w acts by multiplication by − h w ⊂ h of thesemisimple part m w of that Levi subalgebra. By Lemma 5 in [16] the involution w can also be expressed as aproduct of dim h w reflections from the Weyl group of the pair ( m w , h w ), with respect to mutually orthogonalroots. In case of the involution s , s = s γ . . . s γ n is such an expression, and the roots γ , . . . , γ n span the Cartansubalgebra h s .Since m s is the semisimple part of a Levi subalgebra, using elementary transpositions one can reduce the normalordering of the segment β , . . . , β l ( s ) to the form β , . . . , β t , β t +1 , . . . , β t + p , β t + p +1 , . . . , β l ( s ) , (1.6.6)where β t +1 , . . . , β t + p is a normal ordering of the system ∆ + ( m s , h s ) = ∆( m s , h s ) ∩ ∆ s + of positive roots in theroot system ∆( m s , h s ) of the pair ( m s , h s ). Now applying elementary transpositions we can reduce the ordering β t +1 , . . . , β t + p to the form compatible with the decomposition s = s γ . . . s γ n (see Appendix 1).Applying similar arguments to the involution s and using the normal ordering of the system of positive roots∆ + ( m s , h s ) = ∆( m s , h s ) ∩ ∆ s + in the root system ∆( m s , h s ) of the pair ( m s , h s ) inverse to that compatiblewith the decomposition s = s γ n +1 . . . s γ l ′ we finally obtain the following normal ordering of the set ∆ s + β , . . . , β t , β t +1 , . . . , β t + p − n , γ , β t + p − n +2 , . . . , β t + p − n + n , γ ,β t + p − n + n +2 . . . , β t + p − n + n , γ , . . . , γ n , β t + p +1 , . . . , β l ( s ) , . . . , (1.6.7) β , . . . , β q , γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . ,γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) , β q +2 m l ( s − ( l ′ − n )+1 , . . . , β l ( s ) ,β , . . . , β D , where γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . ,γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) is the normal ordering of the system of positive roots ∆ + ( m s , h s ) inverse to that compatible with the decomposition s = s γ n +1 . . . s γ l ′ . We caim that normal ordering (1.6.7) has all the properties listed in the statement of thisproposition, i.e. it is the required ordering (1.6.3). This establishes (iii).The elementary transpositions used above to bring the segments ∆ ss and ∆ ss to the form required in (1.6.3)do not affect the positions of the roots which do not belong to ∆ ss and ∆ ss . We claim that for α ∈ ∆ ( r ) i k , i k > sα > α if sα ∈ ∆ s + .Indeed, if α ∈ (∆ i k ) + , α ∆ i k , sα ∆ i k , sα ∈ (∆ i k ) + this follows from the second property mentioned inLemma 1.6.4.If α ∈ ∆ i k and sα ∈ ∆ s + then sα ∆ ss as s ( s ∆ ss ) = s (∆ ss ) ⊂ ∆ s + since the decomposition s = s s isreduced. Therefore sα > α as the roots from the set ∆ ss form an initial segment in the normal ordering of ∆ s + .If α ∈ (∆ i k ) + , α ∆ i k , sα ∈ ∆ i k then α ∆ ss and sα ∈ ∆ ss as ∆ i k ⊂ ∆ ss . Therefore sα > α as the roots fromthe set ∆ ss ∪ (∆ ) + form a final segment in the normal ordering of ∆ s + and α does not belong to that segment.Finally if α ∈ ∆ i k then sα ∈ − ∆ s + .Moreover, similar arguments together with the fact that all roots from ∆ i k − have zero orthogonal projectionsonto h i k show that the new normal ordering still satisfies the property stated in Lemma 1.6.3. Note that forone–dimensional h i k this property is void.The fact that for any α ∈ ∆ s + , α ∆ and any α ∈ ∆ such that sα ∈ ∆ s + one has sα > α and if sα + α ∈ ∆then sα + α > α is a particular case of the property stated in Lemma 1.6.3 because ∆ s + is the disjoint union ofthe sets (∆ i k ) + . This proves (vii). .5. NORMAL ORDERINGS OF POSITIVE ROOT SYSTEMS α, β ∈ ∆ i k ∩ ∆ s + , α < β and sβ ∈ ∆ s + then the orthogonal projection of sβ onto h i k is obtained by a clockwise rotation with a non–zero angle and by a rescaling with a positive coefficient from theorthogonal projection of α onto h i k .Indeed, observe that the elementary transpositions which we used above to bring the segments formed by theroots from the sets ∆ ss and ∆ ss to the form required in normal ordering (1.6.3) do not affect the positions of otherroots and after this rearrangement the orthogonal projections of roots from ∆ , i k onto h i k still belong to the sectorslabeled ∆ , i k at Figure 3.Therefore by Lemma 1.6.4 if α, β ∈ ∆ i k ∩ ∆ s + , β ∈ ∆ i k and α < β then the orthogonal projection of α onto h i k belongs to the sector labeled ∆ i k at Figure 3. On the other hand since sβ ∈ ∆ s + the orthogonal projection of sβ onto h i k belongs to the upper half plane and does not belong to the sector labeled ∆ i k at Figure 3 as s actson h i k by clockwise rotation by the angle θ i k = 2( ϕ k + ψ k ) > ϕ k . Thus the orthogonal projection of sβ onto h i k is obtained by a clockwise rotation with a non–zero angle and by a rescaling with a positive coefficient from theorthogonal projection of α onto h i k .The case when β ∈ (∆ i k ) + , β ∆ i k , sβ ∈ (∆ i k ) + is treated in a similar way with the help of Lemma 1.6.4.This proves (viii).Next we claim that t = l ( s ) − ( t + p ), i.e. there are equal numbers of roots on the left and on the right fromthe segment β t +1 , . . . , β t + p in the segment β , . . . , β t , β t +1 , . . . , β t + p , β t + p +1 , . . . , β l ( s ) , and t = l ( s ) − p . (1.6.8)Recall that by formula (3.5) in [115], given a reduced decomposition w = s i . . . s i m of a Weyl group element w ,one can represent w as a product of reflections with respect to the roots from the set∆ w − = { β = α i , β = s i α i , . . . , β m = s i . . . s i m − α i m } ,w = s i . . . s i m = s β m . . . s β (1.6.9)Note that β = α i , β = s i α i , . . . , β m = s i . . . s i m − α i m is the initial segment of a normal ordering of the corresponding system of positive roots.Applying this observation to the segment of the normal ordering (1.6.6) consisting of elements from the set ∆ ss one can represent ( s ) − = s as follows s = s β l ( s . . . s β t + p +1 s β t + p . . . s β t +1 s β t . . . s β , (1.6.10)where if s = s i . . . s i l ( s is the corresponding reduced decomposition of s then β = α i , β = s i α i , . . . , β l ( s ) = s i . . . s i l ( s − α i l ( s . (1.6.11)Since β t +1 , . . . , β t + p is a normal ordering of the system of positive roots of the pair ( m s , h s ) and s is thelongest element in the Weyl group of the pair ( m s , h s ) we also have s = s β t + p . . . s β t +1 , and hence by (1.6.10) s = s β l ( s . . . s β t + p +1 s s β t . . . s β . (1.6.12)From the last formula we deduce that s β . . . s β t = ( s β t . . . s β ) − ( s ) − s β l ( s . . . s β t + p +1 s s β t . . . s β . (1.6.13)Now formula (1.6.9) implies that s β . . . s β t = s i t . . . s i , and relations (1.6.11) combined with (1.6.9) yield u ( s ) − s β l ( s . . . s β t + p +1 s u − = s u ( s ) − ( β l ( s ) . . . s u ( s ) − ( β t + p +1 ) == s s it + p +1 ...s il ( s − α il ( s . . . s i t + p +1 = s i t + p +1 . . . s i l ( s , CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES where u = s β . . . s β t = s i t . . . s i . Therefore from formula (1.6.13) we deduce that s i t . . . s i = s i t + p +1 . . . s i l ( s . (1.6.14)Since the decompositions in both sides of (1.6.14) are parts of reduced decompositions they are reduced as well,and we have t = l ( s ) − ( t + p ). This is equivalent to formula (1.6.8).Using a similar formula for the involution s and recalling the definition of the orderings of positive roots of thepairs ( m s , h s ), ( m s , h s ) compatible with decompositions s = s γ . . . s γ n and s = s γ n +1 . . . s γ l ′ (see Appendix 1)we deduce that the number of roots in the segment ∆ m + of normal ordering (1.6.7),∆ m + = γ , β t + p − n +2 , . . . , β t + p − n + n , γ , β t + p − n + n +2 . . . , β t + p − n + n ,γ , . . . , γ n , β t + p +1 , . . . , β l ( s ) , . . . , β , . . . , β q ,γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . , γ l ′ is equal to D − ( l ( s ) − l ′ + D ), where l ( s ) = l ( s ) + l ( s ) is the length of s and D is the number of positive rootsfixed by the action of s . This establishes (iv).Now let α, β ∈ ∆ m + , be any two roots such that α < β . We shall show that the sum α + β cannot be representedas a linear combination P qk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i k < β .Suppose that such a decomposition exists, α + β = P qk =1 c k γ i k . Obviously at least one of the roots α, β mustbelong to the set ∆ + ( m s , h s ) ∩ ∆ m + or to the set ∆ + ( m s , h s ) ∩ ∆ m + for otherwise the set of roots γ i k such that α < γ i k < β is empty.Suppose that α ∈ ∆ + ( m s , h s ) ∩ ∆ m + . The other cases are considered in a similar way.If β ∆ + ( m s , h s ) ∩ ∆ m + then α + β = P qk =1 c k γ i k , and γ i k ≤ γ n . In particular, since α ∈ h s and γ i k ∈ h s if γ i k ≤ γ n , we have β = P qk =1 c k γ i k − α ∈ h s . This is impossible by the definition of the ordering of the set∆ + ( m s , h s ) compatible with the decomposition s = s γ . . . s γ n .If β ∈ ∆ + ( m s , h s ) ∩ ∆ m + then α + β = P qk =1 c k γ i k = P i k ≤ n c k γ i k + P i k >n c k γ i k . This implies α − X i k ≤ n c k γ i k = X i k >n c k γ i k − β. The l.h.s. of the last formula is an element of h s and the r.h.s. is an element h s . Since h ′ = h s + h s is a directvector space decomposition we infer that α = X i k ≤ n,α<γ ik c k γ i k and β = X i k >n,γ ik <β c k γ i k . This is impossible by the definition of the orderings of the sets ∆ + ( m s , h s ) and ∆ + ( m s , h s ) compatible with thedecompositions s = s γ . . . s γ n and s = s γ n +1 . . . s γ l ′ , respectively.Therefore the sum α + β , α < β , α, β ∈ ∆ m + cannot be represented as a linear combination P qk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i k < β . This confirms (v) and completes the proof of the proposition.We shall also need another system of positive roots associated to (the conjugacy class of) the Weyl groupelement s . In order to define it we need to recall the definition of a circular normal ordering of the root system ∆.Let β , β , . . . , β D be a normal ordering of a positive root system ∆ + . Then one can introduce the correspondingcircular normal ordering of the root system ∆ where the roots in ∆ are located on a circle in the following way .5. NORMAL ORDERINGS OF POSITIVE ROOT SYSTEMS β β r r r r r β D − β − β rrrrr - β D ◗◗s◗◗❦ Fig. 4Let α, β ∈ ∆. One says that the segment [ α, β ] of the circle is minimal if it does not contain the opposite roots − α and − β and the root β follows after α on the circle above, the circle being oriented clockwise. In that case onealso says that α < β in the sense of the circular normal ordering, α < β ⇔ the segment [ α, β ] of the circle is minimal . (1.6.15)Later we shall need the following property of minimal segments which is a direct consequence of Proposition3.3 in [59]. Lemma 1.6.6.
Let [ α, β ] be a minimal segment in a circular normal ordering of a root system ∆ . Then if α + β is a root we have α < α + β < β. Note that any segment in a circular normal ordering of ∆ of length equal to the number of positive roots is asystem of positive roots.Now consider the circular normal ordering of ∆ corresponding to the system of positive roots ∆ s + and to itsnormal ordering introduced in Proposition 1.6.1. The segment which consists of the roots α satisfying γ ≤ α < − γ is a system of positive roots in ∆ as its length is equal to the number of positive roots and it is closed under additionof roots by Lemma 1.6.6.The system of positive roots ∆ + introduced this way and equipped with the normal ordering induced by thecircular normal ordering is called the normally ordered system of positive roots associated to the (conjugacy classof) the Weyl group element s ∈ W .We have the following property of the length of s with respect to the sets of simple roots in ∆ s + . Proposition 1.6.7.
For all systems of positive roots ∆ s + the lengths l ( s ) of s with respect to the sets of simpleroots in ∆ s + are the same, and they are equal to the length of s with respect to the set of simple roots in any systemof positive roots ∆ + associated to s .Proof. The first statement is a consequence of the definition of ∆ s + .To prove the second assertion we recall that a root α ∈ ∆ i k belongs to the set (∆ i k ) + = (∆ i k ) + ∩ ∆ s + if andonly if h i k ( α ) >
0. Identifying h R and h ∗ R with the help of the bilinear form one can deduce that α ∈ ∆ i k is in ∆ s + if and only if its orthogonal projection onto h i k is contained in the upper–half plane shown at Figure 3.According to the definition of the set ∆ + ,∆ i k ∩ ∆ + = ((∆ i k ) + \ { α ∈ (∆ i k ) + ∩ ∆ ss : α < γ } ) ∪ {− α : α ∈ (∆ i k ) + ∩ ∆ ss , α < γ } , i.e ∆ i k ∩ ∆ + is obtained from (∆ i k ) + by removing some roots the orthogonal projections of which onto h i k belongto the sector labeled ∆ i k at Figure 3 and by adding the the opposite negative roots the orthogonal projections ofwhich onto h i k belong to the sector labeled s ∆ i k at Figure 3.8 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES
The element s acts on h i k by clockwise rotation with the angle θ i k = 2( ϕ k + ψ k ). Therefore the set ∆ ss ∩ ∆ i k consists of the roots the orthogonal projections of which onto h i k belong to the union of the sectors labeled s ∆ i k and ∆ i k at Figure 3. Together with the description of the set ∆ i k ∩ ∆ + given above it implies that the number ofroots in the set ∆ s ∩ ∆ i k = { α ∈ ∆ i k ∩ ∆ + : sα ∈ ∆ − } , where ∆ s = { α ∈ ∆ + : sα ∈ ∆ − } , is equal to the numberof roots in the set ∆ ss ∩ ∆ i k . From this observation we deduce that the length l ( s ) of s with respect to the systemof simple roots in ∆ s + is the same as the length of s with respect to the system of simple roots in ∆ + , as both ofthem are equal to the cardinality to the set S Mk =0 ∆ ss ∩ ∆ i k (disjoint union) which is the same as the cardinality ofthe set S Mk =0 ∆ s ∩ ∆ i k (disjoint union).The relative positions of the systems of positive rots ∆ s + , ∆ + and of the minimal segments introduced inProposition 1.6.1 are shown at the following picture where all the segments are placed on a circle according to thecircular normal ordering of roots corresponding to normal ordering (1.6.3) of ∆ s + .∆ s + ∆ + ∆ ss ∆ − s − ∆ ss − ∆ − s ∆ ss ∆ − s − ∆ ss − ∆ − s (∆ ) + − (∆ ) + ∆ m + γ γ l ′ γ n γ n + − γ Fig. 5The reader may find this picture useful in combination with Lemma 1.6.6 when adding roots or commutingroots vectors. This picture can be also useful for deriving some formulas containing q-commutators of quantumroot vectors as explained in the next chapter.
A uniform classification of conjugacy classes of Weyl group elements from which one can obtain presentation (1.2.1)was suggested in [16]. .7. BIBLIOGRAPHIC COMMENTS s + associated to (conjugacy classes of) Weyl group elements wassuggested in [98]. It is based on a deep generalization of the results by Coxeter and Steinberg on the properties ofthe Coxeter elements. In our notation this corresponds to the case when γ , . . . , γ l ′ is a set of simple roots in ∆, sothat according to (1.2.1) s is a product of simple reflections, i.e. a Coxeter element. In this case there is a uniqueplane in h R , called a Coxeter plane, on which s acts by rotation by the angle 2 π/h , where h is the Coxeter numberof g . This plane was introduced by Coxeter in book [19], and the pictures of root systems of Lie algebras of highranks which one can find in many textbooks are obtained using orthogonal projections of roots onto these planes.The key observation is that all these projections are non–zero. Coxeter originally applied the above mentionedprocedure to construct regular polytopes.Later in paper [109] Steinberg proved interesting properties of Coxeter elements using the properties of theaction of Coxeter elements on Coxeter planes.The construction of the spectral decomposition for Weyl group elements in Proposition 1.2.1 suggested in [103]is a generalization of similar results on the properties of the Coxeter plane which can be found in [15], Section 10.4.The slices Σ s introduced in [98] are generalizations of the Steinberg cross-sections to the set of conjugacy classesof regular elements in G suggested in [110]. Σ s reduces to a Steinberg cross-section when γ , . . . , γ l ′ is a set ofsimple roots, i.e. when s is a Coxeter element. In this case isomorphism (1.3.1) is stated in [110] without proof.The first proof of this result appeared in [92]. Although that proof is not applicable for the root system of type E .The proof of isomorphism (1.3.1) given in Proposition 1.3.1 is a refined version of the proof of this result presentedin [98].Another construction of the slices Σ s in the case when s are elliptic can be found in [46].The closedness of the varieties N ZsN was justified in [101], Proposition 6.2.In book [106] Slodowy proved Brieskorn’s conjecture on the realization of simple singularities using the adjointquotient of complex semisimple Lie algebras. Although a significant part of Slodowy’s book is devoted to the studyof the conjugation quotient for semisimple algebraic groups and to constructing some its resolutions, he ended upwith a Lie algebra version of the construction of simple singularities and introduced transversal slices for the adjointaction for this purpose. These slices are called now the Slodowy slices. The slices Σ s can be regarded as algebraicgroup analogues of the Slodowy slices.The Lusztig partition was introduced in [70]. Its definition is related to the study of the properties of intersectionsof conjugacy classes in G with Bruhat cells established in [72, 73] where the map Φ W from the set of Weyl groupconjugacy classes to the set of unipotent classes and its one sided inverse Ψ W , which we use in Section 1.4, aredefined using these properties. These properties are also related to the generalized Springer correspondence. Weonly briefly discussed the relevant results in this book.The study of intersections of conjugacy classes in G with Bruhat cells was initiated in [110]. Some resultson these intersections were obtained in [30], and another map from nilpotent orbits in a complex semisimple Liealgebra to conjugacy classes in the Weyl group was defined in [56]. In [72] it is mentioned that this map is likelyto coincide with the map Ψ W introduced in [72].The main result of Theorem 1.5.2 on the dimensions of the slices Σ s is an experimental observation made in[103], Theorem 5.2. Other results of Section 1.5 can also be found in [103].Note that the the slices Σ s listed in Appendix 2 are slightly different from those from Appendix B to [103].The corresponding slices in both sets have the same dimensions. But in this book the roots γ , . . . , γ l ′ in the tablesin Appendix 2 are chosen in such a way that the corresponding root systems ∆ s + satisfy condition (1.6.2). Thealgorithm for constructing the slices Σ s listed in the tables in Appendix B to [103] was modified to fulfill thiscondition. The description of the original algorithm can be found in [103].The ordering of the s –invariant planes in h R according to the angles of rotations by which s acts in the planesas in Theorem 1.5.2 was used in [47] to prove properties of minimal length elements in finite Coxeter groups.Normal orderings of positive root systems of the form ∆ s + described in Proposition 1.6.1 were firstly introducedin [99] where one can also find the construction of normal orderings of positive root systems compatible with Weylgroup involutions from Appendix 1. Later the original definition was refined in [104]. Proposition 1.6.1 is a modifiedversion of Proposition 5.1 in [99] and Proposition 2.2 in [104].Circular orderings of root systems were defined in [58] to describe commutation relations between quantumgroup analogues of root vectors. In [101] this construction was used to modify the positive root systems ∆ s + inorder to construct positive root systems associated to (conjugacy classes of) Weyl group elements which appear inthe end of Section 1.6.0 CHAPTER 1. ALGEBRAIC GROUP ANALOGUES OF SLODOWY SLICES hapter 2
Quantum groups
In this chapter we recall some definitions and results on quantum groups required for the study of q-W–algebras.Besides the standard definitions and results related to quantum groups we shall need some rather non–standardrealizations of the Drinfeld–Jimbo quantum group in terms of which q-W–algebras are defined. These realizationsare related to the definition of the algebraic group analogues of the Slodowy slices in the previous section. We shallconsider the Drinfeld–Jimbo quantum group U h ( g ) defined over the ring of formal power series C [[ h ]], where h isan indeterminate, and some its specializations defined over smaller rings. In this section we remind the definition of the standard Drinfeld-Jimbo quantum group U h ( g ).Let V be a C [[ h ]]–module equipped with the h –adic topology. This topology is characterized by requiring that { h n V | n ≥ } is a base of the neighborhoods of 0 in V , and that translations in V are continuous. In this book all C [[ h ]]–modules are supposed to be complete with respect to this topology.A topological Hopf algebra over C [[ h ]] is a complete C [[ h ]]–module equipped with a structure of C [[ h ]]–Hopfalgebra, the algebraic tensor products entering the axioms of the Hopf algebra are replaced by their completionsin the h –adic topology.The standard quantum group U h ( g ) associated to a complex finite-dimensional semisimple Lie algebra g is atopological Hopf algebra over C [[ h ]] topologically generated by elements H i , X + i , X − i , i = 1 , . . . , l , subject to thefollowing defining relations:[ H i , H j ] = 0 , [ H i , X ± j ] = ± a ij X ± j , X + i X − j − X − j X + i = δ i,j K i − K − i q i − q − i , P − a ij r =0 ( − r (cid:20) − a ij r (cid:21) q i ( X ± i ) − a ij − r X ± j ( X ± i ) r = 0 , i = j, where K i = e d i hH i , e h = q, q i = q d i = e d i h , (cid:20) mn (cid:21) q = [ m ] q ![ n ] q ![ n − m ] q ! , [ n ] q ! = [ n ] q . . . [1] q , [ n ] q = q n − q − n q − q − , with comultiplication defined by∆ h ( H i ) = H i ⊗ ⊗ H i , ∆ h ( X + i ) = X + i ⊗ K − i + 1 ⊗ X + i , ∆ h ( X − i ) = X − i ⊗ K i ⊗ X − i , antipode defined by S h ( H i ) = − H i , S h ( X + i ) = − X + i K i , S h ( X − i ) = − K − i X − i , and counit defined by ε h ( H i ) = ε h ( X ± i ) = 0 . CHAPTER 2. QUANTUM GROUPS
We shall also use the weight–type generators Y i = l X j =1 d i ( a − ) ij H j . Let L ± i = e ± hY i . These elements commute with the quantum simple root vectors X ± i as follows: L i X ± j L − i = q ± δ ij i X ± j . (2.1.1)We also obviously have L i L j = L j L i . (2.1.2)The Hopf algebra U h ( g ) is a quantization of the standard bialgebra structure on g in the sense that U h ( g ) /hU h ( g ) = U ( g ), ∆ h = ∆ (mod h ), where ∆ is the standard comultiplication on U ( g ), and∆ h − ∆ opph h (mod h ) = − δ. Here δ : g → g ⊗ g is the standard cocycle on g , and ∆ opph = σ ∆ h , σ is the permutation in U h ( g ) ⊗ , σ ( x ⊗ y ) = y ⊗ x .Recall that δ ( x ) = (ad x ⊗ ⊗ ad x )2 r ± , r ± ∈ g ⊗ g ,r ± = ± l X i =1 Y i ⊗ H i ± X β ∈ ∆ + ( X β , X − β ) − X ± β ⊗ X ∓ β . (2.1.3)Here X ± β ∈ g ± β are non–zero root vectors of g . The element r ± ∈ g ⊗ g is called a classical r–matrix. One can define a quantum group analogue of the braid group action on g . Let m ij , i = j be equal to 2 , , , a ij a ji is equal to 0 , , ,
3, respectively. The braid group B g associated to g has generators T i , i = 1 , . . . , l , anddefining relations T i T j T i T j . . . = T j T i T j T i . . . for all i = j , where there are m ij T ’s on each side of the equation.Recall that if X ± α i are non–zero simple root vectors of g then one can introduce an action of the braid group B g by algebra automorphisms of g defined on the standard generators as follows: T i ( X ± α i ) = − X ∓ α i , T i ( H j ) = H j − a ji H i ,T i ( X α j ) = 1( − a ij )! ad − a ij X αi X α j , i = j, (2.2.1) T i ( X − α j ) = ( − a ij ( − a ij )! ad − a ij X − αi X − α j , i = j. Similarly, B g acts by algebra automorphisms of U h ( g ) as follows: T i ( X + i ) = − X − i e hd i H i , T i ( X − i ) = − e − hd i H i X + i , T i ( H j ) = H j − a ji H i ,T i ( X + j ) = − a ij X r =0 ( − r − a ij q − ri ( X + i ) ( − a ij − r ) X + j ( X + i ) ( r ) , i = j, (2.2.2) T i ( X − j ) = − a ij X r =0 ( − r − a ij q ri ( X − i ) ( r ) X − j ( X − i ) ( − a ij − r ) , i = j, .2. THE BRAID GROUP ACTION X + i ) ( r ) = ( X + i ) r [ r ] q i ! , ( X − i ) ( r ) = ( X − i ) r [ r ] q i ! , r ≥ , i = 1 , . . . , l. Recall that action (2.2.1) of the generators T i is induced by the adjoint action of certain representatives of theWeyl group elements s i in G . Similarly, action (2.2.2) is induced by conjugation by certain elements of a completionof U h ( g ).More precisely, define a q-exponential by exp ′ q ( x ) = ∞ X k =0 q k ( k − x k [ k ] q ! . Then the automorphism T i in (2.2.2) is given by conjugation by the invertible element (see [88]) T i = exp ′ q − i ( − q − i X − i K i ) exp ′ q − i ( X + i ) exp ′ q − i ( − q i X − i K − i ) q Hi ( Hi +1)2 i = (2.2.3)= exp ′ q − i ( q − i X + i K − i ) exp ′ q − i ( − X − i ) exp ′ q − i ( q i X + i K i ) q Hi ( Hi +1)2 i which belongs to the completion C h [ G ] ∗ of U h ( g ), where C h [ G ] is the restricted dual of U h ( g ), i.e. the Hopf algebragenerated by the matrix elements of finite rank representations of U h ( g ). The inverse of T i in (2.2.3) can be foundusing the identity exp ′ q ( x ) exp ′ q − ( − x ) = 1which implies T − i = q − Hi ( Hi +1)2 i exp ′ q i ( q i X − i K − i ) exp ′ q i ( − X + i ) exp ′ q i ( q − i X − i K i ) = (2.2.4)= q − Hi ( Hi +1)2 i exp ′ q i ( − q i X + i K i ) exp ′ q i ( X − i ) exp ′ q i ( − q − i X + i K − i ) . From formula (2.2.4) we obtain the following relations in C h [ G ] ∗ exp ′ q i ( − X + i ) = exp ′ q − i ( − q i X − i K − i ) q Hi ( Hi +1)2 i T − i exp ′ q − i ( − q − i X − i K i ) = (2.2.5)= exp ′ q − i ( − q i X − i K − i ) q Hi ( Hi +1)2 i exp ′ q − i ( q − i X + i ) T − i . The comultiplication in U h ( g ) induces a comultiplication in C h [ G ] ∗ with respect to which we have∆ h ( T i ) = θ i T i ⊗ T i = T i ⊗ T i θ i , (2.2.6) θ i = exp q i [(1 − q − i ) X + i ⊗ X − i ] , θ i = exp q i [(1 − q − i ) K − i X − i ⊗ X + i K i ] , ∆ h ( T − i ) = θ i − T i ⊗ T i = T i ⊗ T i θ − i , (2.2.7) θ − i = exp q − i [(1 − q i ) X + i ⊗ X − i ] , θ i − = exp q − i [(1 − q i ) K − i X − i ⊗ X + i K i ] . For a reduced decomposition w = s i . . . s i k , T w = T i . . . T i k only depends on w and (2.2.6) implies∆ h ( T w ) = k Y p =1 θ β p T w ⊗ T w = T w ⊗ T w k Y p =1 θ β ′ p , (2.2.8)where for p = 1 , . . . , kX ± β p = T i . . . T i p − X ± i p , X ± β ′ p = T − i k . . . T − i p +1 X ± i p , K β ′ p = T − i k . . . T − i p +1 K i p ,θ β p = exp q βp [(1 − q − β p ) X + β p ⊗ X − β p ] , θ β ′ p = exp q β ′ p [(1 − q − β ′ p ) K − β ′ p X − β ′ p ⊗ X + β ′ p K β ′ p ] . Similarly, for a reduced decomposition w = s i . . . s i k , T w = T − i . . . T − i k only depends on w and (2.2.7) yields∆ h ( T w ) = k Y p =1 θ ′ β p T w ⊗ T w = T w ⊗ T w k Y p =1 θ ′ β ′ p , (2.2.9)4 CHAPTER 2. QUANTUM GROUPS where for p = 1 , . . . , kX ± β p = T − i . . . T − i p − X ± i p , X ± β ′ p ′ = T i k . . . T i p +1 X ± i p , K β p = T − i . . . T − i p − K i p ,θ ′ β ′ p = exp q − β ′ p [(1 − q β ′ p ) X + β ′ p ′ ⊗ X − β ′ p ′ ] , θ ′ β p = exp q − βp [(1 − q β p ) K − β p X − β p ⊗ X + β p K β p ] . If wα i = s i . . . s i k α i = α j for some i and j then T w X ± i = X ± j , T w X ± i = X ± j . (2.2.10) In this section we recall the construction of analogues of root vectors for U h ( g ) in terms of the braid group actionon U h ( g ). Recall that for any reduced decomposition w = s i . . . s i D of the longest element w of the Weyl group W of g the ordering β = α i , β = s i α i , . . . , β D = s i . . . s i D − α i D is a normal ordering in ∆ + , and there is a one–to–one correspondence between normal orderings of ∆ + and reduceddecompositions of w .Fix a reduced decomposition w = s i . . . s i D of w and define the corresponding quantum root vectors in U h ( g )by X ± β k = T i . . . T i k − X ± i k . (2.3.1)Note that one can construct root vectors in the Lie algebra g in a similar way. Namely, the root vectors X ± β k ∈ g ± β k of g can be defined by X ± β k = T i . . . T i k − X ± α ik , (2.3.2)where X ± α ik are as in (2.2.1).The root vectors X ± β satisfy the following relations: X ± α X ± β − q ( α,β ) X ± β X ± α = X α<δ <...<δ n <β C ( k , . . . , k n )( X ± δ ) k ( X ± δ ) k . . . ( X ± δ n ) k n == X α<δ <...<δ n <β C ′ ( k , . . . , k n )( X ± δ ) ( k ) ( X ± δ ) ( k ) . . . ( X ± δ n ) ( k n ) , α < β, (2.3.3)where for α ∈ ∆ + we put ( X ± α ) ( k ) = ( X ± α ) k [ k ] qα ! , k ≥ q α = q d i if the positive root α is Weyl group conjugateto the simple root α i , C ′ ( k , . . . , k n ) ∈ C [ q, q − ], C ( k , . . . , k n ) ∈ P , and P = C [ q, q − ] if g is simply-laced, P = C [ q, q − , q ] if g is of type B l , C l or F , and P = C [ q, q − , q , q ] if g is of type G .Note that by construction X + β (mod h ) = X β ∈ g β ,X − β (mod h ) = X − β ∈ g − β (2.3.4)are root vectors of g .Define an algebra antiautomorphism ψ of U h ( g ) by ψ ( X ± i ) = X ± i , ψ ( H i ) = − H i , ψ ( h ) = h. It satisfies the relations T − i = ψT i ψ and hence for any α ∈ ∆ + ψ ( X ± α ) = X ± α , where X ± β k = T − i . . . T − i k − X ± i k . .3. QUANTUM ROOT VECTORS ψ to relations (2.3.3) one can obtain the following relations for the root vectors X ± β X ± α X ± β − q − ( α,β ) X ± β X ± α = X α<δ <...<δ n <β D ( k , . . . , k n )( X ± δ ) k ( X ± δ ) k . . . ( X ± δ n ) k n == X α<δ <...<δ n <β D ′ ( k , . . . , k n )( X ± δ ) ( k ) ( X ± δ ) ( k ) . . . ( X ± δ n ) ( k n ) , α < β, (2.3.5)where for α ∈ ∆ + we put ( X ± α ) ( k ) = ( X ± α ) k [ k ] qα ! , k ≥ D ( k , . . . , k n ) ∈ P , D ′ ( k , . . . , k n ) ∈ C [ q, q − ].One can also obtain commutation relations between positive and negative root vectors. These relations areknown in some form. For completeness we give a proof of them using (2.3.3) and (2.3.5) only. Lemma 2.3.1.
Let [ − β, α ] , α, β ∈ ∆ + be a minimal segment with respect to the circular normal ordering of ∆ corresponding to a normal ordering β , . . . , β D of ∆ + . Then X + α X − β − X − β X + α = X − β<δ <...<δ n <α C ( k , . . . , k n )( X δ ) k ( X δ ) k . . . ( X δ n ) k n == X − β<δ <...<δ n <α C ′ ( k , . . . , k n )( X δ ) ( k ) ( X δ ) ( k ) . . . ( X δ n ) ( k n ) , (2.3.6) where the inequalities for roots in the sum are with respect to the circular normal ordering in ∆ corresponding tothe normal ordering β , . . . , β D of ∆ + , X δ = X + δ for δ ∈ ∆ + and X δ = X − δ for δ ∈ ∆ − , C ′ ( k , . . . , k n ) ∈ U q ( H ) , U q ( H ) is the C [ q, q − ] –subalgebra of U h ( g ) generated by K ± i , i = 1 , . . . , l , C ( k , . . . , k n ) ∈ P ′ , and P ′ = U q ( H ) if g is simply-laced, P ′ = U q ( H )[ q ] if g is of type B l , C l or F , and P ′ = U q ( H )[ q , q ] if g is of type G .Also X + α X − β − X − β X + α = X − β<δ <...<δ n <α D ( k , . . . , k n )( X δ ) k ( X δ ) k . . . ( X δ n ) k n == X − β<δ <...<δ n <α D ′ ( k , . . . , k n )( X δ ) ( k ) ( X δ ) ( k ) . . . ( X δ n ) ( k n ) , α < β, (2.3.7) where the inequalities for roots in the sum are with respect to the circular normal ordering in ∆ corresponding tothe normal ordering β , . . . , β D of ∆ + , X δ = X + δ for δ ∈ ∆ + and X δ = X − δ for δ ∈ ∆ − , D ′ ( k , . . . , k n ) ∈ U q ( H ) , D ( k , . . . , k n ) ∈ P ′ .Proof. The proof is by induction. We shall consider the first identity in the case when α < β . The others areproved in a similar way.Let w = s i . . . s i D be the reduced decomposition of the longest element of the Weyl group corresponding to thenormal ordering β , . . . , β D of ∆ + .First assume that α = β = α i . Let β = β m = s i . . . s i m − α i m . Then s − i β , . . . , s − i β D , α i is another normalordering of ∆ + , and by (2.3.3) for this normal ordering T − i ( X + α X − β − X − β X + α ) = T − i ( X + i X − β m − X − β m X + i ) == K − i ( − X − i X − s − i β m + q − ( α i ,s − i β m ) X − s − i β m X − i ) == X − s − i β m <δ <...<δ n < − α i K − i C ( k , . . . , k n )( X δ ) k ( X δ ) k . . . ( X δ n ) k n , where C ( k , . . . , k n ) ∈ P , the inequalities for the roots in the sum are with respect to the circular normal orderingof ∆ associated to the ordering s − i β , . . . , s − i β D , α i of ∆ + , and and the quantum root vectors are defined usingthe ordering s − i β , . . . , s − i β D , α i of ∆ + .Now applying T i to the last identity we get X + α X − β − X − β X + α = X − β m <δ <...<δ n ≤− β D K i C ( k , . . . , k n )( X δ ) k ( X δ ) k . . . ( X δ n ) k n == X − β<δ <...<δ n <α K i C ( k , . . . , k n )( X δ ) k ( X δ ) k . . . ( X δ n ) k n , CHAPTER 2. QUANTUM GROUPS where the inequalities for the roots in the sum are with respect to the circular normal ordering associated to theoriginal ordering β , . . . , β D of ∆ + and the quantum root vectors are defined using the ordering β , . . . , β D of ∆ + .This establishes the base of the induction.Now assume that the identity in question is proved for all normal orderings of ∆ + and for all α = β k with k < n for some n > β such that [ − β, α ], α, β ∈ ∆ + is a minimal segment.Let α = β n = s i . . . s i n − α i n , β = β m = s i . . . s i m − α i m , n < m . Then s − i β , . . . , s − i β D , α i is anothernormal ordering of ∆ + , and by the induction hypothesis for this normal ordering with s − i α = s i . . . s i n − α i n , s − i β = β m = s i . . . s i m − α i m we have X + s − i α X − s − i β − X − s − i β X + s − i α == X − s − i β m <δ ′ <...<δ ′ n
For β = P li =1 m i α i , m i ∈ N X ± β is a polynomial in the noncommutative variables X ± i homogeneous in each X ± i of degree m i . Denote by U resq ( g ) the subalgebra in U h ( g ) generated over C [ q, q − ] by the elements K ± i , ( X ± i ) ( k ) , i = 1 , . . . , l, k ≥ . The elements (cid:20) K i ; cr (cid:21) q i = r Y s =1 K i q c +1 − si − K − i q s − − ci q si − q − si , i = 1 , . . . , l, c ∈ Z , r ∈ N belong to U resq ( g ). Denote by U resq ( H ) the subalgebra of U resq ( g ) generated by those elements and by K ± i , i = 1 , . . . , l .Let U P ( n + ) , U P ( n − ) ( U resq ( n + ) , U resq ( n − )) be the P ( C [ q, q − ])–subalgebras of U h ( g ) generated by the X + i andby the X − i , i = 1 , . . . , l (by the ( X + i ) ( r ) and by the ( X − i ) ( r ) , i = 1 , . . . , l , r ≥ X ± β , ( X ± β ) ( r ) we can construct bases for these subalgebras Namely, let ( X ± ) r = ( X ± β ) r . . . ( X ± β D ) r D ,( X ± ) ( r ) = ( X ± β ) ( r ) . . . ( X ± β D ) ( r D ) , r = ( r , . . . r D ) ∈ N D , H k = H k . . . H k l l , k = ( k , . . . , k l ) ∈ N l .Commutation relations (2.3.3), (2.3.5), (2.3.6) and (2.3.7) between quantum root vectors imply the followinglemma. .5. THE UNIVERSAL R–MATRIX Lemma 2.4.2.
The elements ( X + ) r , ( X − ) t ( ( X + ) ( r ) , ( X − ) ( t ) ) for r , t ∈ N D form bases of U P ( n + ) , U P ( n − ) ( U resq ( n + ) , U resq ( n − ) ), respectively.The elements ( X + ) r , ( X − ) t and H k form topological bases of U h ( n + ) , U h ( n − ) and U h ( h ) , respectively.The multiplication defines an isomorphisms of C [ q, q − ] –modules: U resq ( n − ) ⊗ U resq ( H ) ⊗ U resq ( n + ) → U resq ( g ) , (2.4.1) and of complete C [[ h ]] –modules U h ( n − ) ⊗ U h ( h ) ⊗ U h ( n + ) → U h ( g ) , where the tensor products in the leftt hand side are completed in the h –adic topology.Let [ α, β ] = { β p , . . . , β q } be a minimal segment in ∆ + , U P ([ α, β ]) , U P ([ − α, − β ]) ( U resq ([ α, β ]) , U resq ([ − α, − β ]) )the P ( C [ q, q − ]) –subalgebras of U h ( g ) generated by the X + γ and by the X − γ , γ ∈ [ α, β ] (by the ( X + γ ) ( r ) and by the ( X − γ ) ( r ) , γ ∈ [ α, β ] , r ≥ ), respectively. Then the elements ( X ± β p ) r p . . . ( X ± β q ) r q ( ( X ± β p ) ( r p ) . . . ( X ± β q ) ( r q ) ), r i ∈ N form bases of U P ([ α, β ]) , U P ([ − α, − β ]) ( U resq ([ α, β ]) , U resq ([ − α, − β ]) ), respectively.Let U P ([ α, β ]) , U P ([ − α, − β ]) ( U resq ([ α, β ]) , U resq ([ − α, − β ]) ) be the P ( C [ q, q − ]) –subalgebras of U h ( g ) generatedby the X + γ and by the X − γ , γ ∈ [ α, β ] (by the ( X + γ ) ( r ) and by the ( X − γ ) ( r ) , γ ∈ [ α, β ] , r ≥ ), respectively. Then theelements ( X ± β p ) r p . . . ( X ± β q ) r q ( ( X ± β p ) ( r p ) . . . ( X ± β q ) ( r q ) ), r i ∈ N form bases of U P ([ α, β ]) , U P ([ − α, − β ]) ( U resq ([ α, β ]) , U resq ([ − α, − β ]) ), respectively.Let [ α, − β ] = { β p , . . . , β q } , α, β ∈ ∆ + be a minimal segment in ∆ , U P ′ ([ α, − β ]) , U P ′ ([ − α, β ]) ( U resU resq ( H ) ([ α, − β ]) , U resU resq ( H ) ([ − α, β ]) ) the P ′ ( U resq ( H )) –subalgebras of U h ( g ) generated by the X γ ( ( X γ ) ( r ) ), where γ ∈ [ α, − β ] or γ ∈ [ − α, β ] , respectively, and X γ = X + γ if γ ∈ ∆ + , X γ = X − γ if γ ∈ ∆ − . Then the elements ( X β p ) r p . . . ( X β q ) r q ( ( X β p ) ( r p ) . . . ( X β q ) ( r q ) ), r i ∈ N form bases of U P ′ ([ α, − β ]) ( U resU resq ( H ) ([ α, − β ]) ), and the elements ( X − β p ) r p . . . ( X − β q ) r q ( ( X − β p ) ( r p ) . . . ( X − β q ) ( r q ) ), r i ∈ N form bases of U P ′ ([ − α, β ]) ( U resU resq ( H ) ([ − α, β ]) ), respectively.Let U P ′ ([ α, − β ]) , U P ′ ([ − α, β ]) ( U resU resq ( H ) ([ α, − β ]) , U resU resq ( H ) ([ − α, β ]) ) be the P ′ ( U resq ( H )) –subalgebras of U h ( g ) generated by the X γ ( ( X γ ) ( r ) ), where γ ∈ [ α, − β ] or γ ∈ [ − α, β ] , respectively, and X γ = X + γ if γ ∈ ∆ + , X γ = X − γ if γ ∈ ∆ − . Then the elements ( X β p ) r p . . . ( X β q ) r q ( ( X β p ) ( r p ) . . . ( X β q ) ( r q ) ), r i ∈ N form bases of U P ′ ([ α, − β ]) ( U resU resq ( H ) ([ α, − β ]) ), and the elements ( X − β p ) r p . . . ( X − β q ) r q ( ( X − β p ) ( r p ) . . . ( X − β q ) ( r q ) ), r i ∈ N form bases of U P ′ ([ − α, β ]) ( U resU resq ( H ) ([ − α, β ]) ), respectively.Proof. The first four statements of this lemma are just Propositions 8.1.7, 9.1.3 and 9.3.3 in [18]. The proofs ofthe other claims are similar to each other. Consider, for instance, the case of the algebra U resU resq ( H ) ([ α, − β ]).Let [ α, − β ]= { β p , . . . , β q } , α, β ∈ ∆ + be a minimal segment in ∆, U resU resq ( H ) ([ α, − β ]) the U resq ( H )–subalgebra of U h ( g ) generated by the ( X γ ) ( r ) , where γ ∈ [ α, − β ], and X γ = X + γ if γ ∈ ∆ + , X γ = X − γ if γ ∈ ∆ − . We show thatthe elements ( X β p ) ( r p ) . . . ( X β q ) ( r q ) , r i ∈ N form a basis of U resU resq ( H ) ([ α, − β ]).Consider the algebra U q ( g ) = U resq ( g ) ⊗ C [ q,q − ] C ( q ). If x ∈ U resU resq ( H ) ([ α, − β ]) ⊂ U q ( g ) then using commutationrelations (2.3.3) and (2.3.6) one can represent x as a U resq ( H ) ⊗ C [ q,q − ] C ( q )–linear combination of the elements( X β p ) ( r p ) . . . ( X β q ) ( r q ) , r i ∈ N . We can also consider x as an element of U q ( g ) and by the Poincar´e–Birkhoff–Witttheorem for U q ( g ) (see Proposition 9.1.3 in [18]) the above mentioned presentation of x is unique. Now by thePoincar´e–Birkhoff–Witt theorem for U resq ( g ) (see Proposition 9.3.3 in [18] or the third claim of this lemma) thecoefficients in this presentation must belong to U resq ( H ). This completes the proof in the considered case.A basis for U resq ( H ) is a little bit more difficult to describe. We do not need its explicit description. U h ( g ) is a quasitriangular Hopf algebra, i.e. there exists an invertible element R ∈ U h ( g ) ⊗ U h ( g ) (completed tensorproduct), called a universal R–matrix, such that∆ opph ( a ) = R ∆ h ( a ) R − for all a ∈ U h ( g ) , (2.5.1)8 CHAPTER 2. QUANTUM GROUPS and (∆ h ⊗ id ) R = R R , ( id ⊗ ∆ h ) R = R R , (2.5.2)where R = R ⊗ , R = 1 ⊗ R , R = ( σ ⊗ id ) R .From (2.5.1) and (2.5.2) it follows that R satisfies the quantum Yang–Baxter equation R R R = R R R . (2.5.3)For every quasitriangular Hopf algebra we also have( S ⊗ id ) R = ( id ⊗ S − ) R = R − , and ( S ⊗ S ) R = R . (2.5.4)An explicit expression for R may be written by making use of the q–exponential exp q ( x ) = exp ′ q ( qx ) = ∞ X k =0 q k ( k +1) x k [ k ] q !in terms of which the element R takes the form R = Y β exp q β [(1 − q − β ) X − β ⊗ X + β ] exp " h l X i =1 ( Y i ⊗ H i ) , (2.5.5)where the product is over all the positive roots of g , and the order of the terms is such that the α –term appears tothe left of the β –term if α < β with respect to the normal ordering β = α i , β = s i α i , . . . , β D = s i . . . s i D − α i D of ∆ + which is used in the definition of the quantum root vectors X ± β .One can calculate the action of the comultiplication on the root vectors X ± β k in terms of the universal R–matrix.For instance for ∆ h ( X − β k ) one has∆ h ( X − β k ) = e R <β k ( X − β k ⊗ e hβ ∨ ⊗ X − β k ) e R − <β k , (2.5.6)where e R <β k = e R β . . . e R β k − , e R β r = exp q βr [(1 − q − β r ) X + β r ⊗ X − β r ] . The r–matrix r − = − h − ( R − ⊗
1) (mod h ), which is the classical limit of R , coincides with the classicalr–matrix (2.1.3). q-W–algebras will be defined in terms of certain integral forms of non–standard realizations of quantum groupsassociated to Weyl group elements.Let s be an element of the Weyl group W of the pair ( g , h ), and h ′ the orthogonal complement, with respect tothe symmetric bilinear form, to the subspace of h fixed by the natural action of s on h . Let h ′∗ be the image of h ′ in h ∗ under the identification h ∗ ≃ h induced by the symmetric bilinear form on g . The restriction of the naturalaction of s on h ∗ to the subspace h ′∗ has no fixed points. Therefore one can define the Cayley transform s − s ofthe restriction of s to h ′∗ . Denote by P h ′∗ the orthogonal projection operator onto h ′∗ in h ∗ , with respect to thebilinear form. .6. REALIZATIONS OF QUANTUM GROUPS ASSICIATED TO WEYL GROUP ELEMENTS κ ∈ Z be an integer number and U sh ( g ) the topological algebra over C [[ h ]] topologically generated by elements e i , f i , H i , i = 1 , . . . l subject to the relations:[ H i , H j ] = 0 , [ H i , e j ] = a ij e j , [ H i , f j ] = − a ij f j , e i f j − q c ij f j e i = δ i,j K i − K − i q i − q − i ,c ij = κ (cid:16) s − s P h ′∗ α i , α j (cid:17) , K i = e d i hH i , P − a ij r =0 ( − r q rc ij (cid:20) − a ij r (cid:21) q i ( e i ) − a ij − r e j ( e i ) r = 0 , i = j, P − a ij r =0 ( − r q rc ij (cid:20) − a ij r (cid:21) q i ( f i ) − a ij − r f j ( f i ) r = 0 , i = j. (2.6.1) Proposition 2.6.1.
For every solution n ij ∈ C , i, j = 1 , . . . , l of equations d j n ij − d i n ji = c ij (2.6.2) there exists an algebra isomorphism ψ { n ij } : U sh ( g ) → U h ( g ) defined by the formulas: ψ { n ij } ( e i ) = X + i l Y p =1 L n ip p , ψ { n ij } ( f i ) = l Y p =1 L − n ip p X − i , ψ { n ij } ( H i ) = H i . Proof.
The proof of this proposition is by direct verification of defining relations (2.6.1). The most nontrivial partis to verify the deformed quantum Serre relations, i.e. the last two relations in (2.6.1). For instance, the definingrelations of U h ( g ) imply the following relations for ψ { n ij } ( e i ), − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) q i q k ( d j n ij − d i n ji ) ψ { n ij } ( e i ) − a ij − k ψ { n ij } ( e j ) ψ { n ij } ( e i ) k = 0 , for any i = j . Now using equation (2.6.2) we arrive to the quantum Serre relations for e i in (2.6.1).The general solution of equation (2.6.2) is given by n ij = 12 d j ( c ij + s ij ) , (2.6.3)where s ij = s ji .We shall only use the solution for which s ij = 0 for all i, j = 1 , . . . l . Then n ij = 12 d j c ij (2.6.4)From now on we assume that solution (2.6.4) is used to identify U sh ( g ) and U h ( g ).The algebra U sh ( g ) is called the realization of the quantum group U h ( g ) corresponding to the element s ∈ W .Denote by U sh ( n ± ) the subalgebra in U sh ( g ) generated by e i ( f i ) , i = 1 , . . . , l . Let U sh ( h ) be the subalgebra in U sh ( g )generated by H i , i = 1 , . . . , l .We shall construct analogues of quantum root vectors for U sh ( g ). It is convenient to introduce an operator K ∈ End h defined by KH i = l X j =1 n ij d i Y j . (2.6.5)From (2.6.4) we obtain that Kh = κ s − s P h ′ h, h ∈ h . CHAPTER 2. QUANTUM GROUPS
Proposition 2.6.2.
Let s ∈ W be an element of the Weyl group W of the pair ( g , h ) , ∆ the root system of thepair ( g , h ) . Let U sh ( g ) be the realization of the quantum group U h ( g ) associated to s .For any normal ordering of the root system ∆ + the elements e β = ψ − { n ij } ( X + β e hKβ ∨ ) and f β = ψ − { n ij } ( e − hKβ ∨ X − β ) , β ∈ ∆ + lie in the subalgebras U sh ( n + ) and U sh ( n − ) , respectively.The elements f β ∈ U sh ( n − ) , β ∈ ∆ m + generate a subalgebra U sh ( m − ) ⊂ U sh ( g ) such that U sh ( m − ) /hU sh ( m − ) ≃ U ( m − ) , where m − is the Lie subalgebra of g generated by the root vectors X − α , α ∈ ∆ m + .Proof. Fix a normal ordering of the root system ∆ + . Let β = P li =1 m i α i ∈ ∆ + be a positive root, X + β ∈ U h ( g )the corresponding quantum root vector constructed with the help of the fixed normal ordering of ∆ + . Then β ∨ = P li =1 m i d i H i , and so Kβ ∨ = P li,j =1 m i n ij Y j . Now the proof of the first statement follows immediately fromProposition 2.4.1, commutation relations (2.1.1) and the definition of the isomorphism ψ { n ij } .The second assertion is a consequence of (2.3.4).The realizations U sh ( g ) of the quantum group U h ( g ) are related to quantizations of some nonstandard bialgebrastructures on g . At the quantum level changing bialgebra structure corresponds to the so–called Drinfeld twist.The relevant class of such twists described in the following proposition. Proposition 2.6.3.
Let ( A, µ, ı, ∆ , ε, S ) be a Hopf algebra over a commutative ring with multiplication µ , unit ı ,comultiplication ∆ , counit ε and antipode S . Let F be an invertible element of A ⊗ A such that F (∆ ⊗ id )( F ) = F ( id ⊗ ∆)( F ) , ( ε ⊗ id )( F ) = ( id ⊗ ε )( F ) = 1 . (2.6.6) Then, v = µ ( id ⊗ S )( F ) is an invertible element of A with v − = µ ( S ⊗ id )( F − ) . Moreover, if we define ∆ F : A → A ⊗ A and S F : A → A by ∆ F ( a ) = F ∆( a ) F − , S F ( a ) = vS ( a ) v − , then ( A, µ, ı, ∆ F , ε, S F ) is a Hopf algebra denoted by A F and called the twist of A by F . Corollary 2.6.4.
Suppose that A and F are as in Proposition 2.6.3, but assume in addition that A is quasitrian-gular with universal R–matrix R . Then A F is quasitriangular with universal R–matrix R F = F RF − , (2.6.7) where F = σ F . Let P h ′ be the orthogonal projection operator onto h ′ in h with respect to the bilinear form on h . Equip U sh ( g )with the comultiplication ∆ s given by ∆ s ( H i ) = H i ⊗ ⊗ H i , ∆ s ( e i ) = e i ⊗ e − hd i H i + e hκd i s − s P h ′ H i ⊗ e i , ∆ s ( f i ) = f i ⊗ e − hκd i s − s P h ′ H i + hd i H i ⊗ f i , the antipode S s ( x ) given by S s ( e i ) = − e − hκd i s − s P h ′ H i e i e hd i H i , S s ( f i ) = − e hκd i s − s P h ′ H i − hd i H i f i , S s ( H i ) = − H i , and counit defined by ε s ( H i ) = ε s ( X ± i ) = 0 . .6. REALIZATIONS OF QUANTUM GROUPS ASSICIATED TO WEYL GROUP ELEMENTS s is obtained from the standard comultiplication by a Drinfeld twist. Namely, let F = exp ( − h l X i,j =1 n ij d i Y i ⊗ Y j ) ∈ U h ( h ) ⊗ U h ( h ) . (2.6.8)Then ∆ s ( a ) = ( ψ − { n ij } ⊗ ψ − { n ij } ) F ∆ h ( ψ { n ij } ( a )) F − . (2.6.9)Note that the Hopf algebra U sh ( g ) is a quantization of the bialgebra structure on g defined by the cocycle δ ( x ) = (ad x ⊗ ⊗ ad x )2 r s ± , r s ± ∈ g ⊗ g , (2.6.10)where r s ± = r ± + P li =1 κ s − s P h ′ H i ⊗ Y i , and r ± is given by (2.1.3). U sh ( g ) is a quasitriangular topological Hopf algebra with the universal R–matrix R s = F RF − , R s = Q β exp q β [(1 − q − β ) f β ⊗ e β e − hκ s − s P h ′ β ∨ ] × exp h h ( P li =1 ( Y i ⊗ H i ) − P li =1 κ s − s P h ′ H i ⊗ Y i ) i , (2.6.11)where the order of the terms is such that the α –term appears to the left of the β –term if α < β with respect to thenormal ordering of ∆ + with the help of which the quantum root vectors e β , f β are defined in Proposition 2.6.2.Similarly to (2.2.8) one obtains that for a reduced decomposition w = s i . . . s i k and T w = T i . . . T i k onlydepending on w one has from (2.2.8) and (2.6.9)∆ s ( T w ) = k Y p =1 θ sβ p q P li =1 ( − Y i ⊗ KH i + T w Y i ⊗ T w KH i ) T w ⊗ T w = (2.6.12)= T w ⊗ T w q P li =1 ( − T w − Y i ⊗ T w − KH i + Y i ⊗ KH i ) k Y p =1 θ sβ ′ p , where for p = 1 , . . . , ke β p = ψ − { n ij } ( X + β p e hKβ ∨ p ) , f β = ψ − { n ij } ( e − hKβ ∨ p X − β p ) , β p = s i . . . s i p − α i p ,X ± β p = T i . . . T i p − X ± i p , X ± β ′ p = T − i k . . . T − i p +1 X ± i p ,e β ′ p = ψ − { n ij } ( X + β ′ p e hKβ ′ p ∨ ) , f β ′ p = ψ − { n ij } ( e − hKβ ′ p ∨ X − β ′ p ) , K β ′ p = T − i k . . . T − i p +1 K i p , β ′ p = s i k . . . s i p +1 α i p θ sβ p = exp q βp [(1 − q − β p ) e β p e − hκ s − s P h ′ β ∨ p ⊗ f β p ] , θ sβ ′ p = exp q β ′ p [(1 − q − β ′ p ) K − β ′ p e hκ s − s P h ′ β ′ p ∨ f β ′ p ⊗ e β ′ p K β ′ p ] . In the same way, for T w = T − i . . . T − i k only depending on w one has from (2.2.9) and (2.6.9)∆ s ( T w ) = k Y p =1 θ sβ p ′ q P li =1 ( − Y i ⊗ KH i + T w Y i ⊗ T w KH i ) T w ⊗ T w = (2.6.13)= T w ⊗ T w q P li =1 ( − T w − Y i ⊗ T w − KH i + Y i ⊗ KH i ) k Y p =1 θ sβ ′ p ′ , where for p = 1 , . . . , ke β p = ψ − { n ij } ( X + β p e hKβ ∨ p ) , f β p = ψ − { n ij } ( e − hKβ ∨ p X − β p ) , K β p = T − i . . . T − i p − K i p ,X ± β p = T − i . . . T − i p − X ± i p , β p = s i . . . s i p − α i p ,e ′ β ′ p = ψ − { n ij } ( X + β ′ p ′ e hKβ ′ p ∨ ) , f ′ β ′ p = ψ − { n ij } ( e − hKβ ′ p ∨ X − β ′ p ′ ) , β ′ p = s i k . . . s i p +1 α i p , X ± β ′ p ′ = T i k . . . T i p +1 X ± i p ,θ sβ ′ p ′ = exp q − β ′ p [(1 − q β ′ p ) e ′ β ′ p e − hκ s − s P h ′ β ′ p ∨ ⊗ f ′ β ′ p ] , θ sβ p ′ = exp q − βp [(1 − q β p ) K − β p e hκ s − s P h ′ β ∨ p f β p ⊗ e β p K β p ] . CHAPTER 2. QUANTUM GROUPS
In order to define q-W–algebras we shall actually need not the algebras U sh ( g ) themselves but some their formsdefined over certain rings. They are similar to the rational form and the restricted integral form for the standardquantum group U h ( g ). The motivations of the definitions given below will be clear in Section 3.2. The resultsbelow are slight modifications of similar statements for U h ( g ).We start with a very important technical lemma which will play the key role in the definition of q-W–algebras.Below we keep the notation introduced in Section 1.2.Let s ∈ W be an element of the Weyl group. Recall that s can be represented as a product of two involutions, s = s s , where s = s γ . . . s γ n , s = s γ n +1 . . . s γ l ′ , and the roots γ , . . . , γ l ′ form a basis of a subspace h ′∗ ⊂ h ∗ onwhich s acts without fixed points. We shall study the matrix elements of the Cayley transform of the restriction of s to h ′∗ with respect to this basis. Lemma 2.7.1.
Let P h ′∗ be the orthogonal projection operator onto h ′∗ in h ∗ , with respect to the bilinear form.Then the matrix elements of the operator s − s P h ′∗ in the basis γ , . . . , γ l ′ are of the form (cid:18) s − s P h ′∗ γ i , γ j (cid:19) = ε ij ( γ i , γ j ) , (2.7.1) where ε ij = − i < j i = j i > j . Proof.
First we calculate the matrix of the element s with respect to the basis γ , . . . , γ l ′ . We obtain this matrixin the form of the Gauss decomposition of the operator s : h ′∗ → h ′∗ .Let z i = sγ i . Recall that s γ i ( γ j ) = γ j − A ij γ i , A ij = ( γ ∨ i , γ j ). Using this definition the elements z i may berepresented as z i = y i − X k ≥ i A ki y k , where y i = s γ . . . s γ i − γ i . (2.7.2)Using the matrix notation we can rewrite the last formula as follows z i = ( I + V ) ki y k , where V ki = (cid:26) A ki k ≥ i k < i (2.7.3)To calculate the matrix of the operator s : h ′∗ → h ′∗ with respect to the basis γ , . . . , γ l ′ we have to express theelements y i via γ , . . . , γ l ′ . Applying the definition of reflections to (2.7.2) we can pull out the element γ i to theright, y i = γ i − X k
Fix a normal ordering of the system of positive roots ∆ + and let e β , f β be the corresponding quantumroot vectors defined in Proposition 2.6.2. Then the elements f β satisfy the following commutation relations f α f β − q ( α,β )+ κ ( s − s P h ′∗ α,β ) f β f α = X α<δ <...<δ n <β C ( p , . . . , p n ) f ( p ) δ f ( p ) δ . . . f ( p n ) δ n = (2.7.11)= X α<δ <...<δ n <β C ′ ( p , . . . , p n ) f p δ f p δ . . . f p n δ n , α < β, where C ( p , . . . , p n ) ∈ B , C ′ ( p , . . . , p n ) ∈ A .The elements e r = e r β . . . e r D β D , f t = f t D β D . . . f t β , for r = ( r , . . . r D ) , t = ( t , . . . t D ) ∈ N D , form bases of U sq ( n + ) , U sq ( n − ) , respectively, and the multiplication defines an isomorphism of C ( q dr ) –modules: U sq ( n − ) ⊗ U sq ( h ) ⊗ U sq ( n + ) → U sq ( g ) . The elements e r , f t ( e ( r ) = e ( r ) β . . . e ( r D ) β D , f ( t ) = f ( t D ) β D . . . f ( t ) β ) for r , t ∈ N D form bases of U s A ( n + ) , U s A ( n − ) ( U s,res B ( n + ) , U s,res B ( n − ) ), respectively.The multiplication defines an isomorphisms of B –modules: U s,res B ( n − ) ⊗ U s,res B ( h ) ⊗ U s,res B ( n + ) → U s,res B ( g ) . Let [ α, β ] = { β p , . . . , β q } be a minimal segment in ∆ + , U s A ([ α, β ]) , U s A ([ − α, − β ]) ( U s,res B ([ α, β ]) , U s,res B ([ − α, − β ]) )the A ( B ) –subalgebras of U sh ( g ) generated by the e γ and by the f γ , γ ∈ [ α, β ] (by the ( e γ ) ( r ) and by the ( f γ ) ( r ) , γ ∈ [ α, β ] , r ∈ N ), respectively. Then the elements ( e β p ) r p . . . ( e β q ) r q , ( f β q ) r q . . . ( f β p ) r p ( ( e β p ) ( r p ) . . . ( e β q ) ( r q ) , ( f β q ) ( r q ) . . . ( f β p ) ( r p ) ), r i ∈ N form bases of U s A ([ α, β ]) , U s A ([ − α, − β ]) ( U s,res B ([ α, β ]) , U s,res B ([ − α, − β ]) ), respec-tively.The elements f t ( f ( t ) = f ( t D ) β D . . . f ( t ) β ) for t ∈ N D with t i > for at least one i ≥ p form a basis in the rightideal of U s A ( n − ) ( U s,res B ( n − ) ) generated by f γ , γ ∈ [ β p , β D ] (by ( f γ ) ( r ) , γ ∈ [ β p , β D ] , r > ).Proof. Commutation relations (2.7.11) follow from commutation relations (2.3.3), (2.1.1), (2.1.2), Proposition 2.4.1,the definition of the elements e β , f β and the definition of the isomorphism ψ { n ij } .The other statements of this lemma, except for the last one, follow straightforwardly from Lemma 2.4.2 andPropositions 2.6.1 and 2.6.2.For the last statement, using commutation relations (2.7.11) we can represent any element of the right ideal of U s A ( n − ) generated by f γ , γ ∈ [ β p , β D ] as an A –linear combination of the elements f t for t ∈ N D with t i > i ≥ p . This presentation is unique by the Poincar´e–Birkhoff–Witt decomposition for U s A ( n − ) statedabove.Note that a similar result holds for the algebra U sq ( n − ) = U s A ( n − ) ⊗ A C ( q dr ) for the same reasons.We can apply it to represent any element of the right ideal of U s,res B ( n − ) ⊂ U sq ( n − ) generated by ( f γ ) ( r ) , γ ∈ [ β p , β D ], r > C ( q dr )–linear combination of the elements f ( t ) for t ∈ N D with t i > i ≥ p . This presentation is unique and by the uniqueness of the Poincar´e–Birkhoff–Witt decomposition for U s,res B ( n − ) stated above the coefficients in this decomposition belong to B . This completes the proof.A basis for U s,res B ( h ) is a little bit more difficult to describe. We do not need its explicit description. Remark 2.7.3.
Applying the antiautomorphism ω to the elements of the bases constructed in Lemma 2.7.2 andusing (2.7.9) and (2.7.10) we obtain other bases of similar types where the order of the quantum root vectors in theproducts defining the elements of the bases is reversed.By specializing the above constructed bases for q rd = ε rd one can obtain similar bases and similar subalgebrasfor U sε ( g ) and U s,resε ( g ) . Using formulas (2.5.6) and (2.6.9) one can also find that∆ s ( f β k ) = e R s<β k ( e − hκ s − s P h ′ β ∨ k + hβ ∨ k ⊗ f β k + f β k ⊗ e R s<β k ) − = (2.7.12)= G − β k ⊗ f β k + f β k ⊗ X i y i ⊗ x i , CHAPTER 2. QUANTUM GROUPS where G β = e hκ s − s P h ′ β ∨ − hβ ∨ , y i = e − hκ s − s P h ′ γ ∨ xi + hγ ∨ xi y i ,y i ∈ U s A ([ − β k +1 , − β D ]) ∩ U s,res B ([ − β k +1 , − β D ]) ,x i ∈ U s A ([ − β , − β k − ]) ∩ U s,res B ([ − β , − β k − ]) ,y i , x i belong to weight components and have non-zero weights, γ x i is the weight of x i , e R s<β k = e R sβ . . . e R sβ k − , e R sβ r = exp q βr [(1 − q − β r ) e β r e − hκ s − s P h ′ β ∨ ⊗ f β r ] , and ( e R s<β k ) − = ( e R sβ k − ) − . . . ( e R sβ ) − , ( e R sβ r ) − = exp q − βr [(1 − q β r ) e β r e − hκ s − s P h ′ β ∨ ⊗ f β r ] . From (2.7.12) we also obtain∆ s ( f ( n ) β k ) = 1[ n ] q βk ! e R s<β k ( G − β k ⊗ f β k + f β k ⊗ n ( e R s<β k ) − = (2.7.13)= e R s<β k ( n X k =0 q k ( n − k ) β k G − kβ k f ( n − k ) β k ⊗ f ( k ) β k )( e R s<β k ) − == n X k =0 q k ( n − k ) β k G − kβ k f ( n − k ) β k ⊗ f ( k ) β k + X i y ( n ) i ⊗ x ( n ) i , where y ( n ) i = e − hκ s − s P h ′ γ ∨ x ( n ) i + hγ ∨ x ( n ) i y ( n ) i ,y ( n ) i ∈ I >k , x ( n ) i ∈ I
0, and I
From the explicit formulas and the results in [68], Section 5.2 it follows that the braid group elements given by(2.2.3) and (2.2.4) act in U h ( g )–modules topologically free and of finite rank over C [[ h ]] and in U s,res B ( g )–latticesin them.We shall also need the following technical lemma regarding the action of the elements T w and T w on finite rankindecomposable modules. Lemma 2.7.4.
Let V be a finite rank U h ( g ) –module. If T, T ′ are two elements of the braid group B g which act asthe same transformation on h ⊂ U h ( h ) and v is a highest weight vector in V then T v = tT ′ v , where t is a non–zeromultiple of a power of q .Proof. Since v generates a highest weight indecomposable submodule V λ ⊂ V of highest weight λ equal to theweight of v , and V λ is invariant under the braid group action, we can assume without loss of generality that V = V λ .First observe that similarly to the proof of Proposition 4.2 in [75] (see also Proposition 10.1.4 in [18]) one canshow that V ′ = U resq ( g ) v is a U resq ( g )–lattice in V λ in the sense that V ′ ⊗ C [ q,q − ] C [[ h ]] ≃ V λ . This module coincideswith the one defined in Section 4.1 in [70] (see also Proposition 10.1.4 in [18]).It is well known that any element T of the braid group acts as an invertible linear automorphism of V ′ whichcan be specialized to any non–zero numeric value of q in the sense that for any ε ∈ C ∗ T gives rise to a linearautomorphism of U resq ( g ) / ( q − ε ) U resq ( g )–module V ′ ε = V ′ / ( q − ε ) V ′ . It suffices to verify this statement when T = T i for i = 1 , . . . , l , and in this case it follows from the explicit formulas and the results in [68], Section 5.2.Recall that elements of the braid group act as Weyl group elements on h ⊂ U h ( h ). Assume that the actionof T and T ′ on h ⊂ U h ( h ) coincides with the action of a Weyl group element w . Since the C [ q, q − ]–submoduleof V ′ which consists of elements of weight wµ has rank one and T v and T ′ v must belong to this submodule, therelation T v = t ( q ) T ′ v must hold for some rational function t ( q ) of q with poles or zeroes only at zero and infinity.Indeed, if t ( q ) = 0, q = 0 , ∞ then in V ′ q we have T v = 0, i.e. T does not induce an automorphism of V ′ q , and if t − ( q ′ ) = 0, q ′ = 0 , ∞ then in V ′ q ′ we have T ′ v = 0, i.e. T ′ does not induce an automorphism of V ′ q ′ . In both caseswe arrive at a contradiction. Thus t ( q ) must be a non–zero multiple of a power of q . The material presented in Sections 2.1, 2.2, 2.3, 2.4 and 2.5 is mostly standard and we refer to books [18, 51, 68]for more details and omitted proofs. Formula (2.2.3) can be found in [88].Realizations of quantum groups associated to Weyl group elements were introduces in [93] in the case of Coxeterelements and in [99] in general.Lemma 2.7.1 is a generalization of the result of Exercise 3 in Chapter V, §
6, [9].Specializations of quantum groups similar to those which appear in this book were considered in [101]. In thisbook we introduce slightly different specializations of quantum groups in order to use restricted specializations aswell. hapter 3 q-W–algebras
In this chapter we introduce q–W–algebras and study the structure of their the quasi–classical versions, Poissonq-W–algebras. In the next chapter similar results will be obtained for q-W–algebras.As we briefly mentioned in the introduction the naive definition of q-W–algebras as Hecke type algebras Hk ( A, B, χ ) requires some modification. In fact the main ingredient of the definition of q-W–algebras is theadjoint action of the quantum group on itself, and they are defined using a B –subalgebra C B [ G ∗ ] of the quantumgroup the restriction of the adjoint action to which is locally finite. When q is specialized to ε ∈ C ∗ which is not aroot of unity the algebra C B [ G ∗ ] becomes the locally finite part of the quantum group with respect to the adjointaction which was introduced and studied by Joseph.The algebra C B [ G ∗ ] is a quantization of the algebra of regular functions on a Poisson manifold G ∗ which isisomorphic to G as a manifold and the Poisson structure of which is closely related to that of the Poisson–Lie group G ∗ dual to a quasitriangular Poisson–Lie group G .After recalling basic facts on Poisson–Lie groups in Section 3.1 we introduce an algebra C [ G ∗ ] of functions on G ∗ in Section 3.2, its quantization C B [ G ∗ ] ⊂ U sh ( g ) and the subalgebra C B [ G ∗ ] ⊂ C B [ G ∗ ].A special choice of the bialgebra structure entering the definitions of C [ G ∗ ], C B [ G ∗ ] ⊂ U sh ( g ) and C B [ G ∗ ] iscrucial for the definition of q-W–algebras. It depends on the choice of a Weyl group element s ∈ W and ensuresthat one can define a subalgebra C B [ M + ] ⊂ C B [ G ∗ ] equipped with a non–trivial character, so that the q-W–algebra W s B ( G ) can be defined as the result of a quantum constrained reduction with respect to the subalgebra C B [ M + ].Next, in Section 3.4 we proceed with the study of the specialization W s ( G ) of the algebra W s B ( G ) at q rd = 1.We recall that W s ( G ) is naturally a Poisson algebra which can be regarded as the algebra of regular functions ona reduced Poisson manifold which is also an algebraic variety. Poisson reduction works well for differential Poissonmanifolds. Therefore it is easier firstly to describe the reduced Poisson structure on the algebra of C ∞ –functionson the reduced Poisson manifold and then to recover the structure of the algebraic variety on it. This is done inProposition 3.4.3 and Theorem 3.4.5.In Section 3.5 we define a projection operator Π into the algebra W s ( G ). In Theorem 3.5.6, which is central inthis chapter, we obtain a formula for the operator Π suitable for quantization. This formula plays the key role inthe proof of Theorem 4.7.2 describing a localization of the algebra W s B ( G ) in terms of a quantum counterpart ofthe operator Π. Miraculously the formula for Π from Theorem 3.5.6 can be directly extrapolated to the quantumcase. In this section we recall some notions related to Poisson–Lie groups. These facts will be needed for the study ofPoisson q-W–algebras.Let G be a finite-dimensional Lie group equipped with a Poisson bracket, g its Lie algebra. G is called aPoisson–Lie group if the multiplication G × G → G is a Poisson map. A Poisson bracket satisfying this axiom isdegenerate and, in particular, is identically zero at the unit element of the group. Linearizing this bracket at theunit element defines the structure of a Lie algebra in the space T ∗ e G ≃ g ∗ . The pair ( g , g ∗ ) is called the tangentbialgebra of G .Lie brackets in g and g ∗ satisfy the following compatibility condition:690 CHAPTER 3. Q-W–ALGEBRAS
Let δ : g → g ∧ g be the dual of the commutator map [ , ] ∗ : g ∗ ∧ g ∗ → g ∗ . Then δ is a 1-cocycle on g (withrespect to the adjoint action of g on g ∧ g ).Let c kij , f abc be the structure constants of g , g ∗ with respect to the dual bases { e i } , { e i } in g , g ∗ . The compatibilitycondition means that c sab f iks − c ias f skb + c kas f sib − c kbs f sia + c ibs f ska = 0 . This condition is symmetric with respect to exchange of c and f . Thus if ( g , g ∗ ) is a Lie bialgebra, then ( g ∗ , g ) isalso a Lie bialgebra.The following proposition shows that the category of finite-dimensional Lie bialgebras is isomorphic to thecategory of finite-dimensional connected simply connected Poisson–Lie groups. Proposition 3.1.1. If G is a connected simply connected finite-dimensional Lie group, every bialgebra structureon g is the tangent bialgebra of a unique Poisson structure on G which makes G into a Poisson–Lie group. Let G be a finite-dimensional Poisson–Lie group, ( g , g ∗ ) the tangent bialgebra of G . The connected simplyconnected finite-dimensional Poisson–Lie group corresponding to the Lie bialgebra ( g ∗ , g ) is called the dual Poisson–Lie group and denoted by G ∗ .( g , g ∗ ) is called a factorizable Lie bialgebra if the following conditions are satisfied:1. g is equipped with a non–degenerate invariant scalar product ( · , · ).We shall always identify g ∗ and g by means of this scalar product.2. The dual Lie bracket on g ∗ ≃ g is given by [ X, Y ] ∗ = 12 ([ rX, Y ] + [ X, rY ]) , X, Y ∈ g , (3.1.1) where r ∈ End g is a skew symmetric linear operator (classical r-matrix). r satisfies the modified classical Yang-Baxter identity: [ rX, rY ] − r ([ rX, Y ] + [ X, rY ]) = − [ X, Y ] , X, Y ∈ g . (3.1.2)Define operators r ± ∈ End g by r ± = 12 ( r ± id ) . We shall need some properties of the operators r ± . Denote by b ± and n ∓ the image and the kernel of the operator r ± : i ± = Im r ± , k ∓ = Ker r ± . (3.1.3) Proposition 3.1.2.
Let ( g , g ∗ ) be a factorizable Lie bialgebra. Then(i) i ± ⊂ g is a Lie subalgebra, the subspace k ± is a Lie ideal in i ± , i ⊥± = k ± .(ii) k ± is an ideal in g ∗ .(iii) i ± is a Lie subalgebra in g ∗ . Moreover i ± = g ∗ / k ± .(iv) ( i ± , i ∗± ) is a subbialgebra of ( g , g ∗ ) and ( i ± , i ∗± ) ≃ ( i ± , i ∓ ) . The canonical paring between i ∓ and i ± is givenby ( X ∓ , Y ± ) ± = ( X ∓ , r − ± Y ± ) , X ∓ ∈ i ∓ ; Y ± ∈ i ± . (3.1.4)The classical Yang–Baxter equation implies that r ± , regarded as a mapping from g ∗ into g , is a Lie algebrahomomorphism. Moreover, r ∗ + = − r − , and r + − r − = id. Put d = g ⊕ g (direct sum of two copies). The mapping g ∗ → d : X ( X + , X − ) , X ± = r ± X (3.1.5)is a Lie algebra embedding. Thus we may identify g ∗ with a Lie subalgebra in d .Naturally, embedding (3.1.5) extends to a homomorphism G ∗ → G × G, L ( L + , L − ) . We shall identify G ∗ with the corresponding subgroup in G × G . .2. QUANTIZATION OF POISSON–LIE GROUPS AND THE DEFINITION OF Q-W–ALGEBRAS In this section we introduce the main object of this book, q-W–algebras. We start by defining the relevant Poisson–Lie groups and their quantizations. We consider algebras defined over the ring B since later in our constructionthe restricted specialization of the quantum group U sh ( g ) defined over B will play the key role.Let g be a finite-dimensional complex semisimple Lie algebra, h ⊂ g its Cartan subalgebra. Let s ∈ W be anelement of the Weyl group W of the pair ( g , h ) and ∆ + the system of positive roots associated to s . Observe thatcocycle (2.6.10) equips g with the structure of a factorizable Lie bialgebra, where the scalar product is given by thesymmetric bilinear form. Using the identification End g ∼ = g ⊗ g the corresponding r–matrix may be represented as r s = P + − P − + κ s − s P h ′ , where P + , P − and P h ′ are the orthogonal projection operators onto the nilradical n + corresponding to ∆ + , theopposite nilradical n − , and h ′ , respectively, in the direct sum g = n + + h ′ + h ′⊥ + n − , and h ′⊥ is the orthogonal complement to h ′ in h with respect to the symmetric bilinear form.Let G be the connected simply connected semisimple Poisson–Lie group with the tangent Lie bialgebra ( g , g ∗ ), G ∗ the dual Poisson–Lie group.Observe that G is an algebraic group (see e.g. § r s + = P + + κ s − s P h ′ + 12 P h , r s − = − P − + κ s − s P h ′ − P h , where P h is the orthogonal projection operator onto h ⊂ g with respect to the symmetric bilinear form, and hencethe subspaces i ± and k ± defined by (3.1.3) coincide with the Borel subalgebras b ± in g corresponding to ∆ ± andtheir nilradicals n ± , respectively. Therefore every element ( L + , L − ) ∈ G ∗ may be uniquely written as( L + , L − ) = ( n + , n − )( h + , h − ) , (3.2.1)where n ± ∈ N ± , h + = exp (( κ s − s P h ′ + id ) x ) , h − = exp (( κ s − s P h ′ − id ) x ) , x ∈ h . In particular, G ∗ is a solvablesubgroup in G × G . In general G ∗ does not need to be algebraic.Our main object will be a certain algebra of functions on G ∗ , C [ G ∗ ]. This algebra may be explicitly describedas follows. Let π V be a finite-dimensional representation of G . Then matrix elements of π V ( L ± ) are well–definedfunctions on G ∗ , and C [ G ∗ ] is the subspace in C ∞ ( G ∗ ) generated by matrix elements of π V ( L ± ), where V runsthrough all finite–dimensional representations of G . The elements L ± ,V = π V ( L ± ) may be viewed as elements ofthe space C [ G ∗ ] ⊗ End V . For every two finite–dimensional g –modules V and W we denote r s + V W = ( π V ⊗ π W ) r s + ,where r s + is regarded as an element of g ⊗ g . Proposition 3.2.1. C [ G ∗ ] is a Poisson subalgebra in the Poisson algebra C ∞ ( G ∗ ) , the Poisson brackets of theelements L ± ,V are given by { L ± ,V , L ± ,W } = − r s ± V W , L ± ,V L ± ,W ] , { L − ,V , L + ,W } = − r s ± V W , L − ,V L + ,W ] , (3.2.2) where L ± ,V = L ± ,V ⊗ I W , L ± ,W = I V ⊗ L ± ,W , and I X is the unit matrix in X .Moreover, the map ∆ : C [ G ∗ ] → C [ G ∗ ] ⊗ C [ G ∗ ] dual to the multiplication in G ∗ , ∆( L ± ,Vij ) = X k L ± ,Vik ⊗ L ± ,Vkj , (3.2.3) is a homomorphism of Poisson algebras, and the map S : C [ G ∗ ] → C [ G ∗ ] , S ( L ± ,Vij ) = ( L ± ,V ) − ij is an antihomomorphism of Poisson algebras. CHAPTER 3. Q-W–ALGEBRAS
Remark 3.2.2.
Recall that a Poisson–Hopf algebra is a Poisson algebra which is also a Hopf algebra such thatthe comultiplication is a homomorphism of Poisson algebras and the antipode is an antihomomorphism of Poissonalgebras. According to Proposition 3.2.1 C [ G ∗ ] is a Poisson–Hopf algebra. Now we construct a quantization of the Poisson–Hopf algebra C [ G ∗ ]. For any finite rank representation π V : U s,res B ( g ) → V res , where V is a finite rank representation of U h ( g ), one can define an action of elements H i , i = 1 , . . . , l on V res by requiring that H i acts on weight vectors of weight λ by multiplication by λ ( H i ). Then fromthe definition of the R–matrix R s it follows that q L ± ,V given by q L − ,V = ( id ⊗ π V ) R s − = ( id ⊗ π V S s ) R s , q L + ,V = ( id ⊗ π V ) R s . are well–defined invertible elements of U sh ( g ) ⊗ End B ( V res ).If we fix a basis in V res , q L ± ,V may be regarded as matrices with matrix elements ( q L ± ,V ) ij being elements of U sh ( g ).We denote by C B [ G ∗ ] the B –Hopf subalgebra in U sh ( g ) generated by matrix elements of ( q L ± ,V ) ± , where V runs through all finite rank representation of U h ( g ).From the definition of the elements q L ± ,V and from formula (2.7.17) it follows that C B [ G ∗ ] is the B –subalgebrain U sh ( g ) generated by the elements q ± ( Y i − κ s − s P h ′ Y i ) , q ± ( − Y i − κ s − s P h ′ Y i ) , i = 1 , . . . , l, ˜ f β = (1 − q − β ) f β , ˜ e β =(1 − q β ) e β e hβ ∨ , β ∈ ∆ + (see Section 1.4 in [25] and Theorem 4.6 in [24] for a similar result in case of the quantumgroup associated to the standard bialgebra structure).From the Yang–Baxter equation for R s we get relations between q L ± ,V , R V W q L ± ,W q L ± ,V = q L ± ,V q L ± ,W R V W , (3.2.4) R V W q L + ,W q L − ,V = q L − ,V q L + ,W R V W . (3.2.5)By q L ± ,W , q L ± ,V we understand the following matrices in V res ⊗ W res with entries being elements of U sh ( g ) q L ± ,V = q L ± ,V ⊗ I W , q L ± ,W = I V ⊗ q L ± ,W , where I X is the unit matrix in X .From (2.5.2) we can obtain the action of the comultiplication on the matrices q L ± ,V :∆ s ( q L ± ,Vij ) = X k q L ± ,Vik ⊗ q L ± ,Vkj (3.2.6)and the antipode, S s ( q L ± ,Vij ) = ( q L ± ,V ) − ij . (3.2.7)Since R s = 1 ⊗ h ) relations (3.2.4) and (3.2.5) imply that the quotient algebra C B [ G ∗ ] / ( q dr − C B [ G ∗ ]is commutative, and one can equip it with a Poisson structure given by { x , x } = 1 dr [ a , a ] q dr − q dr − , (3.2.8)where a , a ∈ C B [ G ∗ ] reduce to x , x ∈ C B [ G ∗ ] / ( q dr − C B [ G ∗ ] mod ( q dr − C B [ G ∗ ] induce a comultiplication and an antipode in C B [ G ∗ ] / ( q dr − C B [ G ∗ ] compatible with the introduced Poisson structure, and the quotient C B [ G ∗ ] / ( q dr − C B [ G ∗ ] becomes a Poisson–Hopf algebra. Proposition 3.2.3.
The Poisson–Hopf algebra C B [ G ∗ ] / ( q dr − C B [ G ∗ ] is isomorphic to C [ G ∗ ] as a Poisson–Hopfalgebra.Proof. Denote by p : C B [ G ∗ ] → C B [ G ∗ ] / ( q dr − C B [ G ∗ ] = C [ G ∗ ] ′ the canonical projection, and let ˜ L ± ,V =( p ⊗ p V )( q L ± ,V ) ∈ C [ G ∗ ] ′ ⊗ End V , where p V : V res → V = V res / ( q dr − V res is the projection of V res onto thecorresponding module V over U res ( g ) = U s,res B ( g ) / ( q dr − U s,res B ( g ) equipped also with the natural action of h as described above. We denote by π V the corresponding representation of U ( g ). .2. QUANTIZATION OF POISSON–LIE GROUPS AND THE DEFINITION OF Q-W–ALGEBRAS ı : C [ G ∗ ] ′ → C [ G ∗ ] , ( ı ⊗ id ) ˜ L ± ,V = L ± ,V is a well–defined linear isomorphism. Indeed, consider, for instance, element ˜ L + ,V . From (2.6.11) it follows that˜ L + ,V = Q β exp [ p ((1 − q − β ) f β ) ⊗ π V ( X β )] ×× ( p ⊗ id ) exp hP li =1 hH i ⊗ π V (( κ s − s P h ′ + id ) Y i ) i . (3.2.9)On the other hand (3.2.1) implies that every element L + may be represented in the form L + = Y β exp[ b β X β ] exp " l X i =1 b i ( κ s − s P h ′ + id ) Y i , b i , b β ∈ C , and hence L + ,V = Q β exp [ b β π V ( X β )] exp hP li =1 b i π V (( κ s − s P h ′ + id ) Y i ) i . Comparing this with (3.2.9) and recalling the definition of ı we deduce that ı is a linear isomorphism. We have toprove that ı is an isomorphism of Poisson–Hopf algebras.Observe that R s = 1 ⊗ − hr s − (mod h ). Therefore from commutation relations (3.2.4), (3.2.5) it followsthat C [ G ∗ ] ′ is a commutative algebra, and the Poisson brackets of matrix elements ˜ L ± ,Vij (see (3.2.8)) are given by(3.2.2), where L ± ,V are replaced by ˜ L ± ,V . The factor dr in formula (3.2.8) normalizes the Poisson bracket in sucha way that bracket (3.2.8) is in agreement with (3.2.2).From (3.2.6) we also obtain that the action of the comultiplication on the matrices ˜ L ± ,V is given by (3.2.3),where L ± ,V are replaced by ˜ L ± ,V . This completes the proof.We shall call the map p : C B [ G ∗ ] → C B [ G ∗ ] / ( q dr − C B [ G ∗ ] = C [ G ∗ ] the quasiclassical limit.Now using the Hopf algebra C B [ G ∗ ] we shall define q-W–algebras. From the definition of the elements q L ± ,V itfollows that the matrix elements of q L ± ,V ± form Hopf subalgebras C B [ B ± ] ⊂ C B [ G ∗ ], and that C B [ G ∗ ] containsthe subalgebra C B [ N + ] generated by elements the ˜ f β = (1 − q − β ) f β , β ∈ ∆ + .Suppose that the positive root system ∆ + and its ordering are associated to s . Denote by C B [ M + ] the subalgebrain C B [ N + ] generated by the elements ˜ f β , β ∈ ∆ m + .The linear subspace of g generated by the root vectors X α ( X − α ), α ∈ ∆ m + is in fact a Lie subalgebra m + ⊂ g ( m − ⊂ g ). By definition ∆ m + ⊂ ∆ + , and hence m ± ⊂ n ± .Note that one can consider n + and m ± as Lie subalgebras in g ∗ via embeddings n + → g ∗ ⊂ g ⊕ g , x ( x, , m + → g ∗ ⊂ g ⊕ g , x ( x, , m − → g ∗ ⊂ g ⊕ g , x (0 , x ) , Using these embeddings the algebraic subgroups N + , M ± ⊂ G corresponding to the algebraic Lie subalgebras n + , m ± ⊂ g can be regarded as Lie subgroups in G ∗ corresponding to the Lie subalgebras n + , m ± ⊂ g ∗ . This waythe algebra C B [ N + ] becomes naturally a quantization of the algebra of regular functions on the subgroup N + ⊂ G ∗ ,and C B [ M + ] becomes a quantization of the algebra of regular functions on the subgroup M + ⊂ G ∗ in the sensethat p ( C B [ N + ]) = C [ N + ] and p ( C B [ M + ]) = C [ M + ]. Here and below for every algebraic variety V we denote by C [ V ] the algebra of regular functions on V .Note that M − can also be regarded as a subgroup in G ∗ corresponding to the Lie subalgebra m − ⊂ g ∗ .The following proposition gives the most important property of the subalgebra C B [ M + ] which plays the keyrole in the definition of q-W–algebras. Proposition 3.2.4.
The defining relations in the subalgebra C B [ M + ] for the generators ˜ f β = (1 − q − β ) f β , β ∈ ∆ m + = { β , . . . , β c } are of the form ˜ f α ˜ f β − q ( α,β )+ κ ( s − s P h ′∗ α,β ) ˜ f β ˜ f α = X α<δ <...<δ n <β C ( k , . . . , k n ) ˜ f k δ ˜ f k δ . . . ˜ f k n δ n , α < β, (3.2.10)4 CHAPTER 3. Q-W–ALGEBRAS where C ( k , . . . , k n ) ∈ B , the products ˜ f k β ˜ f k β . . . ˜ f k c β c form a B –basis of C B [ M + ] and the products ˜ f k β ˜ f k β . . . ˜ f k D β D form a B –basis of C B [ N + ] .If κ = 1 then for any k i ∈ B , i = 1 , . . . , l ′ the map χ sq : C B [ M + ] → B , χ sq ( ˜ f β ) = (cid:26) β
6∈ { γ , . . . , γ l ′ } k i β = γ i , (3.2.11) is a character of C B [ M + ] vanishing on the r.h.s. and on the l.h.s. of relations (3.2.10).Assume that ε d i = 1 , and ε = 1 if g is of type G . Suppose also that there exists n ∈ Z such that ε nd − = 1 .Let κ = nd . Then the algebra C ε [ M + ] = C B [ M + ] / ( q dr − ε dr ) C B [ M + ] , where ε dr is a root of ε of degree dr , isisomorphic to U sε ( m − ) , the elements f r = f r β . . . f r c β c , r i ∈ N , i = 1 , . . . d form a linear basis of U sε ( m − ) , and forany c i ∈ C , i = 1 , . . . , l ′ the map χ sε : U sε ( m − ) → C , χ sε ( f β ) = (cid:26) β
6∈ { γ , . . . , γ l ′ } c i β = γ i , (3.2.12) is a character of U sε ( m − ) .Proof. By Step 1, Theorem 21.1 in [26] the products ˜ f k β ˜ f k β . . . ˜ f k D β D form a B –basis of C B [ N + ].By Lemma 2.7.2 and Remark 2.7.3 any element of C B [ M + ] can be uniquely represented as a C ( q dr )–linearcombination of the elements ˜ f k β ˜ f k β . . . ˜ f k c β c . By the uniqueness of the Poincar´e–Birkhoff–Witt decomposition for C B [ N + ] established above the coefficients of this decomposition must belong to B .From (2.7.11) we also obtain commutation relations (3.2.10) with C ( k , . . . , k n ) ∈ C ( q dr ). As we alreadyproved the products ˜ f k β ˜ f k β . . . ˜ f k c β c form a B –basis of C B [ M + ]. Therefore the coefficients C ( k , . . . , k n ) in (3.2.10)belong to B .Assume that κ = 1. In order to prove that the map χ sq : C B [ M + ] → B defined by (3.2.11) is a character weshow that all relations (3.2.10) for ˜ f α , ˜ f β with α, β ∈ ∆ m + , which are obviously defining relations in the subalgebra C B [ M + ], belong to the kernel of χ sq . By definition the only generators of C B [ M + ] on which χ sq may not vanish are˜ f γ i , i = 1 , . . . , l ′ . By part (v) of Proposition 1.6.1 for any two roots α, β ∈ ∆ m + such that α < β the sum α + β cannot be represented as a linear combination P tk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i t < β . Hence forany two roots α, β ∈ ∆ m + such that α < β the value of the map χ sq on the right hand side of the correspondingcommutation relation (3.2.10) is equal to zero.Therefore it suffices to prove that χ sq ( ˜ f γ i ˜ f γ j − q ( γ i ,γ j )+( s − s P h ′∗ γ i ,γ j ) ˜ f γ j ˜ f γ j ) = k i k j (1 − q ( γ i ,γ j )+( s − s P h ′∗ γ i ,γ j ) ) = 0 , i < j. The last identity holds provided ( γ i , γ j ) + ( s − s P ∗ h ′ γ i , γ j ) = 0 for i < j which is indeed the case by Lemma 2.7.1.Assume now that ε d i = 1, and ε = 1 if g is of type G . Suppose also that there exists n ∈ Z such that ε nd − = 1. Let κ = nd .Under these conditions imposed on ε the map C ε [ M + ] → U sε ( m − ), ˜ f α (1 − q − α ) f α , α ∈ ∆ m + is obviously analgebra isomorphism.By Lemma 2.7.2 and Remark 2.7.3 the elements f r = f r β . . . f r c β c , r i ∈ N , i = 1 , . . . d form a linear basis of U sε ( m − ).From (2.7.11) we obtain the following commutation relations f α f β − ε ( α,β )+ nd ( s − s P h ′∗ α,β ) f β f α = X α<δ <...<δ n <β D ( k , . . . , k n ) f k δ f k δ . . . f k n δ n , α < β, (3.2.13)where D ( k , . . . , k n ) ∈ C .In order to show that the map χ sε : U sε ( m − ) → C defined by (3.2.12) is a character we verify that all relations(3.2.13) for f α , f β with α, β ∈ ∆ m + , which are obviously defining relations in the subalgebra U sε ( m − ), belong tothe kernel of χ sε . By definition the only generators of U sε ( m − ) on which χ sε may not vanish are f γ i , i = 1 , . . . , l ′ . Bypart (v) of Proposition 1.6.1 for any two roots α, β ∈ ∆ m + such that α < β the sum α + β cannot be represented asa linear combination P tk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i t < β . Hence for any two roots α, β ∈ ∆ m + such that α < β the value of the map χ sε on the right hand side of the corresponding commutation relation (3.2.10)is equal to zero. .2. QUANTIZATION OF POISSON–LIE GROUPS AND THE DEFINITION OF Q-W–ALGEBRAS χ sε ( f γ i f γ j − ε ( γ i ,γ j )+ nd ( s − s P h ′∗ γ i ,γ j ) f γ j f γ j ) = c i c j (1 − ε ( γ i ,γ j )+ nd ( s − s P h ′∗ γ i ,γ j ) ) = 0 , i < j. By Lemma 2.7.1 ( s − s P ∗ h ′ γ i , γ j ) = − ( γ i , γ j ) for i < j , and hence χ sε ( f γ i f γ j − ε ( γ i ,γ j )+ nd ( s − s P h ′∗ γ i ,γ j ) f γ j f γ j ) = c i c j (1 − ε ( γ i ,γ j )(1 − nd ) ) = 0for i < j as by the assumption ε nd − = 1. This completes the proof.Next we define the algebra C B [ G ∗ ] and discuss its properties. For any finite rank representation V of U h ( g ),let q L V = q L − ,V − q L + ,V = ( id ⊗ π V ) R s R s . Let C B [ G ∗ ] be the B –subalgebra in C B [ G ∗ ] generated by the matrixentries of q L V , where V runs over all finite rank representations of U sh ( g ). From the definition of R s we have R s R s = Q β P ∞ k =0 q k ( k +1)2 β [(1 − q − β ) e kβ e − hkκ s − s P h ′ β ∨ ⊗ f ( k ) β ] ×× exp h h P li =1 Y i ⊗ H i i Q β P ∞ k =0 q k ( k +1)2 β [(1 − q − β ) e hkκ s − s P h ′ β ∨ − hkβ ∨ f kβ ⊗ e ( k ) β q kβ ∨ ] . (3.2.14)Using this formula one immediately checks that actually C B [ G ∗ ] ⊂ U s A ( g ) ∩ C B [ G ∗ ].Define the right adjoint action of U sq ( g ) on U sq ( g ) by the formulaAd x ( w ) = S s ( x ) wx , (3.2.15)and the left adjoint action of U sq ( g ) on U sq ( g ) byAd ′ x ( w ) = x wS s ( x ) , (3.2.16)where we use the abbreviated Sweedler notation for the coproduct ∆ s ( x ) = x ⊗ x , x, w ∈ U sq ( g ).Let C B [ G ] be the restricted Hopf algebra dual to U s,res B ( g ) which is generated by the matrix elements of finiterank representations of U s,res B ( g ) of the form V res , where V is a finite rank representation of U sh ( g ). Action (3.2.16)induces a left adjoint action Ad of U s,res B ( g ) on C B [ G ] defined by(Ad xf )( w ) = f (Ad ′ x ( w )) , f ∈ C B [ G ] , x, w ∈ U s,res B ( g ) . (3.2.17)One can also equip finite rank representations of U s,res B ( g ) of the form V res , where V is a finite rank represen-tation of U sh ( g ), with a natural action of C B [ G ∗ ], where the elements q ± ( Y i − κ s − s P h ′ Y i ) , q ± ( − Y i − κ s − s P h ′ Y i ) , i = 1 , . . . l act on a weight vector v λ of weight λ by multiplication by the elements of q ± (( Y i ,λ ∨ ) − κ ( s − s P h ′ Y i ,λ ∨ )) , q ± ( − ( Y i ,λ ∨ ) − κ ( s − s P h ′ Y i ,λ ∨ )) ∈ B , i = 1 , . . . l, respectively, and all the other generators of C B [ G ∗ ] which belong to U s,res B ( g ) act in a natural way. Thereforeadjoint action (3.2.17) can be extended to an action of C B [ G ∗ ], where elements x ∈ C B [ G ∗ ] act by the same formula(3.2.17).Note that by Lemma 2.2 in [53] Ad x ( wz ) = Ad x ( w )Ad x ( z ) . (3.2.18)For ε ∈ C ∗ we define C ε [ G ] = C B [ G ] / ( q dr − ε dr ) C B [ G ], C ε [ G ∗ ] = C B [ G ∗ ] / ( q dr − ε dr ) C B [ G ∗ ], where ε dr isa root of ε of degree dr . Proposition 3.2.5. If ε is not a root of unity the algebra C ε [ G ∗ ] can be identified with the Ad locally finite part U sε ( g ) fin of U sε ( g ) , U sε ( g ) fin = { x ∈ U sε ( g ) : dim(Ad U sε ( g )( x )) < + ∞} , where the adjoint action of the algebra U sε ( g ) on itself is defined by formula (3.2.15).Moreover, the map C B [ G ] → C B [ G ∗ ] , f ( id ⊗ f )( R s R s ) (3.2.19) is an isomorphism of U s,res B ( g ) and C B [ G ∗ ] –modules with respect to the adjoint actions Ad defined by (3.2.15) and(3.2.17), respectively. In particular, C B [ G ∗ ] is stable under the adjoint action of U s,res B ( g ) and C B [ G ∗ ] . CHAPTER 3. Q-W–ALGEBRAS
Proof.
All statements except for the isomorphism C ε [ G ∗ ] = U sε ( g ) fin follow from the definitions and the discussionabove.It remains to establish the isomorphism C ε [ G ∗ ] = U sε ( g ) fin . Let V ω i , i = 1 , . . . , l be the finite rank representationof U h ( g ) with highest weight ω i , i = 1 , . . . , l . From formula (3.2.14) and from the definition of q L V = ( id ⊗ π V ) R s R s it follows that the matrix element ( id ⊗ v ∗ i ) R s R s ( id ⊗ v i ) of q L V ωi corresponding to the highest weight vector v i of V resω i and to the lowest weight vector v ∗ i ∈ V resω i ∗ of the dual representation V resω i ∗ , normalized in such a way that v ∗ i ( v i ) = 1, coincides with L i . This implies that L i are elements of the algebra C ε [ G ∗ ] ⊂ U sε ( g ) as well.Denote by H ⊂ C ε [ G ∗ ] ⊂ U sε ( g ) the subalgebra generated by the elements L i ∈ C ε [ G ∗ ], i = 1 , . . . , l . Similarly toTheorem 7.1.6 and Lemma 7.1.16 in [52] one can obtain that U sε ( g ) fin = Ad U sε ( g ) H . Since C ε [ G ∗ ] is stable underthe adjoint action we have an inclusion, U sε ( g ) fin ⊂ C ε [ G ∗ ].On the other hand the adjoint action of U sε ( g ) on C ε [ G ] is locally finite by the definition of C ε [ G ] and of theadjoint action. Using isomorphism (3.2.19) we deduce that the adjoint action of U sε ( g ) on C ε [ G ∗ ] is locally finiteas well. Hence C ε [ G ∗ ] ⊂ U sε ( g ) fin , and C ε [ G ∗ ] = U sε ( g ) fin .Now we are ready to define q-W–algebras. In the rest of this section we assume that κ = 1. Denote by I B theleft ideal in C B [ G ∗ ] generated by the kernel of χ sq , and by ρ χ sq the canonical projection C B [ G ∗ ] → C B [ G ∗ ] /I B = Q ′B .Let Q B be the image of C B [ G ∗ ] under the projection ρ χ sq .We shall need the following property of the algebras C B [ G ∗ ], C B [ G ∗ ] and C B [ G ] and of C B [ G ∗ ] /I B and Q B . Proposition 3.2.6. C B [ G ∗ ] , C B [ G ∗ ] , C B [ G ] , C B [ G ∗ ] /I B and Q B are free B –modules.Proof. Recall that C B [ G ∗ ] is the B –subalgebra in U sh ( g ) generated by the elements q ± ( Y i − κ s − s P h ′ Y i ) , q ± ( − Y i − κ s − s P h ′ Y i ) , i = 1 , . . . , l , ˜ f β = (1 − q − β ) f β , ˜ e β = (1 − q β ) e β e hβ ∨ , β ∈ ∆ + . The subalgebra of C B [ G ∗ ] generated by the elements q ± ( Y i − κ s − s P h ′ Y i ) , q ± ( − Y i − κ s − s P h ′ Y i ) , i = 1 , . . . , l is in turn a subalgebra of the B –subalgebra U ′B ( h ) ⊂ U sh ( g ) gener-ated by the elements U i = q dr Y i , U − i , i = 1 , . . . , l . The last algebra is obviously B –free with a basis consisting ofthe products U n . . . U n l l , n , . . . , n l ∈ Z . Since B is a principal ideal domain the subalgebra in U ′B ( h ) generated bythe elements q ± ( Y i − κ s − s P h ′ Y i ) , q ± ( − Y i − κ s − s P h ′ Y i ) , i = 1 , . . . , l is also B –free by Theorem 6.5 in [87]. Denote by V i , i ∈ N elements of some B –basis of this subalgebra.Now from Step 1 of the proof of the Theorem in Section 12.1 in [26] it follows that the elements˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k β with n j , k j , i ∈ N , j = 1 , . . . , D form a B –basis in C B [ G ∗ ]. Using this basis and the definition of C B [ G ∗ ] /I B one immediately sees that the classes of the elements ˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 with n j , k m , i ∈ N , j = 1 , . . . , D , m = c + 1 , . . . , D form a B –basis in C B [ G ∗ ] /I B .Since B is a principal ideal domain the subalgebra C B [ G ∗ ] ⊂ C B [ G ∗ ] is also B –free by Theorem 6.5 in [87], andisomorphism (3.2.19) implies that C B [ G ] is B –free.Finally, since B is a principal ideal domain the B –submodule Q B ⊂ C B [ G ∗ ] /I B is B –free. Remark 3.2.7.
The fact that the algebra C q [ G ] = C B [ G ] ⊗ B C ( q dr ) is C ( q dr ) –free can be proved in a morestraightforward way. Namely, according to the results of Section 7 in [69] C q [ G ] is generated by the matrix elementsof finite-dimensional U sq ( g ) ≃ U q ( g ) –modules. By Theorem 10.1.7 in [18] all such modules are completely reducible,the irreducible components being highest weight modules V qλ generated by highest weight vectors of integral dominantweights λ ∈ P + . Therefore C q [ G ] ≃ M λ ∈ P + V qλ ∗ ⊗ V qλ . This decomposition and the fact that all V qλ are C ( q dr ) –free imply that C q [ G ] is C ( q dr ) –free. Now observe that we have an inclusion [ C B [ M + ] , Ker χ sq ] ⊂ Ker χ sq . Using this inclusion, formula (3.2.15), thefact that ∆ s ( C B [ M + ]) ⊂ C B [ B + ] ⊗ C B [ M + ] (see formula (2.7.12)) we deduce that the adjoint action of C B [ M + ] on C B [ G ∗ ] induces an action on Q ′B and on Q B which we also call the adjoint action and denote it by Ad.Let B ε s be the trivial representation of C B [ M + ] given by the counit. Consider the space W s B ( G ) of Ad–invariantsin Q B , W s B ( G ) = Hom C B [ M + ] ( B ε s , Q B ) . (3.2.20) .2. QUANTIZATION OF POISSON–LIE GROUPS AND THE DEFINITION OF Q-W–ALGEBRAS Proposition 3.2.8. W s B ( G ) is isomorphic to the subspace of all v + I B ∈ Q B such that mv ∈ I B (or [ m, v ] ∈ I B )in C B [ G ∗ ] for any m ∈ I B , where v ∈ C B [ G ∗ ] is any representative of v + I B ∈ Q B .Multiplication in C B [ G ∗ ] induces a multiplication on the space W s B ( G ) .Proof. For the proof we shall firstly derive the formula for the adjoint action of the generators ˜ f β . From (2.7.12)using linear independence of weight components and the fact that C B [ B + ] is a Hopf algebra we obtain∆ s ( ˜ f β k ) = G − β k ⊗ ˜ f β k + ˜ f β k ⊗ X i ˜ y i ⊗ ˜ x i , (3.2.21)where G β = e hκ s − s P h ′ β ∨ − hβ ∨ ∈ C B [ B + ] , ˜ y i = e − hκ s − s P h ′ γ ∨ xi + hγ ∨ xi ˜ y i , ˜ y i ∈ C B ([ − β k +1 , − β D ]) , ˜ x i ∈ C B ([ − β , − β k − ]) , ˜ y i , ˜ x i belong to weight components and have non-zero weights, γ x i is the weight of ˜ x i , C B ([ − β k +1 , − β D ])( C B ([ − β , − β k − ]))is the subalgebra in C B [ N + ] generated by ˜ f β k +1 , . . . , ˜ f β D ( ˜ f β , . . . , ˜ f β k − ).By (3.2.21) we also have S s ( ˜ f β ) = − G β ˜ f β − X i S s (˜ y i )˜ x i . Combining this formula with (3.2.21) and using (3.2.15) we deduceAd ˜ f β k w = − G β k [ ˜ f β k , w ] − X i S s (˜ y i )[˜ x i , w ]The induced action of the elements ˜ f β k ∈ C B [ M + ], β k ∈ ∆ m + , on Q ′B takes the formAd ˜ f β k v = − G β k ( ˜ f β k − χ sq ( ˜ f β k )) v − X i S s (˜ y i )(˜ x i − χ sq (˜ x i )) v, β k ∈ ∆ m + . (3.2.22)We have to show that W s B ( G ) is isomorphic to the subspace of all v ∈ Q B ⊂ Q ′B such that mv = 0 in Q ′B forany m ∈ I B .The left ideal I B is generated by elements x − χ sq ( x ), x ∈ C B [ M + ]. Therefore by (3.2.22) if for some v ∈ Q B mv = 0 in Q ′B for any m ∈ I B then v is invariant with respect to the adjoint action of all generators ˜ f β k of C B [ M + ],and hence v ∈ W s B ( G ).Now assume that v ∈ W s B ( G ). We shall prove that mv = 0 in Q ′B for any m ∈ I B .Since the left ideal I B is generated by elements x − χ sq ( x ), x ∈ C B [ M + ] it suffices to show that ( x − χ sq ( x )) v = 0 forany x ∈ C B [ M + ]. We shall prove this statement by induction using the subalgebras C B ([ − β , − β k ]), k = 1 , . . . , c ,so that C B ([ − β , − β c ]) = C B [ M + ].Observe that β is a simple root, and hence from (3.2.22) we obtain0 = Ad ˜ f β v = − G β ( ˜ f β − χ sq ( ˜ f β )) v. Since the element G β ∈ C B [ N + ] is invertible this implies( ˜ f β − χ sq ( ˜ f β )) v = 0 , i.e. ( x − χ sq ( x )) v = 0 in Q ′B for any x ∈ C B ([ − β , − β ]) as the subalgebra C B ([ − β , − β ]) is generated by ˜ f β .Now suppose that for some 1 < k ≤ c ( x − χ sq ( x )) v = 0 in Q ′B for any x ∈ C B ([ − β , − β k − ]). Then from (3.2.22)we obtain 0 = Ad ˜ f β k v = − G β k ( ˜ f β k − χ sq ( ˜ f β k )) v − X i S s (˜ y i )(˜ x i − χ sq (˜ x i )) v = − G β k ( ˜ f β k − χ sq ( ˜ f β k )) v CHAPTER 3. Q-W–ALGEBRAS since ˜ x i ∈ C B ([ − β , − β k − ]). The previous identity and the fact that the element G β k ∈ C B [ N + ] is invertible yield( ˜ f β k − χ sq ( ˜ f β k )) v = 0 . Now observe that by Proposition 3.2.4 any element x of C B ([ − β , − β k ]) can be uniquely represented in the form x = ˜ f β k z + z ′ , where z, z ′ ∈ C B ([ − β , − β k − ]). Therefore xv = ( ˜ f β k z + z ′ ) v = ( ˜ f β k χ sq ( z ) + χ sq ( z ′ )) v = ( χ sq ( ˜ f β k ) χ sq ( z ) + χ sq ( z ′ )) v = χ sq ( ˜ f β k z + z ′ ) v = χ sq ( x ) v. This establishes the induction step and proves the first claim of this proposition.From the second description of the space W s B ( G ) it follows that if v , v ∈ C B [ G ∗ ] are any representatives ofelements v + I q , v + I q ∈ W s B ( G ) then the formula( v + I q )( v + I q ) = v v + I q defines a multiplication in W s B ( G ). This completes the proof.The space W s B ( G ) equipped with the multiplication opposite to the one defined in the previous proposition iscalled the q-W–algebra associated to (the conjugacy class of) the Weyl group element s ∈ W .In conclusion we obtain some results on the structure of Q B . Consider the Lie algebra L B associated to theassociative algebra C B [ M + ], i.e. L B is the Lie algebra which is isomorphic to C B [ M + ] as a linear space, and theLie bracket in L B is given by the usual commutator of elements in C B [ M + ].Define an action of the Lie algebra L B on the space C B [ G ∗ ] /I B : m · ( x + I B ) = ρ χ sq ([ m, x ]) . (3.2.23)where x ∈ C B [ G ∗ ] is any representative of x + I B ∈ C B [ G ∗ ] /I B and m ∈ C B [ M + ]. The algebra W s B ( G ) can beregarded as the intersection of the space of invariants with respect to action (3.2.23) with the subspace Q B ⊂ C B [ G ∗ ] /I B .Note also that since χ sq is a character of C B [ M + ] the ideal I B is stable under the action of C B [ M + ] on C B [ G ∗ ]by commutators.Denote by B χ sq the rank one representation of the algebra C B [ M + ] defined by the character χ sq . Using thedescription of the algebra W s B ( G ) in terms of action (3.2.23) and the isomorphism C B [ G ∗ ] /I B = C B [ G ∗ ] ⊗ C B [ M + ] B χ sq one can also define the algebra W s B ( G ) as the intersection W s B ( G ) = Hom C B [ M + ] ( B χ sq , C B [ G ∗ ] ⊗ C B [ M + ] B χ sq ) ∩ Q B . Using Frobenius reciprocity we also haveHom C B [ M + ] ( B χ sq , C B [ G ∗ ] ⊗ C B [ M + ] B χ sq ) = End C B [ G ∗ ] ( C B [ G ∗ ] ⊗ C B [ M + ] B χ sq ) . Hence the algebra W s B ( G ) acts on the space C B [ G ∗ ] ⊗ C B [ M + ] B χ sq from the right by operators commuting with thenatural left C B [ G ∗ ]–action on C B [ G ∗ ] ⊗ C B [ M + ] B χ sq . By the definition of W s B ( G ) this action preserves Q B and bythe above presented arguments it commutes with the natural left C B [ G ∗ ]–action on Q B .Thus Q B is a C B [ G ∗ ]– W s B ( G ) bimodule equipped also with the adjoint action of C B [ M + ]. By (3.2.18) the adjointaction satisfies Ad x ( yv ) = Ad x ( y )Ad x ( v ) , x ∈ C B [ M + ] , y ∈ C B [ G ∗ ] , v ∈ Q B , (3.2.24)and ∆ s ( x ) = x ⊗ x .Denote by 1 ∈ Q B the image of the element 1 ∈ C B [ G ∗ ] in the quotient Q B under the canonical projection C B [ G ∗ ] → Q B . Obviously 1 is the generating vector for Q B as a module over C B [ G ∗ ]. Using formula (3.2.24) andrecalling that Q B is a C B [ G ∗ ]– W s B ( G ) bimodule, for x ∈ C B [ M + ] , y ∈ C B [ G ∗ ], and for a representative w ∈ C ε [ G ∗ ]of an element w + I B ∈ W s B ( G ) we haveAd x ( wy
1) = Ad x ( yw
1) = Ad x ( y )Ad x ( w
1) == Ad x ( y ) ε s ( x ) w x ( y ) w w Ad x ( y . Since Q B is generated by the vector 1 over C B [ G ∗ ] the last relation implies that the action of W s B ( G ) on Q B commutes with the adjoint action.We can summarize the results of the discussion above in the following proposition. .3. POISSON REDUCTION Proposition 3.2.9.
The space Q B is naturally equipped with the structure of a left C B [ G ∗ ] –module, a right C B [ M + ] –module via the adjoint action and a right W s B ( G ) –module in such a way that the left C B [ G ∗ ] –action and the right C B [ M + ] –action commute with the right W s B ( G ) –action and compatibility condition (3.2.24) is satisfied. Finally we remark that by specializing q to a particular value ε ∈ C , ε = 0, one can define a complex associativealgebra C ε [ G ∗ ] = C B [ G ∗ ] / ( q dr − ε dr ) C B [ G ∗ ], its subalgebra C ε [ M + ] with a nontrivial character χ sε and thecorresponding W–algebra W sε ( G ) = Hom C ε [ M + ] ( C ε s , Q ε ) , (3.2.25)where C ε s is the trivial representation of the algebra C ε [ M + ] induced by the counit, Q ε = Q B /Q B ( q dr − ε dr ).Obviously, for generic ε we have W sε ( G ) = W s B ( G ) / ( q dr − ε dr ) W s B ( G ). In this section we recall basic facts on Poisson reduction. They will be used in the next section to describe Poissonq-W–algebras as reduced Poisson algebras.Let
M, B, B ′ be Poisson manifolds. Two Poisson surjections M π ′ ւ π ց B ′ B form a dual pair if the pullback π ′ ∗ C ∞ ( B ′ ) is the centralizer of π ∗ C ∞ ( B ) in the Poisson algebra C ∞ ( M ). In thiscase the sets B ′ b = π ′ (cid:0) π − ( b ) (cid:1) , b ∈ B are Poisson submanifolds in B ′ called reduced Poisson manifolds.Fix an element b ∈ B . Then the algebra of functions C ∞ ( B ′ b ) may be described as follows. Let I b be the idealin C ∞ ( M ) generated by elements π ∗ ( f ) , f ∈ C ∞ ( B ) , f ( b ) = 0. Denote M b = π − ( b ). Then the algebra C ∞ ( M b )is simply the quotient of C ∞ ( M ) by I b . Denote by P b : C ∞ ( M ) → C ∞ ( M ) /I b = C ∞ ( M b ) the canonical projectiononto the quotient. Lemma 3.3.1.
Suppose that the map f f ( b ) is a character of the Poisson algebra C ∞ ( B ) . Then one can definean action of the Poisson algebra C ∞ ( B ) on the space C ∞ ( M b ) by f · ϕ = P b ( { π ∗ ( f ) , ˜ ϕ } ) , (3.3.1) where f ∈ C ∞ ( B ) , ϕ ∈ C ∞ ( M b ) and ˜ ϕ ∈ C ∞ ( M ) is a representative of ϕ in C ∞ ( M ) such that P b ( ˜ ϕ ) = ϕ .Moreover, C ∞ ( B ′ b ) is the subspace of invariants in C ∞ ( M b ) with respect to this action.Proof. Let ϕ ∈ C ∞ ( M b ). Choose a representative ˜ ϕ ∈ C ∞ ( M ) such that P b ( ˜ ϕ ) = ϕ . Since the map f f ( b ) is acharacter of the Poisson algebra C ∞ ( B ), Hamiltonian vector fields of functions π ∗ ( f ) , f ∈ C ∞ ( B ) are tangent tothe submanifold M b . Therefore the right hand side of (3.3.1) only depends on ϕ but not on the representative ˜ ϕ ,and hence formula (3.3.1) defines an action of the Poisson algebra C ∞ ( B ) on the space C ∞ ( M b ).Using the definition of the dual pair we obtain that ϕ = π ′∗ ( ψ ) for some ψ ∈ C ∞ ( B ′ b ) if and only if P b ( { π ∗ ( f ) , ˜ ϕ } ) = 0 for every f ∈ C ∞ ( B ).Finally we obtain that C ∞ ( B ′ b ) is exactly the subspace of invariants in C ∞ ( M b ) with respect to action (3.3.1). Definition 3.3.2.
The algebra C ∞ ( B ′ b ) is called a reduced Poisson algebra. We also denote it by C ∞ ( M b ) C ∞ ( B ) . Remark 3.3.3.
Note that the description of the algebra C ∞ ( M b ) C ∞ ( B ) obtained in Lemma 3.3.1 is independent ofboth the manifold B ′ and the projection π ′ . Observe also that the Hamiltonian vector fields of functions π ∗ ( f ) , f ∈ C ∞ ( B ) are tangent to M b , and hence the reduced space B ′ b may be identified with across–section of the action ofthe Poisson algebra C ∞ ( B ) on M b by Hamiltonian vector fields in the case when this action is free. In particular,in this case B ′ b may be regarded as a submanifold in M b . In the case when the map f f ( b ) is a character of the Poisson algebra C ∞ ( B ) the Poisson structure onthe algebra C ∞ ( B ′ b ) can be explicitly described as follows. Let ϕ , ϕ ∈ C ∞ ( M b ) C ∞ ( B ) . Choose representatives˜ ϕ , ∈ C ∞ ( M ) such that P b ( ˜ ϕ , ) = ϕ , . Then { ϕ , ϕ } = { ˜ ϕ , ˜ ϕ } mod I b . (3.3.2)0 CHAPTER 3. Q-W–ALGEBRAS
By Lemma 3.3.1 the class in C ∞ ( M ) /I b = C ∞ ( M b ) of the function in right hand side of this formula is C ∞ ( B )–invariant and independent of the choice of the representatives ˜ ϕ , ∈ C ∞ ( M ).An important example of dual pairs is provided by Poisson group actions. Recall that a (local) left Poissongroup action of a Poisson–Lie group A on a Poisson manifold M is a (local) left group action A × M → M whichis also a Poisson map (as usual, we suppose that A × M is equipped with the product Poisson structure).If the space M/A is a smooth manifold, there exists a unique Poisson structure on
M/A such that the canonicalprojection M → M/A is a Poisson map.Let a be the Lie algebra of A . Denote by h· , ·i the canonical paring between a ∗ and a . A map µ : M → A ∗ iscalled a moment map for a (local) left Poisson group action M × A → M if L b X ϕ = h µ ∗ ( θ A ∗ ) , X i ( ξ ϕ ) , (3.3.3)where θ A ∗ is the universal left–invariant Maurer–Cartan form on A ∗ , X ∈ a , b X is the corresponding vector field on M and ξ ϕ is the Hamiltonian vector field of ϕ ∈ C ∞ ( M ). Proposition 3.3.4. ([67], Theorem 4.9)
Let A × M → M be a left (local) Poisson group action of a Poisson–Liegroup A on a Poisson manifold M with moment map µ : M → A ∗ . Denote by Π A ∗ the Poisson tensor of A ∗ . Thenthere exists a right invariant bivector field Λ on A ∗ such that Π µ = Π A ∗ + Λ is a Poisson tensor on A ∗ and themap µ : M → A ∗ µ is Poisson, where A ∗ µ the manifold A ∗ equipped with the Poisson structure associated to Π µ . From the definition of the moment map it follows that if
M/A is a smooth manifold then the canonical projection M → M/A and the moment map µ : M → A ∗ form a dual pair.The main example of Poisson group actions is the so–called dressing action. The dressing action may bedescribed as follows. Proposition 3.3.5.
Let G be a connected simply connected Poisson–Lie group with factorizable tangent Lie bial-gebra, G ∗ the dual group. Then there exists a unique left local Poisson group action G × G ∗ → G ∗ , (( L + , L − ) , g ) g ◦ ( L + , L − ) , such that the identity mapping µ : G ∗ → G ∗ is the moment map for this action.Moreover, let q : G ∗ → G be the map defined by q ( L + , L − ) = L − − L + . Then q ( g ◦ ( L + , L − )) = gL − L − g − . (3.3.4)The notion of Poisson group actions may be generalized as follows. Let A × M → M be a Poisson group actionof a Poisson–Lie group A on a Poisson manifold M . A subgroup K ⊂ A is called admissible if the set C ∞ ( M ) K of K -invariants is a Poisson subalgebra in C ∞ ( M ). If space M/K is a smooth manifold, we may identify the algebras C ∞ ( M/K ) and C ∞ ( M ) K . Hence there exists a Poisson structure on M/K such that the canonical projection M → M/K is a Poisson map.
Proposition 3.3.6. ([90], Theorem 6; [67], § Let ( a , a ∗ ) be the tangent Lie bialgebra of a Poisson–Lie group A . A connected Lie subgroup K ⊂ A with Lie algebra k ⊂ a is admissible if the annihilator k ⊥ of k in a ∗ is a Liesubalgebra k ⊥ ⊂ a ∗ . We shall need the following particular example of dual pairs arising from Poisson group actions.Let A × M → M be a left (local) Poisson group action of a Poisson–Lie group A on a manifold M . Supposethat this action possesses a moment map µ : M → A ∗ . Let K be an admissible subgroup in A . Denote by k theLie algebra of K . Assume that k ⊥ ⊂ a ∗ is a Lie subalgebra in a ∗ . Suppose also that there is a splitting a ∗ = t + k ⊥ (direct sum of vector spaces), and that t is a Lie subalgebra in a ∗ . Then the linear space k ∗ is naturally identifiedwith t .Assume that A ∗ = T K ⊥ as a manifold, where K ⊥ , T are the Lie subgroups of A ∗ corresponding to the Liesubalgebras k ⊥ , t ⊂ a ∗ , respectively. For any a ∗ = tk ⊥ ∈ A ∗ with k ⊥ ∈ K ⊥ , t ∈ T denote π K ⊥ ( a ∗ ) = k ⊥ , π T ( a ∗ ) = t . This defines maps π K ⊥ : A ∗ → K ⊥ , π T : A ∗ → T . .3. POISSON REDUCTION Proposition 3.3.7.
Suppose that for any k ⊥ ∈ K ⊥ the transformation t → t , (3.3.5) t (Ad( k ⊥ ) t ) t , where the subscript t stands for the t –component with respect to the decomposition a ∗ = t + k ⊥ , is invertible.Define a map µ : M → T by µ = π T µ. Then(i) µ ∗ ( C ∞ ( T )) is a Poisson subalgebra in C ∞ ( M ) , and hence one can equip T with a Poisson structure suchthat µ : M → T is a Poisson map.(ii) Moreover, the algebra C ∞ ( M ) K is the centralizer of µ ∗ ( C ∞ ( T )) in the Poisson algebra C ∞ ( M ) . Inparticular, if M/K is a smooth manifold the maps M π ւ µ ց ,M/K T (3.3.6) form a dual pair.Proof. (i) We claim that multiplication in A ∗ gives rise to a right Poisson group action A ∗ µ × A ∗ → A ∗ µ . Indeed, for g ∈ A ∗ denote by l g , r g the left (right) translation by g on A ∗ . By the definition of Π µ Π µ ( gh ) = Π A ∗ ( gh ) + Λ( gh ) = l g ∗ Π A ∗ ( h ) + r h ∗ Π A ∗ ( g ) + r h ∗ Λ( g ) = l g ∗ Π A ∗ ( h ) + r h ∗ Π µ ( g ) , (3.3.7)where we used the fact that Π A ∗ ( gh ) = l g ∗ Π A ∗ ( h ) + r h ∗ Π A ∗ ( g ) as A ∗ is a Poisson–Lie group and that Λ is rightinvariant. Identity (3.3.7) is equivalent to the fact that multiplication in A ∗ gives rise to a right Poisson groupaction A ∗ µ × A ∗ → A ∗ µ .Since k ⊂ a is a Lie subalgebra and k ⊥⊥ = k the subgroup K ⊥ ⊂ A ∗ is admissible. Therefore restricting theaction A ∗ µ × A ∗ → A ∗ µ to K ⊥ we deduce that C ∞ ( A ∗ µ ) K ⊥ is a Poisson subalgebra in C ∞ ( A ∗ µ ), where the action of K ⊥ is induced by the action of K ⊥ ⊂ A ∗ on A ∗ by right translations.Now recall that A ∗ = T K ⊥ as a manifold, and hence C ∞ ( A ∗ µ ) K ⊥ = C ∞ ( T ). Thus T naturally becomes aPoisson manifold and the map π T : A ∗ µ → T becomes Poisson. We deduce that the map µ = π T µ is Poisson as thecomposition of the Poisson maps µ : M → A ∗ µ and π T : A ∗ µ → T .(ii) By the definition of the moment map we have L b X ϕ = h µ ∗ ( θ A ∗ ) , X i ( ξ ϕ ) , (3.3.8)where X ∈ a , b X is the corresponding vector field on M and ξ ϕ is the Hamiltonian vector field of ϕ ∈ C ∞ ( M ).Since A ∗ = T K ⊥ the pullback of the left–invariant Maurer–Cartan form µ ∗ ( θ A ∗ ) may be represented as follows µ ∗ ( θ A ∗ ) = Ad ( π K ⊥ µ ) − ( µ ∗ θ T ) + ( π K ⊥ µ ) ∗ θ K ⊥ , where ( π K ⊥ µ ) ∗ θ K ⊥ ∈ k ⊥ .Now let X ∈ k . Then h ( π K ⊥ µ ) ∗ θ K ⊥ ) , X i = 0 and formula (3.3.8) takes the form L b X ϕ = h Ad ( π K ⊥ µ ) − ( µ ∗ θ T ) , X i ( ξ ϕ ) == h Ad ( π K ⊥ µ ) − ( θ T ) , X i ( µ ∗ ( ξ ϕ )) . (3.3.9)Since by the assumption transformation (3.3.5) is invertible, L b X ϕ = h Ad ( π K ⊥ µ ) − ( θ T ) , X i ( µ ∗ ( ξ ϕ )) = 0 for any X ∈ k if and only if µ ∗ ( ξ ϕ ) = 0. Thus the function ϕ ∈ C ∞ ( M ) is K –invariant if and only if { ϕ, µ ∗ ( ψ ) } = 0 forany ψ ∈ C ∞ ( T ). This completes the proof.From the previous proposition, from Lemma 3.3.1 and Remark 3.3.3 we immediately obtain the followingcorollary.2 CHAPTER 3. Q-W–ALGEBRAS
Corollary 3.3.8.
Suppose that the conditions of Proposition 3.3.7 are satisfied. Let t ∈ T be such that the map f f ( t ) is a character of the Poisson algebra C ∞ ( T ) . Then the action of K on M induces a (local) action on µ − ( t ) and a (local) action on C ∞ ( µ − ( t )) given by X ◦ ϕ = h Ad ( π K ⊥ µ ) − ( θ T ) , X i ( µ ∗ ( ξ ˜ ϕ )) , X ∈ k , ϕ ∈ C ∞ ( µ − ( t )) , where ˜ ϕ is any representative of ϕ ∈ C ∞ ( µ − ( t )) = C ∞ ( M ) /I t in C ∞ ( M ) . The algebra C ∞ ( µ − ( t )) K of invariantswith respect to this action is isomorphic to the reduced Poisson algebra C ∞ ( µ − ( t )) C ∞ ( T ) . In this section we realize the quasiclassical limit of the algebra W s B ( G ) as the algebra of functions on a reducedPoisson manifold. In this section we always assume that κ = 1.Denote by χ s the character of the Poisson subalgebra C [ M + ] such that χ s ( p ( x )) = χ sq ( x ) mod ( q dr −
1) forevery x ∈ C B [ M + ].Note that the image of the algebra C B [ G ∗ ] under the projection p : C B [ G ∗ ] → C B [ G ∗ ] / (1 − q dr ) C B [ G ∗ ] is acertain subalgebra of C [ G ∗ ] that we denote by C [ G ∗ ]. By Proposition 3.2.5 we have C [ G ∗ ] ≃ C [ G ] as algebras. Let I = p ( I B ) be the ideal in C [ G ∗ ] generated by the kernel of χ s . Then by the discussion after formula (3.2.23) thePoisson algebra W s ( G ) = W sq ( G ) / ( q dr − W sq ( G ) is the subspace of all x + I ∈ Q , Q = Q B / (1 − q dr ) Q B ⊂ C [ G ∗ ] /I , such that { m, x } ∈ I for any m ∈ C [ M − ], and the Poisson bracket in W s ( G ) takes the form { ( x + I ) , ( y + I ) } = { x, y } + I , x + I, y + I ∈ W s ( G ). We shall also write W s ( G ) = ( C [ G ∗ ] /I ) C [ M + ] ∩ Q , where the action of thePoisson algebra C [ M + ] on the space C [ G ∗ ] /I is defined as follows x · ( v + I ) = ρ χ s ( { x, v } ) , (3.4.1) v ∈ C [ G ∗ ] is any representative of v + I ∈ C [ G ∗ ] /I and x ∈ C [ M + ].One can describe the space of invariants ( C [ G ∗ ] /I ) C [ M + ] with respect to this action by analyzing the relatedunderlying manifolds and varieties. First observe that the algebra ( C [ G ∗ ] /I ) C [ M + ] is a particular example of thereduced Poisson algebra introduced in Lemma 3.3.1 .Indeed, recall that according to (3.2.1) any element ( L + , L − ) ∈ G ∗ may be uniquely written as( L + , L − ) = ( n + , n − )( h + , h − ) , (3.4.2)where n ± ∈ N ± , h + = exp (( − s P h ′ + P h ′⊥ ) x ) , h − = exp (( s − s P h ′ − P h ′⊥ ) x ) , x ∈ h .Formula (3.2.1) and a decomposition of elements of N + into products of elements which belong to one–dimensional subgroups corresponding to roots also imply that any element L + can be represented in the form L + = Q β exp [ b β X β ] × exp hP li =1 b i ( − s P h ′ + P h ′⊥ ) H i i , b i , b β ∈ C , (3.4.3)where the product over roots is taken according to the normal ordering associated to s .Now define a moment map µ M − : G ∗ → M + by µ M − ( L + , L − ) = m + , (3.4.4)where for L + given by (3.4.3) m + is defined as follows m + = Y β ∈ ∆ m + exp [ b β X β ] , and the product over roots is taken according to the normal order in the segment ∆ m + .Note that by definition C [ M + ] = { ϕ ∈ C [ G ∗ ] : ϕ = ϕ ( m + ) } . Therefore C [ M + ] is generated by the pullbacks ofregular functions on M + with respect to the map µ M − . Since C [ M + ] is a Poisson subalgebra in C [ G ∗ ], and regularfunctions on M + are dense in C ∞ ( M + ) on every compact subset, we can equip the manifold M + with the Poissonstructure in such a way that µ M − becomes a Poisson mapping. .4. POISSON REDUCTION AND Q-W–ALGEBRAS u ∈ M + be the element defined by u = l ′ Y i =1 exp [ t i X γ i ] , t i = k i (mod ( q dr − , (3.4.5)where the product over roots is taken according to the normal order in the segment ∆ m + .By Proposition 3.2.3 the elements L ± ,V = ( p ⊗ p V )( q L ± ,V ) belong to the space C [ G ∗ ] ⊗ End V , where p V : V res → V = V res / ( q dr − V res is the projection of the finite rank U s,res B ( g )–module V res onto the corresponding g –module V , and the map C B [ G ∗ ] / ( q dr − C B [ G ∗ ] → C [ G ∗ ] , L ± ,V L ± ,V is an isomorphism. In particular, from (2.6.11) it follows that L + ,V = Q β exp [ p ((1 − q − β ) f β ) ⊗ π V ( X β )] × ( p ⊗ id ) exp hP li =1 hH i ⊗ π V (( − s P h ′ + P h ′⊥ ) Y i ) i . (3.4.6)From (3.4.6) and the definition of χ s we obtain that χ s ( ϕ ) = ϕ ( u ) for every ϕ ∈ C [ M + ]. χ s naturally extendsto a character of the Poisson algebra C ∞ ( M + ).Now applying Lemma 3.3.1 we can define a reduced Poisson algebra C ∞ ( µ − M − ( u )) C ∞ ( M + ) as follows. Denote by I u the ideal in C ∞ ( G ∗ ) generated by elements µ ∗ M − ψ, ψ ∈ C ∞ ( M + ) , ψ ( u ) = 0. Let P u : C ∞ ( G ∗ ) → C ∞ ( G ∗ ) /I u = C ∞ ( µ − M − ( u )) be the canonical projection. Define the action of C ∞ ( M + ) on C ∞ ( µ − M − ( u )) by ψ · ϕ = P u ( { µ ∗ M − ψ, ˜ ϕ } ) , (3.4.7)where ψ ∈ C ∞ ( M − ) , ϕ ∈ C ∞ ( µ − M − ( u )) and ˜ ϕ ∈ C ∞ ( G ∗ ) is a representative of ϕ such that P u ˜ ϕ = ϕ . The reducedPoisson algebra C ∞ ( µ − M − ( u )) C ∞ ( M + ) is the algebra of C ∞ ( M + )–invariants in C ∞ ( µ − M − ( u )) with respect to action(3.4.7). The reduced Poisson algebra is naturally equipped with a Poisson structure induced from C ∞ ( G ∗ ) asdescribed in (3.3.2). Lemma 3.4.1.
Let q ( µ − M − ( u )) be the closure of q ( µ − M − ( u )) in G with respect to Zariski topology. Then Q ≃ C [ q ( µ − M − ( u ))] , and the algebra W s ( G ) is isomorphic to the algebra of regular functions on q ( µ − M − ( u )) pullbacks ofwhich under the map q are invariant with respect to the action (3.4.7) of C ∞ ( M + ) on C ∞ ( µ − M − ( u )) , i.e. W s ( G ) = C [ q ( µ − M − ( u ))] ∩ C ∞ ( µ − M − ( u )) C ∞ ( M + ) , where C [ q ( µ − M − ( u ))] is regarded as a subalgebra in C ∞ ( µ − M − ( u )) using the map q ∗ : C ∞ ( q ( µ − M − ( u ))) → C ∞ ( µ − M − ( u )) and the imbedding C [ q ( µ − M − ( u ))] ⊂ C ∞ ( q ( µ − M − ( u ))) .Proof. First observe that by the definition µ − M − ( u ) is a submanifold in G ∗ and that I = C [ G ∗ ] ∩ I u . Thereforeby the definition of the algebra C [ G ∗ ] and of the map µ M − the quotient C [ G ∗ ] /I is identified with the algebra offunctions on µ − M − ( u ) generated by the restrictions of elements of C [ G ∗ ] to µ − M − ( u ).Also by the definition Q ⊂ C [ G ∗ ] /I is the algebra generated by the restrictions to µ − M − ( u ) of the pullbacks ofelements from the algebra of regular functions C [ G ] under the map q : G ∗ → G . Therefore Q = C [ q ( µ − M − ( u ))].From these observations we deduce that W s ( G ) = ( C [ G ∗ ] /I ) C [ M − ] ∩ Q = ( C [ G ∗ ] /I ) C [ M − ] ∩ C [ q ( µ − M − ( u ))].Since C [ M − ] is dense in C ∞ ( M − ) on every compact subset in M − we have C ∞ ( µ − M + ( u )) C ∞ ( M − ) ∼ = C ∞ ( µ − M + ( u )) C [ M − ] . Now observe that action (3.4.7) of elements from C [ M − ] coincides with action (3.4.1) when restricted to ele-ments from C [ G ∗ ] /I , and hence W s ( G ) = ( C [ G ∗ ] /I ) C [ M − ] ∩ C [ q ( µ − M − ( u ))] = C ∞ ( µ − M + ( u )) C [ M − ] ∩ C [ q ( µ − M − ( u ))] = C ∞ ( µ − M − ( u )) C ∞ ( M − ) ∩ C [ q ( µ − M + ( u ))]. This completes the proof.4 CHAPTER 3. Q-W–ALGEBRAS
We shall realize the algebra C ∞ ( µ − M − ( u )) C ∞ ( M + ) as the algebra of functions on a reduced Poisson manifold. Inthis construction we use the dressing action of the Poisson–Lie group G on G ∗ .Consider the restriction of the dressing action G × G ∗ → G ∗ to the subgroup M − ⊂ G . Let G ∗ /M − be thequotient of G ∗ with respect to the dressing action of M − , π : G ∗ → G ∗ /M − the canonical projection. Note thatthe space G ∗ /M − is not a smooth manifold. However, it turns out that the subspace π ( µ − M − ( u )) ⊂ G ∗ /M − is asmooth manifold. More precisely, we have the following lemma. Lemma 3.4.2.
The preimage µ − M − ( u ) ⊂ G ∗ is locally stable under the (locally defined) dressing action of M − , andthe algebra C ∞ ( µ − M − ( u )) M − is isomorphic to C ∞ ( µ − M − ( u )) C ∞ ( M + ) .Proof. The proof will be based on Corollary 3.3.8.First observe that according to part (iv) of Proposition 3.1.2 ( i − , i + ) = ( b − , b + ) is a subbialgebra of ( g , g ∗ ).Therefore B − is a Poisson–Lie subgroup in G .By Proposition 3.3.5 and by the definition of the moment map we have for any X ∈ b − , ϕ ∈ C ∞ ( G ∗ ) L b X ϕ ( L + , L − ) = ( θ G ∗ ( L + , L − ) , X )( ξ ϕ ) = ( r − µ ∗ B − ( θ B + ) , X )( ξ ϕ ) , (3.4.8)where b X is the corresponding vector field on G ∗ , ξ ϕ is the Hamiltonian vector field of ϕ ∈ C ∞ ( G ∗ ), and the map µ B − : G ∗ → B + is defined by µ B − ( L + , L − ) = L + . Now from Proposition 3.1.2 (iv) and the definition of themoment map it follows that µ B − is a moment map for the restriction of the dressing action to the subgroup B − .Next observe that the complementary subset to ∆ m + in ∆ + is a minimal segment ∆ m + . Now using Proposition3.1.2 (iv) the subspace m ⊥− in b + can be identified with the linear subspace in b + spanned by the Cartan subalgebra h and by the root subspaces corresponding to the roots from the minimal segment ∆ m + . Using the fact that theadjoint action of h normalizes root subspaces and Lemma 1.6.6 we deduce that m ⊥− ⊂ b + is a Lie subalgebra, andhence M − ⊂ B − is an admissible subgroup.Moreover, the dual group B + can be uniquely factorized as B + = M + M ⊥− , where M ⊥− ⊂ B + is the Lie subgroupcorresponding to the Lie subalgebra m ⊥− ⊂ b + , and M + ⊂ B + is the Lie subgroup corresponding to the Liesubalgebra m + .Now observe that m ⊥− = h + m ⊥− ⊂ b + (direct sum of vector spaces), where m ⊥− is the Lie subalgebra generatedby the root vectors corresponding to the roots from the minimal segment ∆ m + . The Lie subalgebra m − is generatedby the root subspaces corresponding to the roots from the minimal segment ∆ m + . Since all root subspaces areinvariant under the adjoint action of h and the restriction of the adjoint action of the root vectors correspondingto the roots from the minimal segment ∆ m + is nilpotent we deduce that for any m + ∈ m + , k ⊥ ∈ M ⊥− , k ⊥ = hk ⊥ , h ∈ H , k ⊥ = exp( x ), x ∈ m ⊥− one has(Ad( hk ⊥ )( m + )) m + = Ad h ((Ad k ⊥ ( m + )) m + ) = Ad h ((exp(ad x )( m + )) m + ) = Ad h ((Id + V )( m + )) , where the subscript m + stands for the m + –component in the direct vector space decomposition b + = m + + m ⊥− ,and V is a linear nilpotent transformation of m + .The maps Ad h and Id + V are obviously invertible. Hence for any k ⊥ ∈ M ⊥− the map m + → m + , m + (Ad( hk ⊥ )( m + )) m + is invertible as well.We conclude that all the conditions of Corollary 3.3.8 are satisfied with A = B − , K = M − , A ∗ = B + , T = M + , K ⊥ = M ⊥− , µ = µ B − . It follows that the preimage µ − M − ( u ) ⊂ G ∗ is locally stable under the (locally defined)dressing action of M − , and the algebra C ∞ ( µ − M − ( u )) M − is isomorphic to C ∞ ( µ − M − ( u )) C ∞ ( M + ) . This completes theproof.Observe that by (3.3.4) under the map q : G ∗ → G , q ( L + , L − ) = L − − L + the dressing action becomes theaction of G on itself by conjugations. Consider the restriction of this action to the subgroup M + . Denote by π q : G → G/M − the canonical projection onto the quotient with respect to this action. We shall see that π q ( q ( µ − M − ( u ))) is an algebraic variety and C [ π q ( q ( µ − M − ( u )))] ≃ W s ( G ). We shall also obtain an explicit descriptionof the variety q ( µ − M − ( u )) and of the quotient π q ( q ( µ − M − ( u ))). .4. POISSON REDUCTION AND Q-W–ALGEBRAS µ − M − ( u ) of the moment map µ M − under the map q . Let X α ( t ) = exp( tX α ) ∈ G , t ∈ C be the one–parameter subgroup in the algebraic group G corresponding to root α ∈ ∆. Recall that for any α ∈ ∆ + and any t = 0 the element s α ( t ) = X α ( t ) X − α ( − t − ) X α ( t ) ∈ G (3.4.9)is a representative for the reflection s α corresponding to the root α . Denote by s ∈ G the following representativeof the Weyl group element s ∈ W , s = s γ ( t ) . . . s γ l ′ ( t l ′ ) , (3.4.10)where the numbers t i are defined in (3.4.5), and we assume that t i = 0 for any i .We shall also use the following representatives for s and s s = s γ ( t ) . . . s γ n ( t n ) , s = s γ n +1 ( t n +1 ) . . . s γ l ′ ( t l ′ ) . The following Proposition is an improved version of Proposition 7.2 in [102] suitable for the purposes of quan-tization.
Proposition 3.4.3.
Let q : G ∗ → G be the map defined by q ( L + , L − ) = L − − L + . Suppose that the numbers t i defined in (3.4.5) are not equal to zero for all i . Then q ( µ − M − ( u )) ⊂ N − sH Z + M − = N − sH M − Z + = ( N − ∩ N ) Z − sH M − Z + = (3.4.11)= ( N − ∩ N ) Z − sH Z + M − ⊂ N sZN, where H is the subgroup corresponding to the orthogonal complement h ′⊥ of h ′ in h with respect to the symmetricbilinear form on g , Z ± = Z ∩ N ± . The closure q ( µ − M − ( u )) of q ( µ − M − ( u )) with respect to the Zariski topology is alsocontained in N sZN .Proof.
Using definition (3.4.4) of the map µ M − we can describe the preimage µ − M − ( u ) as follows µ − M − ( u ) = { ( uyh + , n − h − ) | n − ∈ N − , h ± = e r s ± x , x ∈ h , y ∈ N ∆ + \ ∆ m + } , (3.4.12)where for any additively closed subset of roots Ξ ⊂ ∆ we denote by N Ξ the subgroup in G generated by theone–parameter subgroups corresponding to the roots from Ξ. Therefore q ( µ − M − ( u )) = { h − − n − − uyh + | n − ∈ N − , h ± = e r s ± x , x ∈ h , y ∈ N ∆ + \ ∆ m + } . (3.4.13)First we show that for any y ∈ N ∆ + \ ∆ m + and n − ∈ N − the element n − − uy belongs to N − sM Z + M − . Fix thecircular normal ordering on ∆ corresponding to the normal ordering of ∆ + associated to s .In the proof we shall frequently use the following lemma. Lemma 3.4.4.
Let [ α, β ] ⊂ ∆ be a minimal segment and assume that [ α, β ] = [ α, γ ] ∪ [ δ, β ] , where the segments [ α, γ ] and [ δ, β ] are disjoint and minimal as well. Then any element m ∈ N [ α,β ] can be uniquely factorized as m = g g = g ′ g ′ , g , g ′ ∈ N [ α,γ ] , g , g ′ ∈ N [ δ,β ] . Moreover, if δ = β then for any m ′ ∈ N [ α,γ ] and any t ∈ C onehas m ′ X β ( t ) = X β ( t ) m ′′ , where m ′′ ∈ N [ α,γ ] .Proof. The proof is obtained by straightforward application of Chevalley’s commutation relations between one–parameter subgroups and Lemma 1.6.6.Since the roots γ , . . . , γ n are mutually orthogonal the adjoint action of s γ i ( t i ), i = 1 , . . . , n on each of the rootsubspaces g γ j , j = 1 , . . . , n, j = i is given by multiplication by a non–zero constant. Therefore there are non–zeroconstants c , . . . , c n such that X γ k ( c k ) s γ . . . s γ k − = s γ . . . s γ k − X γ k ( − t − k ), k = 2 , . . . , n , and we define c = − t − .Obviously we have X γ ( t ) . . . X γ n ( t n ) = X − γ ( − c ) . . . X − γ n ( − c n ) X − γ n ( c n ) . . . X − γ ( c ) X γ ( t ) . . . X γ n ( t n ) =6 CHAPTER 3. Q-W–ALGEBRAS = n X − γ n ( c n ) . . . X − γ ( c ) X γ ( t ) . . . X γ n ( t n ) , n = X − γ ( − c ) . . . X − γ n ( − c n ) ∈ N ∆ n − , where ∆ n − = { α ∈ ∆ − : − γ ≤ α ≤ − γ n } .Using the relation X − γ ( − t − ) X γ ( t ) = X γ ( − t ) s γ one can rewrite the last identity as follows X γ ( t ) . . . X γ n ( t n ) = n X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) s γ X γ ( t ) . . . X γ n ( t n ) . (3.4.14)Now we can write X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) == X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) X − γ ( − c ) . . . X − γ n ( − c n ) X − γ n ( c n ) . . . X − γ ( c ) . The product X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) X − γ ( − c ) . . . X − γ n ( − c n ) belongs to the subgroup of G generatedby the one–parameter subgroups corresponding to roots from the set ∆ = { α ∈ ∆ : − γ ≤ α ≤ γ , s α = − α } . By Lemma 1.6.6 the minimal segment { α ∈ ∆ : − γ ≤ α ≤ γ } is closed under addition of roots andthe set of roots on which s acts by multiplication by − is closed under addition of roots. Assume that the order of roots in ∆ is induced by the circular nor-mal ordering of ∆. Using Lemma 1.6.6 and the fact that ∆ is closed under addition of roots the element X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) X − γ ( − c ) . . . X − γ n ( − c n ) can be represented as a product of elements from one–parameter subgroups corresponding to roots from ∆ ordered in the way described above. Denote the intersectionof ∆ with ∆ + by ∆ = { α ∈ ∆ + : α ≤ γ , s α = − α } and let ∆ − = ∆ ∩ ∆ − = { α ∈ ∆ − : − γ ≤ α, s α = − α } .Then the above mentioned factorization yields X − γ n ( c n ) . . . X − γ ( c ) X γ ( − t ) X − γ ( − c ) . . . X − γ n ( − c n ) = n ′ x ′ , where n ′ ∈ N ∆ − , x ′ ∈ N ∆ .Substituting the last relation into (3.4.14) and using the definition of c and the orthogonality of roots γ and γ we obtain X γ ( t ) . . . X γ n ( t n ) = n x ′ X − γ n ( c n ) . . . X − γ ( c ) s γ X − γ ( − t − ) X γ ( t ) . . . X γ n ( t n ) , where n = n n ′ ∈ N ∆ s − , ∆ s − = { α ∈ ∆ − : s α = − α } .Now we can use the relation X − γ ( − t − ) X γ ( t ) = X γ ( − t ) s γ , the orthogonality of roots γ and γ , andapply similar arguments to get X γ ( t ) . . . X γ n ( t n ) = n x ′ X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) s γ s γ X γ ( t ) . . . X γ n ( t n ) , a = 0 . (3.4.15)We can also write X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) == X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) X − γ ( − c ) . . . X − γ n ( − c n ) X − γ n ( c n ) . . . X − γ ( c ) . The product X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) X − γ ( − c ) . . . X − γ n ( − c n ) belongs to the subgroup of G generated by theone–parameter subgroups corresponding to roots from the set ∆ = { α ∈ ∆ : − γ ≤ α ≤ γ , s α = − α } . By Lemma1.6.6 the minimal segment { α ∈ ∆ : − γ ≤ α ≤ γ } is closed under addition of roots and the set of roots on which s acts by multiplication by − is closed under addition of roots.Assume that the order of roots in ∆ is induced by the circular normal ordering of ∆. Using Lemma 1.6.6 and thefact that ∆ is closed under addition of roots the element X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) X − γ ( − c ) . . . X − γ n ( − c n )can be represented as a product of elements from one–parameter subgroups corresponding to roots from ∆ orderedin the way described above. Denote the intersection of ∆ with ∆ + by ∆ = { α ∈ ∆ + : α ≤ γ , s α = − α } andlet ∆ − = ∆ ∩ ∆ − = { α ∈ ∆ − : s α = − α, − γ ≤ α } . Then the above mentioned factorization yields X − γ n ( c n ) . . . X − γ ( c ) X γ ( a ) X − γ ( − c ) . . . X − γ n ( − c n ) = n ′ x ′′ , (3.4.16)where n ′ ∈ N ∆ − , x ′′ ∈ N ∆ .Substituting the last relation into (3.4.15) and using the definition of c and the orthogonality of roots γ , γ and γ we obtain X γ ( t ) . . . X γ n ( t n ) = n x ′ n ′ x ′′ X − γ n ( c n ) . . . X − γ ( c ) s γ s γ X γ ( − t − ) X γ ( t ) . . . X γ n ( t n ) (3.4.17) .4. POISSON REDUCTION AND Q-W–ALGEBRAS N ∆ ⊂ N ∆ we deduce x ′ n ′ x ′′ ∈ N ∆ . Therefore using arguments applied above to obtain (3.4.16) we get x ′ n ′ x ′′ = n ′′ x ′ , x ′ ∈ N ∆ , n ′′ ∈ N ∆ − , and (3.4.17) takes the form X γ ( t ) . . . X γ n ( t n ) = n x ′ X − γ n ( c n ) . . . X − γ ( c ) s γ s γ X − γ ( − t − ) X γ ( t ) . . . X γ n ( t n ) , where n = n n ′′ ∈ N ∆ s − .We can proceed in a similar way to obtain the following representation X γ ( t ) . . . X γ n ( t n ) = n e xs γ . . . s γ n = n e xs , n ∈ N ∆ s − , e x ∈ N ∆ n + , (3.4.18)where ∆ n + = { α ∈ ∆ + : α ≤ γ n , s α = − α } = { α ∈ ∆ + : γ ≤ α ≤ γ n } .Note that s acts by multiplication by − n − = − ∆ n + = s (∆ n + ). Then N ∆ n − = ( s ) − N ∆ n + s ⊂ N ∆ s − and (3.4.18) can be rewritten in the following form X γ ( t ) . . . X γ n ( t n ) = ns ( s ) − e xs = ns n ′ , n ∈ N ∆ s − , n ′ = ( s ) − e xs ∈ N ∆ n − . (3.4.19)Similarly one has X − γ n +1 ( t n +1 ) . . . X − γ l ′ ( t l ′ ) = n ′′ s γ n +1 . . . s γ l ′ n ′′′ = n ′′ s n ′′′ , n ′′ ∈ N ∆ s − , n ′′′ ∈ N ∆ l ′− , (3.4.20)where ∆ s − = { α ∈ ∆ − : s α = − α } , and ∆ l ′ − = { α ∈ ∆ − : − γ n +1 ≤ α ≤ − γ l ′ } .Combining (3.4.19) and (3.4.20), using the definition of the circular normal ordering of the root system ∆associated to s , Lemmas 1.6.6, 3.4.4 and commutation relations between one–parameter subgroups correspondingto roots one can obtain n − − uy = n − − ns n ′ n ′′ s n ′′′ y = ks gs n ′′′ y, g ∈ N ∆ s − \ ∆ , k ∈ N − . (3.4.21)By Proposition 1.6.1 s ∆ ss ⊂ ∆ s + \ (∆ ss ∪ ∆ ss ∪ ∆ ) ⊂ ∆ + \ ∆ , and hence s ∆ ss ⊂ ∆ m + ⊂ (cid:0) ∆ s + \ (∆ ss ∪ ∆ ss ∪ ∆ ) (cid:1) ∩ (∆ + \ ∆ ) , (3.4.22)Now the minimal segment ∆ s + \ ∆ can be represented as the following disjoint union∆ s + \ ∆ = (cid:0) ∆ s + \ (∆ ss ∪ ∆ ) (cid:1) ∪ ∆ ss . (3.4.23)Note that since the decomposition s = s s is reduced we have s (∆ s + \ (∆ ss ∪ ∆ )) ⊂ ∆ s + \ (∆ ss ∪ ∆ ) ⊂ ∆ + \ ∆ . (3.4.24)Observe that the segments ∆ ss , ∆ s + \ (∆ ss ∪ ∆ ) are minimal with respect to the circular normal ordering on∆ associated to s . Thus from (3.4.23) and Lemma 3.4.4 we deduce that the element g ∈ N ∆ s − \ ∆ from formula(3.4.21) can be uniquely decomposed as the product g = g ′ g ′′ , where g ′ ∈ N − ( ∆ s + \ (∆ ss ∪ ∆ ) ), and g ′′ ∈ N − ∆ ss . By(3.4.22), (3.4.24) we have s g ′ ( s ) − ∈ N − and ( s ) − g ′′ s ∈ M − , and (3.4.21) takes the form n − − uy = ks g ′ g ′′ s n ′′′ y = ks g ′ ( s ) − s ( s ) − g ′′ s n ′′′ y = k ′ s b ny, b n ∈ M − , k ′ ∈ N − . (3.4.25)The element b ny belongs to the subgroup N ∆ ′ , ∆ ′ = { α ∈ ∆ : γ l ′ < α ≤ − γ l ′ } . The images of all roots fromthe set (∆ ′ \ ∆ ) ∩ ∆ + under the action of s belong to ∆ − . The set complimentary to (∆ ′ \ ∆ ) ∩ ∆ + in ∆ ′ is − ∆ m + ∪ (∆ ∩ ∆ + ). Therefore using the decomposition of elements N ∆ ′ as products of elements from one–parametersubgroups corresponding to roots and recalling Lemma 3.4.4 one can get b ny = y ′ z + m , where z + ∈ Z + = Z ∩ N + , sy ′ s − ∈ N − , m ∈ M − , and n − − uy = k ′′ sz + m = k ′′ z ′ + sm, m ∈ M − , k ′′ ∈ N − , z + , z ′ + = sz + s − ∈ Z + . (3.4.26)Hence n − − uy ∈ N − sZ + M − .8 CHAPTER 3. Q-W–ALGEBRAS
Now we show that N − sZ + M − = N − sM − Z + = ( N − ∩ N ) Z − sM − Z + = ( N − ∩ N ) Z − sZ + M − . Note that the elements of the subgroup N − ∩ N are transformed to M − by the conjugation by s − . Therefore usinga decomposition of elements N − into products of elements from one–parameter subgroups corresponding to rootsand Lemma 3.4.4 one can obtain that any k ′′ ∈ N − can be represented in the form k ′′ = nz − k ′′′ , n ∈ N ∩ N − , z − ∈ Z − = Z ∩ N − , k ′′′ ∈ N − ∩ N , s − k ′′′ s ∈ M − . Therefore for any m ∈ M − , z + ∈ Z + , we have k ′′ smz + = nz − sm ′ z + , n ∈ N ∩ N − , m ′ = s − k ′′′ sm ∈ M − , z − ∈ Z − , and hence N − sM − Z + ⊂ ( N ∩ N − ) Z − sM − Z + . The opposite inclusion is obvious. Thus N − sM − Z + = ( N ∩ N − ) Z − sM − Z + . Similarly one obtains N − sZ + M − = N − sM − Z + = ( N ∩ N − ) Z − sZ + M − and obviously ( N ∩ N − ) Z − sZ + M − ⊂ ( N − ∩ N ) sZM − ⊂ N sZN .Next we prove that for any n − ∈ N − , x ∈ h and y ∈ N ∆ + \ ∆ m + we have h − − n − − uyh + ∈ ( N ∩ N − ) Z − sH M − Z + ⊂ N sZN , where h ± = e r s ± x , i.e. q ( µ − M − ( u )) ⊂ ( N ∩ N − ) Z − sH M − Z + ⊂ N sZN .Let H ′ ⊂ H be the subgroup corresponding to the Lie subalgebra h ′ ⊂ h . Recall that we denote by H ⊂ H thesubgroup corresponding to the orthogonal complement h ′⊥ of h ′ in h with respect to the symmetric bilinear formon g . Note that h ′⊥ is the space of fixed points for the action of s on h . We obviously have H = H ′ H . From thedefinition of r s ± it follows that for any h ∈ H and h ′ ∈ H ′ elements h + = h h ′ and h − = h − s ( h ′ ) are of the form h ± = e r s ± x for some x ∈ h and all elements h ± are obtained in this way.Next observe that the space ( N ∩ N − ) Z − sH M − Z + is invariant with respect to the following action of thesubgroup of H × H formed by elements of the form ( h + , h − ) = ( h h ′ , h − s ( h ′ )):( h + , h − ) ◦ L = h − − Lh + , h = h + = h h ′ , h − = h − s ( h ′ ) . (3.4.27)Indeed, let L = vz − skwz + , v ∈ ( N ∩ N − ) , w ∈ M − , z ± ∈ Z ± , k ∈ H be an element of ( N ∩ N − ) Z − sH M − Z + .Then ( h + , h − ) ◦ L = h − − vz − h − h − − skh + h − wz + h + = h − − vz − h − skh h − wz + h + (3.4.28)since s − h − − s = h h ′− . The r.h.s. of the last equality belongs to ( N ∩ N − ) Z − sH M − Z + because H normalizes( N ∩ N − ), M − and Z ± .Comparing action (3.4.27) with (3.4.13) and recalling that for any n − ∈ N − and y ∈ N ∆ + \ ∆ m + one has n − − uy ∈ ( N ∩ N − ) Z − sM − Z + ⊂ ( N ∩ N − ) Z − sH M − Z + we deduce q ( µ − M − ( u )) ⊂ ( N ∩ N − ) Z − sH M − Z + .Similarly, using the fact that H normalizes the subgroups N − , N ∩ N − , M − and Z ± of G and recalling thatfor any n − ∈ N − and y ∈ N ∆ + \ ∆ m + we also have n − − uy ∈ N − sZ + M − = N − sM − Z + = ( N ∩ N − ) Z − sZ + M − =( N ∩ N − ) Z − sM − Z + , one can show that q ( µ − M − ( u )) ⊂ N − sH Z + M − = N − sH M − Z + = ( N ∩ N − ) Z − sH M − Z + =( N ∩ N − ) Z − sH Z + M − = ( N ∩ N − ) sZ − H Z + M − ⊂ N sZN , where the last inclusion follows from the inclusions Z − H Z + ⊂ Z , M − ⊂ N .The set q ( µ − M − ( u )) is not Zariski closed in G . But by Proposition 1.3.1 N sZN is Zariski closed in G . Thereforethe Zariski closure q ( µ − M − ( u )) is contained in N sZN .Now we are in a position to describe the closure q ( µ − M − ( u )), the quotient π q ( q ( µ − M − ( u ))) and the algebra W s ( G ). Theorem 3.4.5.
Suppose that the numbers t i defined in (3.4.5) are not equal to zero for all i . Let N ′ s ⊂ N − ∩ N s be the subgroup generated by one–parameter subgroups corresponding to the roots α ∈ ∆ s − \ ∆ satisfying thecondition − γ l ′ < α , M s − = M − ∩ N s . Then q ( µ − M − ( u )) = N − sZM − = N − sZM s − is invariant under conjugationsby elements of M − , the conjugation action of M − on q ( µ − M − ( u )) is free, the quotient π q ( q ( µ − M − ( u ))) is a smoothvariety isomorphic to N ′ s sZM s − ≃ Σ s = sZN s , π q ( q ( µ − M − ( u ))) ≃ N ′ s sZM s − . Moreover, the conjugation action M − × N ′ s sZM s − → N − sZM − (3.4.29) is an isomorphism of varieties, and hence the algebra C [ q ( µ − M − ( u ))] is isomorphic to C [ M − ] ⊗ C [ N ′ s sZM s − ] .The Poisson algebra W s ( G ) is isomorphic to the Poisson algebra of regular functions on N ′ s sZM s − , W s ( G ) = C [ π q ( q ( µ − M − ( u )))] = C [ N ′ s sZM s − ] ≃ C [Σ s ] . Thus the algebra W s B ( G ) is a non–commutative deformation of thealgebra of regular functions on the transversal slice Σ s ≃ N ′ s sZM s − . .4. POISSON REDUCTION AND Q-W–ALGEBRAS Proof.
Firstly, as we observed in Lemma 3.4.2 the preimage µ − M − ( u ) is locally stable under the (locally defined)dressing action of M − , and hence q ( µ − M − ( u )) ⊂ N sZN is (locally) stable under the action of M − ⊂ N on N sZN by conjugations. Since the conjugation action of N on N sZN is free the (locally defined) conjugation action of M − on q ( µ − M − ( u )) is (locally) free as well.Now observe that by Proposition 3.4.3 q ( µ − M − ( u )) ⊂ N sZN . Since by Theorem 1.3.1 the conjugation action of N on N sZN is free and regular, sZN s being a cross–section for this action, and q ( µ − M − ( u )) is closed, the inducedaction of M − ⊂ N on q ( µ − M − ( u )) is globally defined and is free as well. Therefore the quotient π q ( q ( µ − M − ( u ))) is asmooth variety.Next observe that the definitions of M s − and of N ′ s and Lemma 3.4.4 imply the following factorization N s = M s − N ′ s . Thus if szn s ∈ sZN s , z ∈ Z , n s ∈ N s then n s can be uniquely factorized as n s = m s n ′ s , m s ∈ M s − , n ′ s ∈ N ′ s and we have szn s = szm s n ′ s . Conjugating this element by n ′ s we deduce that szn s is uniquely conjugated to the element n ′ s szm s ∈ N ′ s sZM s − and hence N ′ s sZM s − ≃ sZN s = Σ s is a cross–section for the action of N on N sZN as well.We show now that the closure of q ( µ − M − ( u )) contains the varieties N − sZM − and N ′ s sZM s − . Recall that by(3.4.11) q ( µ − M − ( u )) ⊂ N − sH Z + M − = N − H Z + sM − . Observe that the proof of presentation (3.4.26), formula(3.4.27) and the definition of q ( µ − M − ( u )) imply that it contains elements of the form ksn for some n ∈ M − ⊂ N and arbitrary k ∈ N − H Z + . This also follows from the fact that q ( µ − M − ( u )) is closed with respect to the rightmultiplication by arbitrary elements from Z + and with respect to the left multiplication by arbitrary elements from N − , as Z + ⊂ N ∆ + \ ∆ m + , and q ( µ − M − ( u )) is closed with respect to the right multiplication by arbitrary elementsfrom N ∆ + \ ∆ m + , and q ( µ − M − ( u )) is also closed with respect to the restriction of action (3.4.27) to H .Now recall that ¯ h ( α ) > α ∈ ∆ s + \ ∆ and ¯ h ( α ) = 0 for α ∈ ∆ , and hence the C ∗ –action on G induced byconjugations by the elements h ( t ) from the one–parameter subgroup generated by − ¯ h ∈ h is contracting on N andfixes all elements of Z . Applying action (3.4.27) with h = h ( t ) to the elements ksn with arbitrary k ∈ N − H Z + weimmediately deduce, with the help of (3.4.27), that the M − –component n can be contracted to the identity elementusing the above defined contracting action, and the closure of q ( µ − M − ( u )) contains the variety N − Zs as the closureof Z − H Z + is Z .By definition q ( µ − M − ( u )) is closed with respect to the left multiplication by arbitrary elements from N − . Recallalso that M − ⊂ N freely acts on q ( µ − M − ( u )) by conjugations. Therefore q ( µ − M − ( u )) also contains the variety N − ZsM − = N − sZM − . In particular, N ′ s sZM s − ⊂ q ( µ − M − ( u )).Note that N − sH Z + M − ⊂ N − ZsM − ⊂ q ( µ − M − ( u )) ⊂ N − sH Z + M − . This implies after taking closures that N − ZsM − = q ( µ − M − ( u )) = N − sH Z + M − . But the variety N − ZsM − is closed. Indeed, let N ′ s − = N s − ∩ N − , N ′ = { n ∈ N : s − ns ∈ N } . Thesedefinitions and Lemma 3.4.4 imply that N ∩ N − = N ′ s − N ′ . Note that s normalizes Z and Z normalizes N ′ . Observealso that s − N ′ sM − = N α ≤− γ l ′ , where N α ≤− γ l ′ is the subgroup of N generated by one–parameter subgroupscorresponding to the roots α ∈ ∆ s − from the segment defined by the condition α ≤ − γ l ′ . Therefore every element nzsm ∈ N − ZsM − = ( N ∩ N − ) ZsM − , n ∈ N ∩ N − , z ∈ Z , m ∈ M − with n = n n , n ∈ N ′ s − , n ∈ N ′ can berepresented as follows nzsm = n n zsm = n sz ′ s − n ′ sm ∈ N ′ s − sZN α ≤− γ l ′ , z ′ ∈ Z, n ′ ∈ N ′ , and any element of N ′ s − sZN α ≤− γ l ′ can be obtained this way. Thus N − ZsM − = ( N ∩ N − ) ZsM − = N ′ s − sZN α ≤− γ l ′ .But the variety N ZN is closed by Corollary 1.3.2 and s − N ′ s − sZN α ≤− γ l ′ ⊂ N ZN CHAPTER 3. Q-W–ALGEBRAS is a closed subvariety of it as s − N ′ s − s ⊂ N , N α ≤− γ l ′ ⊂ N are closed subgroups. Therefore N − ZsM − = ( N ∩ N − ) ZsM − = N ′ s − sZN α ≤− γ l ′ is closed.Hence N − ZsM − = q ( µ − M − ( u )) = N − sH Z + M − . Using Lemma 1.3.1 one can also easily obtain that N − ZsM − = N − ZsM s − .Finally we show that N ′ s ZsM s − is a cross–section for the free conjugation action of M − on N − ZsM − = q ( µ − M − ( u )). Observe that any two points of N ′ s ZsM s − are not M − –conjugate. Indeed, we have an inclusion N ′ s ZsM s − ⊂ q ( µ − M − ( u )), and two points of q ( µ − M − ( u )) can not be M − -conjugate if they are not N –conjugate in N sZN . But N ′ s ZsM s − is a cross–section for the action of N on ZsZN . Thus any two points of N ′ s ZsM s − are not N –conjugate, and hence they are not M − –conjugate. Therefore the closed variety π q ( q ( µ − M − ( u ))) must contain theclosed variety N ′ s ZsM s − ≃ Σ s .From formula (1.6.5) for the cardinality ♯ ∆ m + of the set ∆ m + and from the definitions of q ( µ − M − ( u )) and of N ′ s ZsM s − we deduce that the dimension of the quotient π q ( q ( µ − M − ( u ))) is equal to the dimension of the variety N ′ s ZsM s − ,dim π q ( q ( µ − M − ( u ))) = dim G − M − = 2 D + l − ♯ ∆ m + = 2 D + l − (cid:18) D − l ( s ) − l ′ − D (cid:19) == l ( s ) + 2 D + l − l ′ = dim N s + dim Z = dim sZN s = dim Σ s = dim N ′ s ZsM s − . Since π q is a morphism of varieties π − q ( N ′ s ZsM s − ) = π − q (Σ s ) is a closed smooth subvariety of the smooth variety N − ZsM − = q ( µ − M − ( u )). The number of connected components of π − q ( N ′ s ZsM s − ) is equal to the cardinality of thefinite set Z/Z , where Z is the identity component of Z . Each such component is irreducible.The number of connected components of the smooth closed variety N − ZsM − = q ( µ − M − ( u )) is also equal to thecardinality of the finite set Z/Z . Each such component is irreducible. The last two observations together with theidentity for the dimensions imply π − q ( N ′ s ZsM s − ) = N − ZsM − .Therefore π q ( q ( µ − M − ( u ))) ≃ N ′ s ZsM s − , N ′ s ZsM s − is a cross–section for the action of M − on q ( µ − M − ( u )), and theconjugation action M − × N ′ s ZsM s − → N − ZsM − is an isomorphism of varieties. We conclude that the algebra C [ q ( µ − M − ( u ))] is isomorphic to C [ M − ] ⊗ C [ N ′ s ZsM s − ], C [ q ( µ − M − ( u ))] ∼ = C [ M − ] ⊗ C [ N ′ s ZsM s − ].Now recall that by Lemma 3.4.1 W s ( G ) = C [ q ( µ − M − ( u ))] ∩ C ∞ ( µ − M − ( u )) C ∞ ( M + ) , where C [ q ( µ − M − ( u ))] is regarded as a subalgebra in C ∞ ( µ − M − ( u )) using the map q ∗ : C ∞ ( q ( µ − M − ( u ))) → C ∞ ( µ − M − ( u ))and the imbedding C [ q ( µ − M − ( u ))] ⊂ C ∞ ( q ( µ − M − ( u ))).By Lemma 3.4.2 the algebra C ∞ ( µ − M − ( u )) M − is isomorphic to C ∞ ( µ − M − ( u )) C ∞ ( M + ) , and hence W s ( G ) = C [ q ( µ − M − ( u ))] ∩ C ∞ ( µ − M − ( u )) C ∞ ( M + ) = C [ q ( µ − M − ( u ))] ∩ C ∞ ( µ − M − ( u )) M − . (3.4.30)As we already proved the variety q ( µ − M − ( u )) is stable under the conjugation action of M − , and the map π q : q ( µ − M + ( u )) → π q q ( µ − M + ( u )) is a morphism of varieties. Moreover, under the map q : G ∗ → G the local dressingaction of M − on G ∗ becomes the conjugation action on G . Therefore the map C [ π q q ( µ − M − ( u ))] → C [ q ( µ − M − ( u ))] ∩ C ∞ ( µ − M − ( u )) M − , ψ π ∗ q ψ (3.4.31)is an isomorphism, where C [ q ( µ − M − ( u ))] is regarded as a subalgebra in C ∞ ( µ − M − ( u )) using the map q ∗ : C ∞ ( q ( µ − M + ( u ))) → C ∞ ( µ − M − ( u ))and the imbedding C [ q ( µ − M − ( u ))] ⊂ C ∞ ( q ( µ − M − ( u ))).Combining (3.4.30) and (3.4.31) we obtain that W s ( G ) ∼ = C [ π q q ( µ − M − ( u ))]. This completes the proof. .5. ZHELOBENKO TYPE OPERATORS FOR POISSON Q-W–ALGEBRAS In this section we present the main result of this chapter, a formula for a projection operator Π : C [ N − ZsM − ] → C [ N − ZsM − ] M − onto the subspace of invariants C [ N − ZsM − ] M − which is isomorphic to W s ( G ) as an algebraaccording to Theorem 3.4.5. This formula has a direct quantum analogue which will be introduced in the nextchapter.The operator Π can be defined following the philosophy of [104] where a similar projection operator ontothe subspace C [ N ZsN ] N ⊂ C [ N ZsN ] was defined and studied. More precisely, according to Theorem 3.4.5 any g ∈ N − ZsM − can be uniquely represented in the form g = nn s zsm s n − , n ∈ M − , n s ∈ N ′ s , m s ∈ M s − , z ∈ Z. (3.5.1)If for f ∈ C [ N − ZsM − ] we define Π f ∈ C [ N − ZsM − ] by(Π f )( g ) = f ( n − gn ) = f ( n s zsm s ) (3.5.2)then Π f is an M − –invariant function, and any M − –invariant regular function on N − ZsM − can be obtained thisway. Moreover, by the definition Π = Π, i.e. Π is a projection onto C [ N − ZsM − ] M − .To obtain an explicit formula for the operator Π we firstly find an explicit formula for n in terms of g in(3.5.1). Denote by ω the Chevalley anti–involution on g which is induced by the antiautomorphism ω of U sh ( g ) on U ( g ) ≃ U sh ( g ) /hU sh ( g ). We also denote the corresponding anti–involution of G by the same letter. The followingproposition is an analogue of Proposition 2.11 in [104], where a similar statement was proved for the action of N on N ZsN . Proposition 3.5.1.
Let g = nn s zsm s n − ∈ N − ZsM − . Let α i be the simple roots of a system of positive roots ∆ + associated to s , s i the corresponding simple reflections, β , . . . , β D , β j = s i . . . s i j − α i j a normal orderingof ∆ + corresponding to s . Let ω i , i = 1 , . . . l be the fundamental weights corresponding to ∆ + , v ω i a non–zerohighest weight vector in the irreducible highest weight G –module V ω i of highest weight ω i , ( · , · ) the contravariantnon–degenerate bilinear form on V ω i such that ( v, xw ) = ( ω ( x ) v, w ) for any v, w ∈ V ω i , x ∈ g and ( v ω i , v ω i ) = 1 .Then n can be uniquely factorized as n = X − β ( t ) . . . X − β c ( t c ) and the numbers t i can be found inductively by thefollowing formula t p = c p ( w p v ω ip , g p s − w p − v ω ip )( w p − v ω ip , g p s − w p − v ω ip ) , (3.5.3) where w p = s β p . . . s β , w p − = s β p − . . . s β , c p is a non–zero constant only depending on the choice of the repre-sentative s β p ∈ G and on the choice of the root vector X − β p ∈ g , g p = n − p gn p , n p = X − β ( t ) . . . X − β p − ( t p − ) and it is assumed that n = 1 , w = 1 .Proof. The numbers t p can be found by induction starting with p = 1. We shall establish the induction step. Thecase p = 1 corresponding to the base of the induction can be considered in a similar way.Assume that t , . . . , t p − have already been found. Then g p s − = n − p gn p s − = X − β p ( t p ) . . . X − β c ( t c ) n s zsm s X − β c ( − t c ) . . . X − β p ( − t p ) s − , where n p = X − β ( t ) . . . X − β p − ( t p − ). By Lemma 3.4.4 we can write X − β p ( t p ) . . . X − β c ( t c ) n s = m X − β p ( t p ) , m ∈ N [ − β p +1 , − β D ] . Since ∆ w − p = { β , . . . , β p } we have w − p N [ − β p +1 , − β D ] w p ⊂ N − , and hence( w p v ω ip , g p s − w p − v ω ip ) = ( w p v ω ip , m X − β p ( t p ) zsm s X − β c ( − t c ) . . . X − β p ( − t p ) s − w p − v ω ip ) == ( w p v ω ip , X − β p ( t p ) zsm s X − β c ( − t c ) . . . X − β p ( − t p ) s − w p − v ω ip )as v ω ip is a highest weight vector.Now observe that m s X − β c ( − t c ) . . . X − β p ( − t p ) ∈ M s − N [ − β p , − β c ] . Note that the properties of the normal orderingin ∆ + associated to s imply that M s − = N [ − β k , − β c ] for some k and that the union [ − β p , − β c ] ∪ [ − β k , − β c ] is a minimalsegment, so the subgroup M s − N [ − β p , − β c ] is generated by one–parameter subgroups corresponding to the roots fromthat segment. Thus using Lemma 3.4.4 one can uniquely factorize the element m s X − β c ( − t c ) . . . X − β p ( − t p ) as m s X − β c ( − t c ) . . . X − β p ( − t p ) = m m , m ∈ N [ − β k , − β c ] , m ∈ N [ − β p , − β k − ] , CHAPTER 3. Q-W–ALGEBRAS where it is assumed that N [ − β p , − β k − ] = 1 if p > k −
1. If α ∈ [ β p , β k − ] then sα ∈ ∆ s + and by the propertiesof the normal ordering (1.6.3) in Proposition 1.6.1 we have sα > α , and if sα + α ∈ ∆ s + for α ∈ ∆ then sα + α > α . Observing also that Z is generated by one–parameter subgroups corresponding to roots from ∆ and by the centralizer of s in H which normalizes all one–parameter subgroups corresponding to roots, we deduce zsm s X − β c ( − t c ) . . . X − β p ( − t p ) s − = m zsm s − , m = zsm s − z − ∈ N [ − β p +1 , − β D ] .Now using Lemma 3.4.4 we can uniquely factorize the element X β p ( t p ) m ∈ N [ − β p , − β D ] as X β p ( t p ) m = m X β p ( t p ), m ∈ N [ − β p +1 , − β D ] . Remark that w − p N [ − β p +1 , − β D ] w p ⊂ N − as ∆ w − p = { β , . . . , β p } and hence( w p v ω ip , g p s − w p − v ω ip ) = ( w p v ω ip , X − β p ( t p ) m zsm s − w p − v ω ip ) == ( w p v ω ip , m X − β p ( t p ) zsm s − w p − v ω ip ) == ( w p v ω ip , X − β p ( t p ) zsm s − w p − v ω ip )as v ω ip is a highest weight vector.By the definition of m we have sm s − ∈ N . Denote by δ , . . . , δ D the normal ordering (1.6.3) in ∆ s + corresponding to s , so that β p = δ q for some q . Applying arguments similar to those used above we can factorize sm s − = m m , m ∈ N [ δ ,δ q − ] , m ∈ N [ δ q ,δ D ] . Note that [ δ q , δ D ] ⊂ [ β p , β D ], and each root from this segmentremains positive under the action of w − p − . Therefore we obtain that( w p v ω ip , g p s − w p − v ω ip ) = ( w p v ω ip , X − β p ( t p ) zsm s − w p − v ω ip ) == ( w p v ω ip , X − β p ( t p ) zm m w p − v ω ip ) = ( w p v ω ip , X − β p ( t p ) zm w p − v ω ip ) == ( X β p ( t p ) w p v ω ip , zm w p − v ω ip ) , where we assume that the root vectors X ± β p are chosen in such a way that ω ( X − β p ) = X β p .Since β p ∆ w − p − we infer X β p w p − v ω ip = 0, and hence w p − v ω ip is a highest weight vector for the sl –triplegenerated by the elements X ± β p . Moreover, since w − p − ( − β p ) = − α i p the vector X − α ip = w − p − X − β p w p − is a rootvector corresponding to − α i p , and hence X − β p w p − v ω ip = w p − X − α ip v ω ip = 0 by the definition of v ω ip . Therefore w p − v ω ip is a highest weight vector for the two–dimensional irreducible representation of the sl –triple generatedby the elements X ± β p , and s β p w p − v ω ip is a non–zero lowest weight vector for that representation. Recalling thestandard sl –representation theory, the fact that each weight space is an eigenspace for the action of H and that X β p ( t p ) w p v ω ip = X β p ( t p ) s β p w p − v ω ip we deduce X β p ( t p ) w p v ω ip = X β p ( t p ) s β p w p − v ω ip = s β p w p − v ω ip + t p X β p s β p w p − v ω ip == s β p w p − v ω ip + t p c p w p − v ω ip , where c p is a non–zero constant only depending on the choice of the representative s β p ∈ G and on the choice ofthe root vectors X ± β p . The first term in the last sum has weight − δ q + w p − ω i p , and the second one w p − ω i p .The vector zm w p − v ω ip is a linear combination of vectors of weights of the form w p − ω i p + P q − r =1 c r δ r + ω ,where ω ∈ h ∗ , c r ∈ { , , , . . . } . Since weight spaces corresponding to different weights are orthogonal withrespect to the contravariant bilinear non–degenerate form on V ω ip the only nontrivial contributions to the product( X β p ( t p ) w p v ω ip , zm w p − v ω ip ) come from the products of vectors of weights either − δ q + w p − ω i p or w p − ω i p .In the first case we must have w p − ω i p + P q − r =1 c r δ r + ω = − δ q + w p − ω i p , and hence P q − r =1 c r δ r + ω = − δ q . Inparticular, h ( P q − r =1 c r δ r + ω ) = P q − r =1 c r h ( δ r ) = − h ( δ q ) which is impossible as − h ( δ q ) < c r h ( δ r ) ≥ w p − ω i p + P q − r =1 c r δ r + ω = w p − ω i p or P q − r =1 c r δ r + ω = 0. In particular, h ( P q − r =1 c r δ r + ω ) = P q − r =1 c r h ( δ r ) = 0 which forces c r = 0 for all r as h ( δ r ) > c r ∈ { , , , . . . } , andhence ω = 0 as well.We conclude that the only nontrivial contributions to the product( X β p ( t p ) w p v ω ip , zm w p − v ω ip )come from the products of vectors of weights w p − ω i p . By the above considerations only terms of the form zw p − v ω ip may give contributions of weight w p − ω i p in the weight decomposition of the element zm w p − v ω ip , and this yields( w p v ω ip , g p s − w p − v ω ip ) = ( X β p ( t p ) w p v ω ip , zm w p − v ω ip ) == t p c p ( w p − v ω ip , zw p − v ω ip ) . .5. ZHELOBENKO TYPE OPERATORS FOR POISSON Q-W–ALGEBRAS t p = c p ( w p v ω ip , g p s − w p − v ω ip )( w p − v ω ip , zw p − v ω ip ) . Similar arguments show that( w p − v ω ip , zw p − v ω ip ) = ( w p − v ω ip , g p s − w p − v ω ip ) . Combining the last two identities we obtain formula (3.5.3).Formulas (3.5.3) have no direct quantum analogues, and for the purposes of quantization we shall need otherformulas for the coefficients t p introduced in the previous proposition. These formulas express the coefficients t p interms of other matrix elements of finite-dimensional irreducible representations of G . These matrix elements canbe defined by specializing the results of Lemma 2.3 in [25] at q = 1. By this lemma there are integral dominantweights µ p ( µ ′ p ), p = 1 , . . . , D and elements v p ∈ V µ p ( v ′ p ∈ V µ ′ p ) such that( v p , X ( n D ) − β D . . . X ( n ) − β v µ p ) = ( X ( n D ) − β D . . . X ( n ) − β = X − β p , (3.5.4)( v ′ p , X ( n ) − β . . . X ( n D ) − β D v µ ′ p ) = ( X ( n ) − β . . . X ( n D ) − β D = X − β p , where for α ∈ ∆, k ∈ N X ( k ) α = X kα k ! . Proposition 3.5.2.
Let g = nn s zsm s n − ∈ N − ZsM − be an element of N − ZsM − represented as in (3.5.1), where n = X − β ( t ) . . . X − β c ( t c ) , g p = n − p gn p , n p = X − β ( t ) . . . X − β p − ( t p − ) . Then g p ∈ N [ − β p , − β D ] ZsM s − , g p = X − β p ( r p ) . . . X − β D ( r D ) zsm ′ s , m ′ s ∈ M s − , (3.5.5) r p = t p , and the numbers t p can be found inductively by the following formula t p = ( v p , g p s − v µ p )( v µ p , g p s − v µ p ) = ( v ′ p , g p s − v µ ′ p )( v µ ′ p , g p s − v µ ′ p ) . (3.5.6) Proof.
We shall prove this proposition by induction. In the proof we shall use the notation and arguments fromthe proof of the previous proposition. Firstly we show that g ∈ N − ZsM s − . Indeed, using Lemma 3.4.4 we canrepresent any element m ∈ M − as m = m m , m ∈ N [ − β , − β k − ] , m ∈ M s − . Next, if α ∈ [ β , β k − ] then sα ∈ ∆ s + and by the properties of the normal ordering (1.6.3) in Proposition 1.6.1 we have sα > α , and if sα + α ∈ ∆ s + for α ∈ ∆ then sα + α > α , and in both cases sα, sα + α ∈ ∆ + . Observing also that Z is generated bythe one–parameter subgroups corresponding to roots from ∆ and by the centralizer of s in H which normalizesall one–parameter subgroups corresponding to roots, we deduce zsm s − ∈ N − Z for any z ∈ Z . Therefore zsm = zsm s − sm ∈ N − ZsM s − , so ZsM − ⊂ N − ZsM s − , and hence g ∈ N − ZsM − ⊂ N − ZsM s − .Now we shall prove the first two statements by induction. Assume that g p = X − β p ( r p ) . . . X − β D ( r D ) zsm ′ s , z ∈ Z, m ′ s ∈ M s − . (3.5.7)Note that in the case when p = 1 this expression is already established as it directly follows from the fact that g ∈ N − ZsM s − proved above.We show first that r p = t p . By Lemma 3.4.4 we can write X − β p ( r p ) . . . X − β D ( r D ) = m X − β p ( r p ) , m ∈ N [ − β p +1 , − β D ] . Since ∆ w − p = { β , . . . , β p } we have w − p N [ − β p +1 , − β D ] w p ⊂ N − , and hence( w p v ω ip , g p s − w p − v ω ip ) = ( w p v ω ip , m X − β p ( r p ) zsm ′ s s − w p − v ω ip ) =( w p v ω ip , X − β p ( r p ) zsm ′ s s − w p − v ω ip )4 CHAPTER 3. Q-W–ALGEBRAS as v ω ip is a highest weight vector.Now recall that m ′ s ∈ M s − . Note that the properties of the normal ordering in ∆ + associated to s imply that M s − = N [ − β k , − β c ] for some k .By the definition of m ′ s we have sm ′ s s − ∈ N . Denote by δ , . . . , δ D the normal ordering in ∆ s + correspondingto s , so that β p = δ q for some q . Using arguments similar to those used above we can factorize sm ′ s s − = mm ′ , m ∈ N [ δ ,δ q − ] , m ′ ∈ N [ δ q ,δ D ] . Note that [ δ q , δ D ] ⊂ [ β p , β D ], and each root from this segment remains positive underthe action of w − p − . Therefore we obtain that( w p v ω ip , g p s − w p − v ω ip ) = ( w p v ω ip , X − β p ( t p ) zsm ′ s s − w p − v ω ip ) == ( w p v ω ip , X − β p ( t p ) zmm ′ w p − v ω ip ) = ( w p v ω ip , X − β p ( t p ) zmw p − v ω ip ) == ( X β p ( t p ) w p v ω ip , zmw p − v ω ip ) , where we assume that the root vectors X ± β p are chosen in such a way that ω ( X − β p ) = X β p .Since β p ∆ w − p − we infer X β p w p − v ω ip = 0, and hence w p − v ω ip is a highest weight vector for the sl –triplegenerated by the elements X ± β p . Moreover, since w − p − ( − β p ) = − α i p the vector X − α ip = w − p − X − β p w p − is a rootvector corresponding to − α i p , and hence X − β p w p − v ω ip = w p − X − α ip v ω ip = 0 by the definition of v ω ip . Therefore w p − v ω ip is a highest weight vector for the two–dimensional irreducible representation of the sl –triple generatedby the elements X ± β p , and s β p w p − v ω ip is a non–zero lowest weight vector for that representation. Recalling thestandard sl –representation theory, the fact that each weight space is an eigenspace for the action of H and that X β p ( t p ) w p v ω ip = X β p ( t p ) s β p w p − v ω ip we deduce X β p ( r p ) w p v ω ip = X β p ( r p ) s β p w p − v ω ip = s β p w p − v ω ip + r p X β p s β p w p − v ω ip == s β p w p − v ω ip + r p c p w p − v ω ip , where c p is a non–zero constant only depending on the choice of the representative s β p ∈ G and on the choice ofthe root vector X ± β p . The first term in the last sum has weight − δ q + w p − ω i p , and the second one w p − ω i p .The vector zmw p − v ω ip is a linear combination of vectors of weights of the form w p − ω i p + P q − u =1 c u δ u + ω ,where ω ∈ h ∗ , c u ∈ { , , , . . . } . Since weight spaces corresponding to different weights are orthogonal withrespect to the contravariant bilinear non–degenerate form on V ω ip the only nontrivial contributions to the product( X β p ( r p ) w p v ω ip , zmw p − v ω ip ) come from the products of vectors of weights either − δ q + w p − ω i p or w p − ω i p .In the first case we must have w p − ω i p + P q − u =1 c u δ u + ω = − δ q + w p − ω i p , and hence P q − u =1 c u δ u + ω = − δ q . Inparticular, h ( P q − u =1 c u δ u + ω ) = P q − u =1 c u h ( δ u ) = − h ( δ q ) which is impossible as − h ( δ q ) < c u h ( δ u ) ≥ w p − ω i p + P q − u =1 c u δ u + ω = w p − ω i p or P q − u =1 c u δ u + ω = 0. In particular, h ( P q − u =1 c u δ u + ω ) = P q − u =1 c u h ( δ u ) = 0 which forces c u = 0 for all u as h ( δ u ) > c u ∈ { , , , . . . } , andhence ω = 0 as well.We conclude that the only nontrivial contributions to the product( X β p ( r p ) w p v ω ip , zmw p − v ω ip )come from the products of vectors of weights w p − ω i p . By the above considerations only terms of the form zw p − v ω ip may give contributions of weight w p − ω i p in the weight decomposition of the element zmw p − v ω ip , and this yields( w p v ω ip , g p s − w p − v ω ip ) = ( X β p ( r p ) w p v ω ip , zmw p − v ω ip ) == r p c p ( w p − v ω ip , zw p − v ω ip ) . Therefore r p = c p ( w p v ω ip , g p s − w p − v ω ip )( w p − v ω ip , zw p − v ω ip ) . Similar arguments show that( w p − v ω ip , zw p − v ω ip ) = ( w p − v ω ip , g p s − w p − v ω ip ) . Combining the last two identities we obtain formula (3.5.3) for r p , i. e. t p = r p . At the same time when p = 1this yields formula (3.5.3) for r and establishes the base of induction. .5. ZHELOBENKO TYPE OPERATORS FOR POISSON Q-W–ALGEBRAS g p +1 is defined by putting t p = 0 in the formula g p = X − β p ( t p ) . . . X − β c ( t c ) n s zsm s X − β c ( − t c ) . . . X − β p ( − t p ) , (3.5.8)and along with t p = r p (3.5.7) yields g p +1 = X − β p +1 ( r ′ p +1 ) . . . X − β D ( r ′ D ) zsm ′ s , z ∈ Z, m ′ s ∈ M s − . This establishes the induction step, and hence the first two statements of this proposition are proved.Now we establish the second expression for t p in (3.5.6), the first one can be justified in a similar way. By(3.5.5) we have ( v ′ p , g p s − v µ ′ p ) = ( v ′ p , X − β p ( r p ) . . . X − β D ( r D ) zsm ′ s s − v µ ′ p ) (3.5.9)Recall that sM s − s − ⊂ N by the definition of M s − . Note that ZN ⊂ G is a Lie subgroup with the Lie algebra n + z .Since ZN is generated by the one–parameter subgroups corresponding to the roots from the set ∆ s + ∪ ∆ and by H and the action of G on V µ p is locally finite we can write zsm ′ s s − v µ p = xv µ p , where x ∈ U ( n + z ).Now observe that n + z = (( n ∩ n − ) + z − ) + h ′⊥ + ( z + + ( n ∩ n + )), where z ± = z ∩ n ± and the terms ( n ∩ n ± ) + z ± , h ′⊥ in the direct linear sum in the right hand side are Lie algebras. Therefore by the Poincar´e-Birkhoff-Witttheorem x is a linear combination of terms of the form x − x x + , x ± ∈ U (( n ∩ n ± ) + z ± ) x ∈ U ( h ′⊥ ). Since v µ p is ahighest weight vector only terms with x + ∈ C can contribute to the right hand side of (3.5.9). Rearranging scalarfactors we can assume without loss of generality that for these terms x + = 1. Observe also that the roots α ∈ ∆ − corresponding to the root vectors generating the Lie algebra ( n ∩ n − ) + z − satisfy the condition α > − β c and that − β p ≤ − β c . Therefore (3.5.4) implies that only terms with x − ∈ C can contribute to the right hand side of (3.5.9).Rearranging scalar factors we can assume without loss of generality that for these terms x − = 1. Thus by (3.5.4)we deduce ( v ′ p , g p s − v µ ′ p ) = ( v ′ p , X − β p ( r p ) . . . X − β D ( r D ) x v µ ′ p ) = µ ′ p ( x ) r p . Similarly we obtain ( v µ ′ p , g p s − v µ ′ p ) = ( v µ ′ p , x v µ ′ p ) = µ ′ p ( x ) . The last two identities imply the second expression in (3.5.6).
Remark 3.5.3.
Note that N − ZsM − is a closed subvariety in G . Therefore for each p = 1 , . . . , c the right handside of (3.5.3) is a regular function on N − ZsM − as the composition of the regular function n = X − β ( t ) . . . X − β c ( t c ) t p defined on M − and of isomorphism (3.4.29) of varieties. Hence the denominators in (3.5.3) and (3.5.6) must becanceled. From Proposition 3.5.2 we immediately obtain the following corollary.
Corollary 3.5.4.
Let φ k and ψ k , k = 1 , . . . , D be the regular functions on G defined as follows φ k ( g ) = ( v k , gs − v µ k ) , k = 1 , . . . , D, g ∈ G,ψ k ( g ) = ( v ′ k , gs − v µ ′ k ) , k = 1 , . . . , D, g ∈ G. Then the restrictions of the functions φ k , k = 1 , . . . , p − to N − sZM − = N − sZM s − or the restrictions of thefunctions ψ k ( g ) , k = 1 , . . . , p − to N − sZM − = N − sZM s − generate the vanishing ideal of N [ − β p , − β D ] sZM s − ⊂ N − sZM s − . By repeating verbatim the arguments in the proof of Proposition 3.5.2 one can also obtain another corollary.
Corollary 3.5.5.
The restrictions of the functions φ k , k = 1 , . . . , p − to N − sZHM − = N − sZHM s − or therestrictions of the functions ψ k ( g ) , k = 1 , . . . , p − to N − sZHM − = N − sZHM s − generate the vanishing ideal of N [ − β p , − β D ] sZHM s − ⊂ N − sZHM s − .Let ∆ ′ s = (∆ m + ∩ ∆ s ) ∪ { α ∈ ∆ + : α > γ l ′ } , N ′− = N − ∆ ′ s . Then the restrictions of the functions φ k ( g ) , k =1 , . . . , p − to N − sZHN − = N − sZHN ′− or the restrictions of the functions ψ k ( g ) , k = 1 , . . . , p − to N − sZHN − = N − sZHN ′− generate the vanishing ideal of N [ − β p , − β D ] sZHN ′− ⊂ N − sZHN ′− . CHAPTER 3. Q-W–ALGEBRAS
Observing that in the notation of Proposition 3.5.2 for g = nn s zsm s n − ∈ N − ZsM − we have g c +1 = n s zsm s = n − c +1 gn c +1 , n = n c +1 = X − β ( t ) . . . X − β c ( t c ) and recalling the definition of the operator Π in (3.5.2) we infer thefollowing theorem from Proposition 3.5.2. Theorem 3.5.6.
Let G p , p = 1 , . . . , c be the rational function on G defined by G p ( g ) = ( v p , gs − v µ p )( v µ p , gs − v µ p ) , (3.5.10) and Π p the operator on the space of rational functions on G induced by conjugation by the element exp( − G p X − β p ) , Π p f ( g ) = f (exp( − G p ( g ) X − β p ) g exp( G p ( g ) X − β p )) . (3.5.11) Then the restriction of the composition Π ◦ . . . ◦ Π c to C [ N − ZsM − ] is equal to the projection operator Π onto thesubspace C [ N − ZsM − ] M − of M − –invariant regular functions on N − ZsM − , Π : C [ N − ZsM − ] → C [ N − ZsM − ] M − , Π = Π ◦ . . . ◦ Π c . (3.5.12)This theorem has a quantum counterpart which will be formulated and proved in the next chapter. Corollary3.5.5 and Theorem 3.5.6 are the only results of this chapter which will be used in Chapter 4. The results on Poisson–Lie groups used in this book can be found in [18], [28], [85], [91].Proposition 3.1.1 is stated in [18] as Theorem 1.3.2 and Proposition 3.1.2 and the relevant properties of classicalr-matrices can be found in [7], [89].The result stated in Proposition 3.2.1 can be found in [91], Section 2.Q-W–algebras for realizations of quantum groups associated to Weyl group elements were introduced in [95, 96]in the case of Coxeter elements and in [99] in the general situation. However, in the definitions given in thosepapers other forms of the quantum group are used. The definition of q-W–algebras in this book is more close tothe one given in [101]; it uses the Ad locally finite part of the quantum group (see [52], Chapter 7) which reducesto the algebra of regular functions on G when q = 1. However, in this book we define all algebras over slightlydifferent rings.The exposition in Sections 3.2 and 3.4 follows [99, 101] with some appropriate modifications.The presentation of the results on Poisson reduction in Section 3.3 is close to [97], Section 2.3. More details onthe notion and the properties of dual pairs and Poisson reduction can be found in [90], and for statements relatedto the moment map for Poisson–Lie group actions the reader is referred to [67].The original definition of the Poisson algebras W s ( G ) using the classical Poisson reduction only was given in[98].The definition of the classical Zhelobenko type operator Π in Section 3.5 is a modified version of the definitiongiven in [104]. hapter 4 Zhelobenko type operators forq-W–algebras
In this chapter we define a quantum analogue Π q of the operator Π and apply it to describe q-W–algebras. Observethat the operator Π is defined using the conjugation action and operators of multiplication by the functions G p inthe space C [ N − ZsM − ]. The conjugation action has a natural quantum group analogue, the adjoint action. Butmultiplication by functions in C [ G ] is quite far from the multiplication in the algebra C B [ G ∗ ] which is used in thedefinition of q-W–algebras. However, using isomorphism (3.2.19) of Ad–modules C B [ G ∗ ] and C B [ G ] we can try todescribe q-W–algebras in terms of the space C B [ G ] multiplication in which is more closely related to that of C [ G ].Therefore it is natural to expect that a quantum analogue of the operator Π, if it exists at all, should be definedin terms of the adjoint action and of operators of multiplication in C B [ G ] using appropriate quantum analogues offormulas (3.5.10), (3.5.11) and (3.5.12). We shall see that this conjecture is almost correct. In fact C B [ G ] shouldbe replaced with a certain localization. More precisely, recall that the operator Π is defined using the functions G p given by (3.5.10). Natural analogues of matrix elements which appear in formula (3.5.10) can be defined inthe algebra C B [ G ]. But formula (3.5.10) contains some artificial denominators which are canceled in the formulafor Π (see Remark 3.5.3). It turns out that in the quantum case formulas similar to (3.5.10) make sense but thedenominators in them are not canceled in the formula for Π q , and we are forced to replace C B [ G ] with a localizationcontaining all such denominators. This will also force us to replace the algebra W s B ( G ) with a certain localization W s,loc B ( G ) of it.The main difficulty in defining a quantum analogue Π q of the operator Π is that the proof of the fact thatthe operator defined by (3.5.12) is a projection operator onto W s ( G ) is based on isomorphism (3.4.29) a quantumcounterpart of which does not make sense. Recall that W s B ( G ) is the space of invariants with respect to the adjointaction of C B [ M + ] on Q B . Although quantum analogues of operators (3.5.11) can be defined the proof of the factthat their composition similar to (3.5.12) is a projection operator on the localization W s,loc B ( G ) of W s B ( G ) shouldonly use the algebra structure of C B [ G ], the properties of the adjoint action of C B [ M + ] on Q B , and the structureof Q B . These are the only technical tools in our disposal.Thus our first task is to describe in terms of C B [ G ] the C B [ M + ]–module Q B originally defined using C B [ G ∗ ].In the classical case this would correspond to describing the vanishing ideal of the closed subvariety N − ZsM − ≃ N − ZsM s − ⊂ G as by Lemma 3.4.1 and by Theorem 3.4.5 Q B / ( q dr − Q B ≃ C [ N − ZsM s − ]. It turns out that notall elements of C [ G ] generating the vanishing ideal of N − ZsM s − have nice quantum counterparts in C B [ G ]. Recallthat C [ G ] is P + × P + –graded via the left and the right regular action of H on G . The subvariety N − ZsM s − ⊂ G isclosed and some generators of its vanishing ideal belong to the graded components and some do not. It turns outthat at least some of the generators of the latter type have no nice quantum counterparts. But for our purposesit suffices to replace N − ZsM s − with a larger variety N − ZHsM s − the vanishing ideal of which has a nice quantumcounterpart J B in C B [ G ]. This counterpart is described in Proposition 4.1.2 and its image under the natural map C B [ G ] ≃ C B [ G ∗ ] → Q B is zero.After recollecting some facts on the algebra C B [ G ] and on the adjoint action in Section 4.2 we study properties of J B in Section 4.3. For technical reasons related to the non–commutativity of the algebra C B [ G ] we also introduce andstudy a B –submodule J B ⊂ J B which is a quantum counterpart of the vanishing ideal of N − sZHN ′− ⊃ N − ZHsM s − .In order to show that Π q is a projection operator on W s,loc B ( G ) we shall need more relations which resem-978 CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS ble relations in the algebras C [ N [ − β p , − β D ] sZHM s − ] , C [ N [ − β p , − β D ] sZHN ′− ] of regular functions on the varieties N [ − β p , − β D ] sZHM s − ⊂ N − sZHM s − and N [ − β p , − β D ] sZHN ′− ⊂ N − sZHN ′− introduced in Corollary 3.5.5. Thesealgebras are closely related to the algebras of regular functions on the subvarieties N [ − β p , − β D ] sZM s − ⊂ N − sZM s − .The use of these subvarieties in our setting comes from that fact that elements of N [ − β p , − β D ] sZM s − are obtained fromelements g ∈ N − sZM s − by conjugation by n − p as in Proposition 3.5.2. This conjugation corresponds to applyingthe operator Π ◦ . . . ◦ Π p − in the algebra C [ N − ZsM s − ], so that the vanishing ideal of N [ − β p , − β D ] sZM s − belongs tothe kernel of this operator. For the purposes of quantization we have to replace the varieties N [ − β p , − β D ] sZM s − withthe larger varieties N [ − β p , − β D ] sZHM s − ⊂ N − sZHM s − , N [ − β p , − β D ] sZHN ′− ⊂ N − sZHN ′− , N [ − β p , − β D ] sZM s − ⊂ N [ − β p , − β D ] sZHM s − , N [ − β p , − β D ] sZHN ′− due to the reasons mentioned above, and the vanishing ideals of these ex-tended varieties have well–defined quantum analogues J p B and J p B defined and studied in Section 4.4. Note thatsince we define all our algebras over the ring B , for technical reasons, we shall have to consider a priori slightlylarger ideals I p B = ( J p B ⊗ C ( q dr )) ∩ C B [ G ] and I p B = ( J p B ⊗ C ( q dr )) ∩ C B [ G ].In Section 4.5 we introduce the localizations mentioned above and study their properties and the relevantproperties of the adjoint action. The results obtained in Sections 4.1, 4.2, 4.3, 4.4 and 4.5 are prerequisites for thestudy of the properties of the quantum analogues P p of the operators Π p and of their compositions in Section 4.6,the main properties being summarized in Proposition 4.6.1. In Proposition 4.6.7 we also define quantum analoguesof monomials in variables G p , p = 1 , . . . , c which play a crucial role in the study of equivariant modules over aquantum group and in the proof of the De Concini–Kac–Procesi conjecture.Finally in Section 4.7 we prove that the image of the operator Π q almost coincides with the localization W s,loc B ( G )of the algebra W s B ( G ). In this section we describe a quantum counterpart J B ⊂ C B [ G ] of the vanishing ideal of the variety N − ZHsM s − ⊃ N − ZsM s − containing the closure q ( µ − M − ( u )) = C [ N − ZsM s − ] of the level surface of the moment map µ M − corre-sponding to the element u ∈ M + . As we mentioned in the introduction to this chapter isomorphism (3.2.19) ofAd–modules C B [ G ∗ ] and C B [ G ] plays a central role in the description of J B . However, for technical reasons wereplace the adjoint action of U s,res B ( g ) on C B [ G ] with the twisted adjoint action defined by(Ad xf )( w ) = f ( ω S − s (Ad ′ x ( S s ω w ))) = f (( ω S − s )( x ) wω ( x )) , (4.1.1)where f ∈ C B [ G ] , x, w ∈ U s,res B ( g ), and consider isomorphism (3.2.19) twisted by ω S − s , ϕ : C B [ G ] → C B [ G ∗ ] , f ( id ⊗ f )( id ⊗ ω S − s )( R s R s ) . (4.1.2)If κ = 1 it induces a homomorphism of C B [ M + ]–modules φ : C B [ G ] → Q B , φ ( f ) = ϕ ( f )1 , (4.1.3)where C B [ G ] is equipped with the restriction of action (4.1.1) to C B [ M + ] and Q B with the action induced by theadjoint action Ad of C B [ M + ] and 1 is the image of 1 ∈ C B [ G ∗ ] in Q B under the natural map C B [ G ∗ ] → Q B .The following proposition is a quantum counterpart of Proposition 3.4.11. The proof of Proposition 3.4.11 wasbased on the Chevalley commutation relations between one–parameter subgroups in G as described in Lemma1.3.1 and on formula (3.4.9) for representatives of Weyl group elements in G . In the quantum case instead ofthe Chevalley commutation relations we have commutation relations between quantum root vectors, and the Weylgroup is replaced with the corresponding braid group generators of which are also expressed in terms on generatorsof the quantum group by formula (2.2.3). However, in the quantum case the generators of the braid group do notsquare to identity automorphisms of the quantum group and we are only allowed to use the braid group relations.The action of the braid group on quantum root vectors is also very difficult to control. It is much more complicatedthan the conjugation action of representatives in G of Weyl group elements on root vectors in g . All this bringsadditional complications to the proof of Proposition 4.1.2.As before we assume that a Weyl group element s ∈ W and an ordered system of positive roots ∆ + associatedto s are fixed, and denote by β , . . . , β D the ordered roots in ∆ + as in Section 1.6. .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP U resq ( w ′ ( b + )) be the subalgebra in U resq ( g ) generated by the elements( X − β kl ′ ) ( n kl ′ ) , . . . , ( X − β ) ( n ) , ( X + β kl ′ +1 ) ( n kl ′ +1 ) , . . . , ( X + β D ) ( n D ) , n i ∈ N , i = 1 , . . . , D, where β k l ′ = γ l ′ , and by U resq ( H ).Below we denote the multiplication in the algebra C B [ G ] by ⊗ . We shall also use the following notation forelements of C B [ G ]. Let V res be a U s,res B ( g )–lattice in a finite rank U h ( g )–module V . Recall that there is acontravariant non–degenerate bilinear form ( · , · ) on V such that ( u, xv ) = ( ω ( x ) u, v ) for any u, v ∈ V , x ∈ U h ( g ).Assume that u is such that ( u, w ) ∈ B for any w ∈ V res . Then ( u, · ) is an element of the dual module V res ∗ . Since V and V res are of finite ranks and ( · , · ) is non–degenerate all elements of V res ∗ can be obtained this way. Clearly,for any v ∈ V res ( u, · v ) ∈ C B [ G ], and by the definition C B [ G ] is generated by such elements.Let c β ∈ B , β ∈ ∆ m + be elements such that c β = (cid:26) k i ∈ B if β = γ i , i = 1 , . . . , l ′ . As we observed in the proof of Proposition 3.2.6 the elements ˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k β with n j , k j , i ∈ N , j =1 , . . . , D form a B –basis in C B [ G ∗ ].Clearly, the elements ˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D alsoform a B –basis in C B [ G ∗ ]. Let I k B be the B –submodule in C B [ G ∗ ] generated by the elements˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D , and where at least one k j > j < c + 1. Since these elements are linearlyindependent they form a B –basis in I k B . Proposition 4.1.1.
Let J B be the left ideal in C B [ G ] generated by the elements ( u, · v ) ∈ C B [ G ] , where u is ahighest weight vector in a finite rank representation V of U h ( g ) , and v ∈ V res is such that ( u, T s xv ) = 0 for any x ∈ U resq ( w ′ ( b + )) . Denote I B = ( J B ⊗ B C ( q dr )) ∩ C B [ G ] . Then ϕ ( J B ) ⊂ I k B ∩ C B [ G ∗ ] and ϕ ( I B ) ⊂ I k B ∩ C B [ G ∗ ] .Let Q k B is the image of C B [ G ∗ ] ⊂ C B [ G ∗ ] under the canonical projection C B [ G ∗ ] → C B [ G ∗ ] /I k B . Denote by ∈ Q k B the image of ∈ C B [ G ∗ ] in Q k B .If u is a highest weight vector in a finite rank indecomposable representation V λ of U h ( g ) of highest weight λ such that ( u, u ) = 1 then for any f ∈ C B [ G ] we have in Q k B ϕ ( f ⊗ ( u, · T − s u )) = ϕ (Ad ( q − ( κ s − s P h ′ + id ) λ ∨ )( f )) q ( s − + id )( id − κP h ′ ) λ ∨ q ( s − + id )( id − κP h ′ ) λ ∨ ϕ (Ad ( q ( − κ s − s s − P h ′ + s − ) λ ∨ )( f ))1 , where the classes in the quotient C B [ G ∗ ] /I k B of the elements in the right hand side of (4.1.10) belong to Q k B . Inparticular, φ (( u, · T − s u )) = q ( s − + id )( id − κP h ′ ) λ ∨ , and q ( s − + id )( id − κP h ′ ) λ ∨ should be understood as the class of the element q ( s − + id )( id − κP h ′ ) λ ∨ ∈ C B [ G ∗ ] in thequotient C B [ G ∗ ] /I k B . This class belongs to Q k B .Proof. The proof of this proposition is based on Lemma 4.1.7 which will be proved in the end of this section.First we obtain a useful expression for ( id ⊗ ω S − s )( R s R s ). In order to do that we recall some properties ofuniversal R-matrices, ( S s ⊗ id ) R s = ( id ⊗ S − s ) R s = R s − , ( S s ⊗ S s ) R s = R s , Using the first identity above we can write R s R s = R s ( id ⊗ S s )( R s − ) = ( id ⊗ S − s )( R s − )( id ⊗ S s )( R s − ) = ( id ⊗ S s )(( id ⊗ S − s )( R s − ) ◦ R s − ) , where ( a ⊗ b ) ◦ ( c ⊗ d ) = ac ⊗ db. Now since ω an algebra antiautomorphism we have( id ⊗ ω S − s )( R s R s ) = ( id ⊗ ω )(( id ⊗ S − s )( R s − ) ◦ R s − ) = ( id ⊗ ω S − s )( R s − )( id ⊗ ω )( R s − ) . CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS
Recalling the definition of R s we obtain R s − = exp " − h ( l X i =1 Y i ⊗ H i − l X i =1 κ s − s P h ′ H i ⊗ Y i ) × (4.1.5) × Y β ∈ ∆ + exp q − β [(1 − q β ) f β ⊗ e β e − hκ s − s P h ′ β ∨ ] == Y β ∈ ∆ + exp q − β [(1 − q β ) e h ( κ s − s P h ′ − id ) β ∨ f β ⊗ e β q β ∨ ] ×× exp " − h ( l X i =1 Y i ⊗ H i − l X i =1 κ s − s P h ′ H i ⊗ Y i ) and using the fact that S − s = Ad q ρ ∨ , where ρ is a half of the sum of the positive roots, we also deduce( id ⊗ S − s )( R s − ) = Y β ∈ ∆ + exp q − β [(1 − q β ) q − β ( ρ ∨ ) e β q β ∨ ⊗ e h ( κ s − s P h ′ − id ) β ∨ f β ] × (4.1.6) exp " − h ( l X i =1 ( Y i ⊗ H i ) + l X i =1 κ s − s P h ′ H i ⊗ Y i ) . The order of the terms in the products in the formulas above is such that the α –term appears to the left of the β –term if α > β with respect to the normal ordering of ∆ + .Combining (4.1.5) and (4.1.6) we arrive at the following expression for ( id ⊗ ω S − s )( R s R s )( id ⊗ ω S − s )( R s R s ) = ← Y → exp q − β [(1 − q β ) q − β ( ρ ∨ ) e β q β ∨ ⊗ ω ( e h ( κ s − s P h ′ − id ) β ∨ f β )] ×× exp " h ( l X i =1 ( Y i ⊗ H fi ) + l X i =1 κ s − s P h ′ H i ⊗ Y fi ) × (4.1.7) × exp " h ( l X i =1 ( Y i ⊗ H ri ) − l X i =1 κ s − s P h ′ H i ⊗ Y ri ) ×× ← Y → exp q − β [(1 − q β ) f β ⊗ ω ( e β e − hκ s − s P h ′ β ∨ )] , where in the product ← Y → the upper (the lower) arrow indicates the order of the terms in the first (the second) factor of the tensor productrelative to the normal ordering of ∆ + , and superscripts f ( r ) indicate that the corresponding term appears in thefront (in the rear) of all the other terms in the product.Assume now that u ∈ V has highest weight λ and v ∈ V res is any vector of weight µ such that ( u, · v ) ∈ C B [ G ].Observe that the elements of the subalgebra C B [ M + ] appear in the first factor of the tensor product in formula(4.1.7) on the right. Then, since v is a highest weight vector, we have in Q k B using (4.1.7), the definitions of I k B andof Q k B = Im( C B [ G ∗ ] → C B [ G ∗ ] /I k B ), and the definition of the homomorphism ϕϕ (( u, · v )) = q λ ∨ + κ s − s P h ′ λ ∨ + µ ∨ − κ s − s P h ′ µ ∨ ×× ( id ⊗ f )( l ′ Y → i = 1 exp q − γi [ − q − γ i k i ⊗ ω ( e γ i e − hκ s − s P h ′ γ ∨ i )] × (4.1.8) × ← Y → β > γl ′ exp q − β [(1 − q β ) f β ⊗ ω ( e β e − hκ s − s P h ′ β ∨ )]) , .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP f = ( u, · v ).Now formula (4.1.8), the definitions of the subalgebra U resq ( w ′ ( b + )) and of the elements e β = X + β q Kβ ∨ , andLemma 4.1.7 imply that in Q k B φ (( u, · v )) = X i x i ( u, T s y i v )1 , (4.1.9)where x i ∈ C B [ G ∗ ], y i ∈ U resq ( w ′ ( b + )). Since every element of V res is the sum of its weight components a formulaof type (4.1.9) holds for arbitrary ( u, · v ) ∈ C B [ G ], where u is a highest weight vector.If v is chosen in such a way that ( u, T s xv ) = 0 for any x ∈ U resq ( w ′ ( b + )) we deduce from (4.1.9) that ( u, T s y i v ) =0. Thus φ (( u, · v )) = 0 in Q k B .Now using the properties (∆ s ⊗ id ) R s = R s R s , ( id ⊗ ∆ s ) R s = R s R s , and the fact that ω is a coautomorphism and S s is an anti-coautomorphism we get( id ⊗ ∆ s )( id ⊗ ω S − s )( R s R s ) = ( id ⊗ ω S − s ⊗ ω S − s )( id ⊗ ∆ opps )( R s R s ) == ( id ⊗ ω S − s ⊗ ω S − s )( R s − R s − )( id ⊗ ω ⊗ ω )( R s − R s − ) . From this identity we obtain, similarly to (4.1.9), that for any f ∈ C B [ G ] in Q k B ϕ ( f ⊗ ( u, · v )) = X i x ′ i ϕ ( f ) x ′′ i ( u, T s y ′ i v )1 , where x ′ i , x ′′ i ∈ C B [ G ∗ ], y ′ i ∈ U resq ( w ′ ( b + )). Hence ϕ ( f ⊗ ( u, · v )) = 0 in Q k B by the choice of v , i.e. ϕ ( J B ) ⊂ I k B ∩ C B [ G ∗ ].In order to show that ϕ ( I B ) ⊂ I k B ∩ C B [ G ∗ ] we naturally extend ϕ to and Ad–module isomorphism ϕ : C q [ G ] → C q [ G ∗ ], where C q [ G ] = C B [ G ] ⊗ B C ( q dr ), C q [ G ∗ ] = C B [ G ∗ ] ⊗ B C ( q dr ). By the definition of I B we have ϕ ( I B ) ⊂ ( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ] as obviously ϕ ( J B ⊗ B C ( q dr )) ⊂ I k B ⊗ B C ( q dr ) by the first part of the proof and ϕ ( C B [ G ]) ⊂ C B [ G ∗ ].We also have ( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ] ⊂ ( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ] as C B [ G ∗ ] ⊂ C B [ G ∗ ].Recall that by the definition the elements ˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D , and where at least one k j > j < c + 1 form a B –basis in I k B , and this basis can be completedto a B –basis of C B [ G ∗ ] which consists of the elements ˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D .This implies ( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ] = I k B , and hence ϕ ( I B ) ⊂ ( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ] = (( I k B ⊗ B C ( q dr )) ∩ C B [ G ∗ ]) ∩ C B [ G ∗ ] = I k B ∩ C B [ G ∗ ].Arguments similar to those in the first part of the proof show that if u is a highest weight vector in a finite rankindecomposable representation V λ of U h ( g ) of highest weight λ then in Q k B ϕ ( f ⊗ ( u, · T − s u )) = q ( κ s − s P h ′ + id ) λ ∨ ϕ ( f ) q s − ( − κ s − s P h ′ + id ) λ ∨ q ( s − + id )( id − κP h ′ ) λ ∨ Ad( q ( − κ s − s s − P h ′ + s − ) λ ∨ )( ϕ ( f ))1 == q ( s − + id )( id − κP h ′ ) λ ∨ ϕ (Ad ( q ( − κ s − s s − P h ′ + s − ) λ ∨ )( f ))1 == q ( s − + id )( id − κP h ′ ) λ ∨ ϕ (Ad ( q ( − κ s − s s − P h ′ + s − ) λ ∨ )( f )) . The formula above can also be rewritten as ϕ ( f ⊗ ( u, · T − s u )) = Ad( q − ( κ s − s P h ′ + id ) λ ∨ )( ϕ ( f )) q ( s − + id )( id − κP h ′ ) λ ∨ ϕ (Ad ( q − ( κ s − s P h ′ + id ) λ ∨ )( f )) q ( s − + id )( id − κP h ′ ) λ ∨ . In particular, in Q k B ϕ (( u, · T − s u )) = q ( s − + id )( id − κP h ′ ) λ ∨ . CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS
Note that the classes in the quotient C B [ G ∗ ] /I k B of the elements q ( s − + id )( id − κP h ′ ) λ ∨ ϕ (Ad ( q ( − κ s − s s − P h ′ + s − ) λ ∨ )( f )) , ϕ (Ad ( q − ( κ s − s P h ′ + id ) λ ∨ )( f )) q ( s − + id )( id − κP h ′ ) λ ∨ and q ( s − + id )( id − κP h ′ ) λ ∨ in the right hand sides of the formulas above a priori belong to Q k B . This completes theproof.Now consider the case when κ = 1 and k i ∈ B , i = 1 , . . . , l ′ are defined in (3.2.11). Recall that the elements˜ e n β . . . ˜ e n D β D V i ˜ f k D β D . . . ˜ f k c +1 β c +1 ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D form a B –basis in C B [ G ∗ ]and observe that in the considered case the elements ( ˜ f β c − c β c ) k c . . . ( ˜ f β − c β ) k with n j , k j , i ∈ N , j = 1 , . . . , D ,and where at least one k j > j < c + 1, form a B –basis of Ker χ sq . Therefore in the considered case I k B = I B and Q k B = Q B , and we can apply the previous proposition to get the following statement. Proposition 4.1.2.
Assume that κ = 1 and k i ∈ B are defined in (3.2.11). Then J B and I B lie in the kernel of φ .Moreover, if u is a highest weight vector in a finite rank indecomposable representation V λ of U h ( g ) of highestweight λ such that ( u, u ) = 1 then for any f ∈ C B [ G ] φ ( f ⊗ ( u, · T − s u )) = ϕ (Ad ( q − ( s − s P h ′ + id ) λ ∨ )( f )) q P h ′⊥ λ ∨ q P h ′⊥ λ ∨ φ (Ad ( q ( − s − s s − P h ′ + s − ) λ ∨ )( f )) , where the classes in the quotient C B [ G ∗ ] /I B of the elements in the right hand side of (4.1.10) belong to Q B . Inparticular, φ (( u, · T − s u )) = q P h ′⊥ λ ∨ , and q P h ′⊥ λ ∨ should be understood as the class of the element q P h ′⊥ λ ∨ ∈ C B [ G ∗ ] in the quotient C B [ G ∗ ] /I B . Thisclass belongs to Q B . The rest of this section will be devoted to the proof of Lemma 4.1.7. This proof is in turn split into severalother lemmas.
Lemma 4.1.3.
Let V be a finite rank representation of U h ( g ) , u, v ∈ V . Let w = s i . . . s i D be a reduced decompo-sition of the longest element of the Weyl group W . Then for any β = s i . . . s i k − α i k ∈ ∆ + and k ∈ N ( u, ( X + β ) ( k ) v ) = X p,p ′ ( u, K p,p ′ ( X − β ) ( p ) ( X + β ) ( p ′ ) T β v ) , (4.1.11) where the sum in the right hand side is finite, X ± β = T i . . . T i k − X ± i k , K p,p ′ ∈ C [ q, q − ] ∗ are integer powers of q ,and T β = T i . . . T i k − T − i k T − i k − . . . T − i . Proof.
First we assume that v is a weight vector of weight λ . Then the only nontrivial contribution to the left handside of (4.1.11) comes from the component of u of weight λ + kβ , so we can assume that u has weight λ + kβ .Conjugating (2.2.5) by T i . . . T i k − we get exp ′ q i ( − X + β ) = exp ′ q − i ( − q i X − β K − β ) q Hβ ( Hβ +1)2 i exp ′ q − i ( q − i X + β ) T β , (4.1.12)where K β = q β ∨ , H β = T i . . . T i k − H i k .Evaluating this identity on the matrix element ( u, · v ) and using the commutation relations of the quan-tum group we obtain (4.1.11) for v of weight λ , where the sum in the right hand side is such that all terms K p,p ′ ( X − β ) ( p ) ( X + β ) ( p ′ ) T β v have weight λ + kβ , i.e. p ′ − p − β ∨ ( λ ) = k . Adding identities of type (4.1.11) for weightvectors v we obtain identities of the same type for arbitrary v . .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP s is the longest element in the Weyl group W of the semisimple part m of the standard Levi subalgebrathe positive roots of which in ∆ + form the set∆ = { γ , β t + p − n +2 , . . . , β t + p − n + n , γ , β t + p − n + n +2 . . . , β t + p − n + n , γ , . . . , γ n , (4.1.13) − β t +1 , . . . , − β t + p − n } and s acts on them by multiplication by −
1. The roots in (4.1.13) are ordered as in the normal ordering of ∆ + associated to s .Let s = s i . . . s i p = s i k . . . s i k . . . s i kn s i kn +1 . . . s i p , i k = i be the corresponding reduced decomposition of s , where γ m = s i k . . . s i k . . . s i km − α i km , m = 1 , . . . , n . Since s is an involution we also have the following reduced decomposition s = s i p . . . s i = s i p . . . s i kn +1 s i kn . . . s i k . . . s i k . (4.1.14)Let γ ≤ β q ≤ γ n , β q = s i . . . s i q − α i q . Since s = − W s = s i q . . . s i s s i . . . s i q = s i q . . . s i s i . . . s i p s i . . . s i q = s i q +1 . . . s i p s i . . . s i q , and in the right hand side we obtain a reduced decomposition of s . Now s α i q = − α i q = s i q +1 . . . s i p s i . . . s i q α i q = − s i q +1 . . . s i p s i . . . s i q − α i q , and hence s i q +1 . . . s i p s i . . . s i q − α i q = α i q . (4.1.15)Form the expressions γ m = s i k . . . s i k . . . s i km − α i km , m = 1 , . . . , n we deduce s γ . . . s γ q − = s i . . . s i k − s i k . . . s i kq − − s i kq − s i kq − − . . . s i . In particular, we have reduced decompositions s = s γ . . . s γ n = s i . . . s i k − s i k . . . s i kn − s i kn s i kn − . . . s i (4.1.16)and s = s i kn . . . s i s s i . . . s i kn = s i kn s i kn − . . . s i s i . . . s i k − s i k . . . s i kn − . (4.1.17)Comparing the first expression above with (4.1.14) we obtain the following identity for reduced decompositions s i . . . s i k − s i k . . . s i kn − = s i p . . . s i kn +1 . (4.1.18)As the roots γ m , m = 1 , . . . , n are mutually orthogonal we deduce s γ . . . s γ q − γ q = s i . . . s i k − s i k . . . s i kq − − s i kq − s i kq − − . . . s i s i . . . s i kq − α i kq == s i . . . s i k − s i k . . . s i kq − − s i kq − . . . s i kq − α i kq = γ q = s i . . . s i kq − α i kq . Therefore s i kq − . . . s i s i . . . s i k − s i k . . . s i kq − − s i kq − . . . s i kq − α i kq = α i kq , and s i kq − . . . s i s i . . . s i k − s i k . . . s i kq − − s i kq − . . . s i kq − is a reduced decomposition since it is a part of reduced decomposition (4.1.17).The last two properties and (2.2.10) imply T i kq − . . . T i T i . . . T i k − T i k . . . T i kq − − T i kq − . . . T i kq − X ± i kq = X ± i kq , and hence T γ . . . T γ q − X ± γ q = T i . . . T i k − T i k . . . T i kq − − T − i kq − T − i kq − − . . . T − i T i . . . T i kq − X ± i kq == T i . . . T i k − T i k . . . T i kq − − T i kq − . . . T i kq − X ± i kq = T − i . . . T − i kq − X ± i kq = X ± γ q . (4.1.19)04 CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS
Lemma 4.1.4.
Let V be a finite rank representation of U h ( g ) , u, v ∈ V . Then for any m , . . . , m n ∈ N ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) = ( u, KT γ . . . T γ n v ) , (4.1.20) where K is of the form K = X p , . . . , pkp ′ , . . . , p ′ m K p ,...,p k p ′ ,...,p ′ m ( X − δ ) ( p ) . . . ( X − δ k ) ( p k ) ( X + β ) ( p ′ ) . . . ( X + β m ) ( p ′ m ) , (4.1.21) the finite sum in the right hand side is over all δ i , β j ∈ ∆ such that δ < δ < . . . < δ k , γ ≤ β < β < . . . <β m ≤ γ n , and X − δ i , X + β j are the quantum root vectors of U h ( m ) ⊂ U h ( g ) defined with the help of normal ordering(4.1.13) of ∆ , K p ,...,p n p ′ ,...,p ′ m ∈ U resq ( H ) .Proof. By Lemma 4.1.3 ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) == X c , . . . , cnc ′ , . . . , c ′ n K c ,...,c n c ′ ,...,c ′ n ( u, ( X − γ ) ( c ) ( X + γ ) ( c ′ ) T γ ( X − γ ) ( c ) ( X + γ ) ( c ′ ) T γ . . . ( X − γ n ) ( c n ) ( X + γ n ) ( c ′ n ) T γ n v ) , where the sum in the right hand side is finite and K c ,...,c n c ′ ,...,c ′ n are integer powers of q .Using (4.1.19) and observing that X ± γ = X ± γ as γ the first simple root in the normal ordering of ∆ + associatedto s we obtain ( u, ( X + γ ) ( k ) . . . ( X + γ n ) ( k n ) v ) == X c , . . . , cnc ′ , . . . , c ′ n K c ,...,c n c ′ ,...,c ′ n ( u, ( X − γ ) ( c ) ( X + γ ) ( c ′ ) ( X − γ ) ( c ) ( X + γ ) ( c ′ ) . . . ( X − γ n ) ( c n ) ( X + γ n ) ( c ′ n ) T γ . . . T γ n v ) . Now (4.1.20) follows from the previous identity by bringing the monomials( X − γ ) ( c ) ( X + γ ) ( c ′ ) ( X − γ ) ( c ) ( X + γ ) ( c ′ ) . . . ( X − γ n ) ( c n ) ( X + γ n ) ( c ′ n ) in the right hand side to the form as in the right hand side of (4.1.21) with the help of Lemma 2.4.2. Lemma 4.1.5.
Let V be a finite rank representation of U h ( g ) , u, v ∈ V . Suppose that u is a highest weight vector.Then for any m , . . . , m n ∈ N ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) == ( u, X q ,...,q kn C q ,...,q kn T γ . . . T γ n ( X − β ) ( q ) . . . ( X − β kn ) ( q kn ) v ) , where C q ,...,q kn ∈ C [ q, q − ] .Proof. Denote T = T γ . . . T γ n . From the definition of T γ , . . . , T γ n and (4.1.18) we obtain T = T γ . . . T γ n = T i . . . T i k − T i k . . . T i kn − T − i kn T − i kn − . . . T − i = (4.1.22)= T i p . . . T i kn +1 T − i kn T − i kn − . . . T − i . Therefore for γ ≤ β q ≤ γ n , β q = s i . . . s i q − α i q we have( T ) − X + β q = T i . . . T i kn T − i kn +1 . . . T − i p T − i . . . T − i q − X + i q . (4.1.23)By (4.1.15), (2.2.10) T − i q +1 . . . T − i p T − i . . . T − i q − X + i q = X + i q , and hence T − i kn +1 . . . T − i p T − i . . . T − i q − X + i q = T i kn . . . T i q +1 X + i q . .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP T ) − X + β q = T i . . . T i kn T i kn . . . T i q +1 X + i q (4.1.24)Since u is a highest weight vector, Lemma 4.1.4 and the definition of the bilinear form on V imply( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) = ( u, X p ′ ,...,p ′ m K ,..., p ′ ,...,p ′ m ( X + β ) ( p ′ ) . . . ( X + β m ) ( p ′ m ) T γ . . . T γ n v ) . Combining this with (4.1.24) we get ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) = (4.1.25)= ( u, X p ′ ,...,p ′ m K ,..., p ′ ,...,p ′ kn T γ . . . T γ n ( T i . . . T i kn )(( X + δ ) ( p ′ ) . . . ( X + δ kn ) ( p ′ m ) ) v ) , where for q = 1 , . . . , k n we denote δ q = s i kn . . . s i q +1 α i q , and X + δ q = T i kn . . . T i q +1 X + i q .Let U resq ( m ) be the restricted specialization of U h ( m ), U resq ( m ) the subalgebra of U resq ( m ) generated by( X + β ) ( k ) for simple roots β ∈ ∆ and k ≥
0. Then for q = 1 , . . . , k n and k ≥ X + δ q ) ( k ) ∈ U resq ( m ).Since s = s i kn . . . s i s s i . . . s i kn = s i kn . . . s i s i p . . . s i s i . . . s i kn = s i kn . . . s i s i p . . . s i kn +1 , and in the right hand side we have a reduced decomposition, the elements Y φ kn +1 = T − i kn . . . T − i T − i p . . . T − i kn +2 X + i kn +1 ,Y φ kn +2 = T − i kn . . . T − i T − i p . . . T − i kn +3 X + i kn +2 , . . . ,Y φ p = T − i kn . . . T − i X + i p ,Y φ = T − i kn . . . T − i X + i , . . . ,Y φ kn = X + i kn belong to U resq ( m ) and the products( Y φ kn +1 ) ( q kn +1 ) ( Y φ kn +2 ) ( q kn +2 ) . . . ( Y φ p ) ( q p ) ( Y φ ) ( q ) . . . ( Y φ kn ) ( q kn ) form a C [ q, q − ]–basis of U resq ( m ). Expanding each monomial ( X + δ ) ( p ′ ) . . . ( X + δ m ) ( p ′ kn ) in (4.1.25) with respect tothis basis we get( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) = ( u, X q ,...,q p K q ,...,q p T γ . . . T γ n ( T i . . . T i kn )(( Y φ kn +1 ) ( q kn +1 ) ( Y φ kn +2 ) ( q kn +2 ) . . .. . . ( Y φ p ) ( q p ) ( Y φ ) ( q ) . . . ( Y φ kn ) ( q kn ) ) v ) , (4.1.26)where K q ,...,q p ∈ C [ q, q − ].Now for r = k n + 1 , . . . p ( T i . . . T i kn )( Y φ r ) = T − i p . . . T − i r +1 X + i r , (4.1.27)and for r = 1 , . . . k n ( T i . . . T i kn )( Y φ r ) = T i . . . T i r X + i r = − T i . . . T i r − X − i r K i r = − X − β r K β r , (4.1.28)where K β r = T i . . . T i r − K i r .Since u is a highest weight vector, by Lemma 2.7.4 we have T γ . . . T γ u = ω ( c ) T i p . . . T i u , where c ∈ C [ q, q − ] ∗ .Therefore for any element w ∈ V , x ∈ U h ( g ) we can write( u, T γ . . . T γ n xw ) = c ( u, T i . . . T i p xw ) . Combining this with (4.1.27) yields for r = k n + 1 , . . . p ( u, T γ . . . T γ n ( T i . . . T i kn )( Y φ r ) xw ) = c ( u, T i . . . T i p ( T − i p . . . T − i r +1 )( X + i r ) xw ) == c ( u, ( T i . . . T i r )( X + i r ) T i . . . T i p xw ) = − c ( u, ( T i . . . T i r − )( X − i r K i r ) T i . . . T i p xw ) == − c ( u, X − β r K β r T i . . . T i p xw ) = 0 . CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS
From this formula, (4.1.26), (4.1.28) and the commutation relations between the elements K β r and the quantumroot vectors we deduce ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) v ) == ( u, X q ,...,q kn C q ,...,q kn T γ . . . T γ n ( X − β ) ( q ) . . . ( X − β kn ) ( q kn ) v ) , where C q ,...,q kn ∈ C [ q, q − ].Recall now that according to the results obtained in the proof of Proposition 5.1 in [99] s is the longest elementin the Weyl group W of the semisimple part m of the Levi subalgebra the positive roots of which in ∆ + form theset ∆ = { γ n +1 , β q +2 , . . . , β q + m , γ n +2 , β q + m +2 , . . . , β q + m , γ n +3 , . . . , (4.1.29) γ l ′ , β q + m l ( s +1 , . . . , β q +2 m l ( s − ( l ′ − n ) } , and s acts on them by multiplication by −
1. The roots in (4.1.29) are ordered as in the normal ordering of ∆ + associated to s .As above we denote γ m = s i . . . s i kn +1 . . . s i km − α i km , m = n + 1 , . . . , l ′ .Let w = s i . . . s i kn +1 − , T w = T i . . . T i kn +1 − . Then e ∆ = w − (∆ ) is the set of positive roots of the semisimplepart e m of a standard Levi subalgebra of g , e s = w − s w is the longest element in the Weyl group f W of e m .For any root β ∈ ∆ + denote e X ± β = T − w ( X ± β ) , e X ± β = T − w ( X ± β ) , e T β = T − w T β T w . Similarly to Lemma 4.1.4 we have
Lemma 4.1.6.
Let V be a finite rank representation of U h ( g ) , u, v ∈ V . Then for any m n +1 , . . . , m l ′ ∈ N ( u, ( e X + γ n +1 ) ( m n +1 ) . . . ( e X + γ l ′ ) ( m l ′ ) v ) = ( u, e K e T γ n +1 . . . e T γ l ′ v ) , (4.1.30) where e K is of the form e K = X p , . . . , pkp ′ , . . . , p ′ m M p ,...,p k p ′ ,...,p ′ m ( e X − δ ) ( p ) . . . ( e X − δ k ) ( p k ) ( e X + β ′ ) ( p ′ ) . . . ( e X + β ′ m ) ( p ′ m ) , (4.1.31) the finite sum in the right hand side is over all δ i , β j ∈ ∆ such that δ < δ < . . . < δ k , γ n +1 ≤ β ′ < β ′ < . . . <β ′ m ≤ γ l ′ , M p ,...,p n p ′ ,...,p ′ m ∈ U resq ( H ) .Proof. Similarly to Lemma 4.1.4 we deduce that formula (4.1.30) holds with e K = X p , . . . , pkp ′ , . . . , p ′ m L p ,...,p k p ′ ,...,p ′ m ( e X − δ ) ( p ) . . . ( e X − δ k ) ( p k ) ( e X + β ′ ) ( p ′ ) . . . ( e X + β ′ m ) ( p ′ m ) , (4.1.32)where L p ,...,p n p ′ ,...,p ′ m ∈ U resq ( H ).Let U resq ( e m ) be the restricted specialization of U h ( e m ), U resq ( e m ) the subalgebra of U resq ( e m ) generated by( X + β ) ( k ) for simple roots β ∈ e ∆ and k ≥
0. Then the monomials ( e X − δ ) ( p ) . . . ( e X − δ k ) ( p k ) , δ i , ∈ ∆ , δ < δ < . . . <δ k form a linear basis of U resq ( e m ) and the monomials ( e X − δ ) ( p ) . . . ( e X − δ k ) ( p k ) δ i , ∈ ∆ , δ < δ < . . . < δ k form alinear basis of U resq ( e m ) as well. Thus e K can be rewritten in form (4.1.31) using the latter basis. .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP Lemma 4.1.7.
Let V be a finite rank representation of U h ( g ) , u, v ∈ V . Suppose that u is a highest weight vector.Then for any m , . . . , m l ′ ∈ N ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) == X n ,...,n D F n ,...,n D ( u, T γ . . . T γ n T γ n +1 . . . T γ l ′ ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) v ) == c X n ,...,n D F n ,...,n D ( u, T s ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) v ) == c ′ X n ,...,n D F n ,...,n D ( u, T s ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) v ) , where F n ,...,n D ∈ C [ q, q − ] , c, c ′ ∈ C [ q, q − ] ∗ are integer powers of q up to constant factors.Proof. By Lemmas 4.1.5 and 4.1.6 we have( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) == ( u, X q ,...,q kn C q ,...,q kn T γ . . . T γ n ( X − β ) ( q ) . . . ( X − β kn ) ( q kn ) ×× T w X p , . . . , pkp ′ , . . . , p ′ m M p ,...,p k p ′ ,...,p ′ m ( e X − δ ) ( p ) . . . ( e X − δ k ) ( p k ) ( e X + β ′ ) ( p ′ ) . . . ( e X + β ′ m ) ( p ′ m ) e T γ n +1 . . . e T γ l ′ T − w v ) == ( u, X q ,...,q kn C q ,...,q kn T γ . . . T γ n ( X − β ) ( q ) . . . ( X − β kn ) ( q kn ) ×× X p , . . . , pkp ′ , . . . , p ′ m M p ,...,p k p ′ ,...,p ′ m ( X − δ ) ( p ) . . . ( X − δ k ) ( p k ) T w ( e X + β ′ ) ( p ′ ) . . . ( e X + β ′ m ) ( p ′ m ) e T γ n +1 . . . e T γ l ′ T − w v ) . Observe that ( X − β ) ( k ) , β ≤ β ≤ γ l ′ generate a C [ q, q − ]–subalgebra for which the monomials( X − β kl ′ ) ( q kl ′ ) . . . ( X − β ) ( q ) form a linear basis. Rewriting the monomials( X − β ) ( q ) . . . ( X − β kn ) ( q kn ) ( X − δ ) ( p ) . . . ( X − δ k ) ( p k ) in terms of this basis we arrive at( u, ( X + γ ) ( k ) . . . ( X + γ n ) ( k n ) ( X + γ n +1 ) ( k n +1 ) . . . ( X + γ l ′ ) ( k l ′ ) v ) = (4.1.33)= ( u, T γ . . . T γ n X q , . . . , qkl ′ p ′ , . . . , p ′ m N q ,...,q kl ′ p ′ ,...,p ′ m ( X − β kl ′ ) ( q kl ′ ) . . . ( X − β ) ( q ) T w ( e X + β ′ ) ( p ′ ) . . .. . . ( e X + β ′ m ) ( p ′ m ) e T γ n +1 . . . e T γ l ′ T − w v ) , where N q ,...,q kl ′ p ′ ,...,p ′ m ∈ C [ q, q − ].Now observe that if γ n < β ≤ γ l ′ then β ∆ . The root β can be written in the form β = nα + c , where α ∆ is a simple root, n > c is a linear combination of other simple roots. Since W is parabolic in W , s ∈ W and m ⊂ g is a standard Levi subalgebra we deduce s β = nα + c ′ , where c ′ is a linear combination of othersimple roots. This implies s β ∈ ∆ + . Thus ω ( T γ . . . T γ n ( X − β )) u = 0 for otherwise the vector ω ( T γ . . . T γ n ( X − β )) u has weight greater than that of u which is impossible as u is a highest weight vector. Applying this observation in(4.1.33) we deduce ( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) = (4.1.34)08 CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS = ( u, T γ . . . T γ n X q , . . . , qkl ′ p ′ , . . . , p ′ m K q ,...,q kn p ′ ,...,p ′ m ( X − β kn ) ( q kn ) . . . ( X − β ) ( q ) T w ( e X + β ′ ) ( p ′ ) . . . ( e X + β ′ m ) ( p ′ m ) e T γ n +1 . . . e T γ l ′ T − w v ) , where K q ,...,q kn p ′ ,...,p ′ m ∈ C [ q, q − ].Denote e T = e T γ n +1 . . . e T γ l ′ . For γ n +1 ≤ β q ≤ γ l ′ we have β q = ws k n +1 . . . s i q − α i q , q = k n +1 , . . . , k l ′ . Similarlyto (4.1.24) we infer ( e T ) − e X + β q = T i kn +1 . . . T i kl ′ T i kl ′ . . . T i q +1 X + i q . (4.1.35)Denote e X + δ q = T i kl ′ . . . T i q +1 X + i q , q = k n +1 , . . . k l ′ . Then from (4.1.34) and (4.1.35) we obtain( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) = (4.1.36)= ( u, T γ . . . T γ n X q , . . . , qkl ′ p ′ kn +1 , . . . , p ′ kl ′ K q ,...,q kn p ′ kn +1 ,...,p ′ kl ′ ( X − β kn ) ( q kn ) . . . ( X − β ) ( q ) T w e T γ n +1 . . . e T γ l ′ ( T i kn +1 . . . T i kl ′ ) ×× (( e X + δ kn +1 ) ( p ′ kn +1 ) . . . ( e X + δ kl ′ ) ( p ′ kl ′ ) ) T − w v ) == ( u, T γ . . . T γ n X q , . . . , qkl ′ p ′ kn +1 , . . . , p ′ kl ′ K q ,...,q kn p ′ kn +1 ,...,p ′ kl ′ ( X − β kn ) ( q kn ) . . . ( X − β ) ( q ) T w e T γ n +1 . . . e T γ l ′ T i kn +1 . . . T i kl ′ ×× ( e X + δ kn +1 ) ( p ′ kn +1 ) . . . ( e X + δ kl ′ ) ( p ′ kl ′ ) T − w ′ v ) , where K q ,...,q kn p k ′ n +1 ,...,p ′ kl ′ ∈ C [ q, q − ] and T w ′ = T w T i kn +1 . . . T i kl ′ = T i . . . T i kl ′ . Now if w = s i . . . s i kn +1 . . . s i kl ′ s i kl ′ +1 . . . s i p ′ . . . s i D is the reduced decomposition of the longest element of the Weyl group corresponding to the normal ordering in ∆ + associated to s , so that e s = s i kn +1 . . . s i kl ′ s i kl ′ +1 . . . s i p ′ is the corresponding reduced decomposition of e s , then,similarly to (4.1.22), we have e T γ n +1 . . . e T γ l ′ = T i p ′ . . . T i kl ′ +1 T − i kl ′ T − i kl ′ − . . . T − i kn +1 , and hence e T γ n +1 . . . e T γ l ′ T i kn +1 . . . T i kl ′ = T i p ′ . . . T i kl ′ +1 . (4.1.37)Observe that T = T w e T γ n +1 . . . e T γ l ′ T i kn +1 . . . T i kl ′ = T i . . . T i kn +1 − T i p ′ . . . T i kl ′ +1 is the braid group element corresponding to the reduced decomposition s i . . . s i kn +1 − s i p ′ . . . s i kl ′ +1 (4.1.38)which is a part of the reduced decomposition obtained from the reduced decomposition s i . . . s i kn +1 − s i kn +1 . . . s i kl ′ s i kl ′ +1 . . . s i p ′ by inverting the part e s = s i kn +1 . . . s i kl ′ s i kl ′ +1 . . . s i p ′ . This inversion gives a reduced decomposition again because e s = − f W .Now for β ≤ β q ≤ β k n we have T − ( X − β q ) = T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 T − i q . . . T − i T i . . . T i q − X − i q = (4.1.39) .0. A QUANTUM ANALOGUE OF THE LEVEL SURFACE OF THE MOMENT MAP T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 T − i q X − i q == − T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 ( X + i q K i q ) = T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 ( X + i q ) R q , where R q = T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 ( K i q ) . Since s i . . . s i kn +1 − s i p ′ . . . s i kl ′ +1 is a reduced decomposition T − i kl ′ +1 . . . T − i p ′ T − i kn +1 − . . . T − i q +1 ( X + i q ) ( k ) ∈ U resq ( n + ) , where U resq ( n + ) is the subalgebra of U resq ( g ) generated by ( X + i ) ( k ) i = 1 , . . . , l , k = 0 , , . . . . Applying thisobservation, (4.1.39), commutation relations between elements R q and the quantum root vectors, and the fact that u is a weight vector we obtain from (4.1.36)( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) == ( u, T γ . . . T γ n T w e T γ n +1 . . . e T γ l ′ T i kn +1 . . . T i kl ′ XT − w ′ v ) = (4.1.40)= ( u, T γ . . . T γ n T γ n +1 . . . T γ l ′ T w T i kn +1 . . . T i kl ′ XT − w ′ v ) == ( u, T γ . . . T γ n T γ n +1 . . . T γ l ′ T w ′ XT − w ′ v ) , X ∈ U resq ( n + ) , where we also used the fact that all monomials ( e X + δ kn +1 ) ( p ′ kn +1 ) . . . ( e X + δ kl ′ ) ( p ′ kl ′ ) in (4.1.34) belong to U resq ( n + ) since s i kn +1 . . . s i kl ′ is a reduced decomposition.Now consider the reduced decomposition w = s i kl ′ . . . s i s p kl ′ +1 . . . s p D , the corresponding normal ordering β ′ k l ′ , . . . , β ′ β ′ p kl ′ +1 , . . . , β ′ p D of ∆ + , the quantum root vectors X + β ′ q = ( T − i kl ′ . . . T − i q − X + i q ≤ q ≤ k l ′ T − w ′ T − p kl ′ +1 . . . T − p q − X + p q k l ′ + 1 ≤ q ≤ D , and the linear basis of U resq ( n + ),( X + β ′ kl ′ ) ( n kl ′ ) . . . ( X + β ′ ) ( n ) ( X + β ′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′ D ) ( n D ) . (4.1.41)Consider also another reduced decomposition w = s i kl ′ +1 . . . s i D s p . . . s p kl ′ , the corresponding normal ordering β ′ k l ′ , . . . , β ′ β ′ p kl ′ +1 , . . . , β ′ p D of ∆ + , the quantum root vectors X + β ′′ q = (cid:26) T w ′′ T i p . . . T p q − X + p q ≤ q ≤ k l ′ T i kl ′ +1 . . . T i q − X + i q k l ′ + 1 ≤ q ≤ D , where T w ′′ = T i kl ′ +1 . . . T i D , and the linear basis of U resq ( n + ),( X + β ′′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′′ D ) ( n D ) ( X + β ′′ ) ( n ) . . . ( X + β ′′ kl ′ ) ( n kl ′ ) . (4.1.42)We can represent the monomials ( X + β ′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′ D ) ( n D ) using basis (4.1.42),( X + β ′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′ D ) ( n D ) = (4.1.43)= X q ,...q D C q ,...q D ( X + β ′′ kl ′ +1 ) ( q kl ′ +1 ) . . . ( X + β ′′ D ) ( q D ) ( X + β ′′ ) ( q ) . . . ( X + β ′′ kl ′ ) ( q kl ′ ) , where C q ,...q D ∈ C [ q, q − ]. Applying T w ′ to this identity we get( e X + β ′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( e X + β ′ D ) ( n D ) = (4.1.44)10 CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS = X q ,...q D C q ,...q D ( X + β kl ′ +1 ) ( q kl ′ +1 ) . . . ( X + β D ) ( q D ) ( e X + β ′′ ) ( q ) . . . ( e X + β ′′ kl ′ ) ( q kl ′ ) , where for k l ′ + 1 ≤ q ≤ D e X + β ′ q = T w ′ X + β ′ q = T − p kl ′ +1 . . . T − p q − X + p q ∈ U resq ( n + ) , (4.1.45)and for 1 ≤ q ≤ k l ′ e X + β ′′ q = T w ′ X + β ′′ q = T w ′ T w ′′ T i p . . . T p q − X + p q = T w T i p . . . T p q − X + p q ∈ U resq ( n − ) U q ( H ) , (4.1.46) U resq ( n − ) U q ( H ) is the algebra generated by U resq ( n − ) and by U q ( H ), and we used the fact that T i p . . . T p q − X + p q ∈ U resq ( n + ) and T w U resq ( n + ) ⊂ U resq ( n − ) U q ( H ). Observing that the elements e X + β ′′ q ∈ U resq ( n − ) U q ( H ), 1 ≤ q ≤ k l ′ have strictly negative weights we deduce from (4.1.44), (4.1.45), (4.1.46) and (2.4.1) in Lemma 2.4.2 that (4.1.43)takes the form( X + β ′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′ D ) ( n D ) = X q kl ′ +1 ,...q D C q kl ′ +1 ,...q D ( X + β ′′ kl ′ +1 ) ( q kl ′ +1 ) . . . ( X + β ′′ D ) ( q D ) , (4.1.47)where C q kl ′ +1 ,...q D ∈ C [ q, q − ]. Recalling basis (4.1.41) we infer that every element of U resq ( n + ) is a C [ q, q − ]–linearcombination of monomials of the form( X + β ′ kl ′ ) ( n kl ′ ) . . . ( X + β ′ ) ( n ) ( X + β ′′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′′ D ) ( n D ) . (4.1.48)Kostant’s formula shows that they form a linear basis of U resq ( n + ).Now for 1 ≤ q ≤ k l ′ we have T w ′ X + β ′ q = T w ′ T − i kl ′ . . . T − i q − X + i q = T i . . . T i q X + i q = − T i . . . T i q − X − i q K i q = − X − β q K β q , where K β q = T i . . . T i q − K i q , and for k l ′ + 1 ≤ q ≤ D T w ′ X + β ′′ q = T w ′ T i kl ′ +1 . . . T i q − X + i q = X + β q . The last two identities and commutation relations between elements K β q and quantum root vectors imply T w ′ (cid:18) ( X + β ′ kl ′ ) ( n kl ′ ) . . . ( X + β ′ ) ( n ) ( X + β ′′ kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β ′′ D ) ( n D ) (cid:19) == Q n ,...,n kl ′ ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) , where Q n ,...,n kl ′ is a monomial in K ± , . . . K ± l .Recalling basis (4.1.48), and using the previous formula we deduce that in (4.1.40) T w ′ XT − w ′ = X n ,...,n D F ′ n ,...,n D ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) , where F ′ n ,...,n D ∈ U q ( H ), and (4.1.40) takes the form( u, ( X + γ ) ( m ) . . . ( X + γ n ) ( m n ) ( X + γ n +1 ) ( m n +1 ) . . . ( X + γ l ′ ) ( m l ′ ) v ) = (4.1.49)= X n ,...,n D F n ,...,n D ( u, T γ . . . T γ n T γ n +1 . . . T γ l ′ ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) v ) , where F n ,...,n D ∈ C [ q, q − ]. This proves the first formula in the statement of this lemma.To justify the last two formulas in the statement of the lemma we observe that T γ l ′ . . . T γ n +1 T γ n . . . T γ , T − s and T − s act as the same transformations of h ⊂ U h ( h ) and apply Lemma 2.7.4. This completes the proof. .2. QUANTIZED ALGEBRA OF REGULAR FUNCTIONS ON A POISSON–LIE GROUP Corollary 4.1.8.
The products ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) or ( X + β kl ′ +1 ) ( n kl ′ +1 ) . . . ( X + β D ) ( n D ) ( X − β kl ′ ) ( n kl ′ ) . . . ( X − β ) ( n ) form a U resq ( H ) –basis in the subalgebra U resq ( w ′ ( b + )) in U resq ( g ) generated over U resq ( H ) by the elements ( X − β kl ′ ) ( n kl ′ ) , . . . , ( X − β ) ( n ) , ( X + β kl ′ +1 ) ( n kl ′ +1 ) , . . . , ( X + β D ) ( n D ) , n i ∈ N , i = 1 , . . . , D. In this short section we give several formulas related to the adjoint action and commutation relations in the algebra C B [ G ].Firstly, following [10], Theorem I.8.16 we recall the commutation relations in the algebra C B [ G ] which followfrom the fact that U sh ( g ) is quasitriangular. Namely, if V , V ′ are finite rank representations of U h ( g ), ( V ) η , ( V ′ ) ρ ,( V ) β , ( V ′ ) γ their weight subspaces of weights η, ρ, β and γ , respectively, and v ∈ ( V ) η , v ∈ ( V ′ ) ρ , u ∈ ( V ) β , u ∈ ( V ′ ) γ then evaluating the identity ∆ opps ( x ) R s = R s ∆ s ( x ) on the matrix element ( u, · v ) ⊗ ( u , · v ) and recallingformula (2.6.11) we obtain q (( κ s − s P h ′ + id ) η ∨ ,ρ ∨ ) ( u , · v ) ⊗ ( u, · v ) + X ν,i ( u , · u ν,i v ) ⊗ ( u, · u − ν,i v ) = (4.2.1)= q (( κ s − s P h ′ + id ) β ∨ ,γ ∨ ) ( u, · v ) ⊗ ( u , · v ) + X ν,i q (( κ s − s P h ′ + id )( β ∨ + ν ∨ ) ,γ ∨ − ν ∨ ) ( ω ( u − ν,i ) u, · v )) ⊗ ( ω ( u ν,i ) u , · v ) , where u − ν,i = c − ν,i f ( n ) β . . . f ( n D ) β D , n β + . . . + n D β D = ν,u ν,i = c ν,i e n β . . . e n D β D , n β + . . . + n D β D = ν,c ± ν,i ∈ B , and similarly evaluating the identity ∆ opps ( x ) R s − = R s − ∆ s ( x ) on the matrix element ( u, · v ) ⊗ ( u , · v )we get q (( κ s − s P h ′ − id ) η ∨ ,ρ ∨ ) ( u , · v ) ⊗ ( u, · v ) + X ν,i ( u , · u ′− ν,i v ) ⊗ ( u, · u ′ ν,i v ) == q (( κ s − s P h ′ − id ) β ∨ ,γ ∨ ) ( u, · v ) ⊗ ( u , · v ) + X ν,i q (( κ s − s P h ′ − id )( β ∨ + ν ∨ ) ,γ ∨ − ν ∨ ) ( ω ( u ′ ν,i ) u, · v )) ⊗ ( ω ( u ′− ν,i ) u , · v ) , where u ′− ν,i = c ′− ν,i f ( n D ) β D . . . f ( n ) β , n β + . . . + n D β D = ν,u ′ ν,i = c ′ ν,i e n D β D . . . e n β , n β + . . . + n D β D = ν,c ′± ν,i ∈ B .If v = T s − v λ ∈ ( V ) s − λ , v = T s − v µ ∈ ( V ′ ) s − µ , u ∈ ( V ) β , u = v µ ∈ ( V ′ ) µ , where v λ ∈ V and v µ ∈ V ′ arehighest weight vectors, then the previous identity yields q (( κ s − s P h ′ − id ) λ ∨ ,µ ∨ ) ( v µ , · T s − v µ ) ⊗ ( u, · T s − v λ ) = (4.2.2)= q (( κ s − s P h ′ − id ) β ∨ ,µ ∨ ) ( u, · T s − v λ ) ⊗ ( v µ , · T s − v µ ) . The next lemma shows how the adjoint action behaves with respect to the multiplication in C B [ G ].12 CHAPTER 4. ZHELOBENKO TYPE OPERATORS FOR Q-W–ALGEBRAS
Lemma 4.2.1.
For any f, g ∈ C B [ G ] , x ∈ U s,res B ( g ) we have Ad x ( f ⊗ g ) = Ad x f ⊗ g (( ω S − s )( x ) · ω x ) , where ∆ s x = (∆ s ⊗ id )∆ s x = x ⊗ x ⊗ x in the Sweedler notation.In particular, Ad f β ( f ⊗ g ) = f ⊗ g (( ω S − s )( f β ) · ) + Ad f β f ⊗ g ( G β · ) + X i Ad x i f ⊗ g (( ω S − s )( y i ) · )+ (4.2.3)+ f ( G β · G − β ) ⊗ g ( G β · ω ( f β )) + X i Ad y i f ⊗ g (( ω S − s )( y i ) · ω x i ) == f ⊗ g (( ω S − s )( f β ) · ) + Ad f β f ⊗ g ( G β · ) + f ( G β · G − β ) ⊗ g ( G β · ω ( f β ))++ X i Ad y i f ⊗ g ( G β · ω x i ) + X i Ad x i f ⊗ g (( ω S − s )( y i ) · ω x i ) and Ad f ( n ) β ( f ⊗ g ) = n X k =0 n − k X p =0 q − k ( n − k ) − p ( n − k − p ) β Ad ( G − kβ f ( p ) β )( f ) ⊗ g ( ω S − s ( G − k − pβ f ( n − k − p ) β ) · ω ( f ( k ) β ))++ n − X k =0 X i q − k ( n − k ) β Ad ( G − kβ x ( n − k ) i )( f ) ⊗ g (( ω S − s )( G − kβ y ( n − k ) i ) · ω ( f ( k ) β ))+ (4.2.4)+ X i Ad ( y ( n ) i )( f ) ⊗ g (( ω S − s )( y ( n ) i ) · ω ( x ( n ) i )) , where G β , x i , y i , x ( p ) i , y ( p ) i are defined in (2.7.12) and (2.7.13), and ∆ s ( x i ) = x i ⊗ x i , ∆ s ( y i ) = y i ⊗ y i , ∆ s ( y ( p ) i ) = y ( p ) i ⊗ y ( p ) i in the Sweedler notation.Proof. Denote using the Sweedler notation∆ s x = (∆ s ⊗ id ⊗ id )(∆ s ⊗ id )∆ s x = x ⊗ x ⊗ x ⊗ x and observe that the definition of Ad ′ implies that for any x, z ∈ U s,res B ( g )∆ opps (Ad ′ xz ) = ( x ⊗ x )( z ⊗ z )( S s x ⊗ S s x ) = Ad ′ x z ⊗ x z S s x . Let z = S s ω y , y ∈ U s,res B ( g ). Then, since ω S − s is an algebra homomorphism and a coalgebra anti-homomorphism, we deduce∆ s ω S − s (Ad ′ xS s ω y ) = ( ω S − s ⊗ ω S − s )∆ opps (Ad ′ xz ) = ( ω S − s ⊗ ω S − s )Ad ′ x z ⊗ x z S s x == ( ω S − s ⊗ ω S − s )Ad ′ x ( S s ω )( y ) ⊗ x ( S s ω )( y ) S s x = ω S − s Ad ′ x (( S s ω )( y )) ⊗ ( ω S − s )( x ) y ω x . Evaluating the last identity on f ⊗ g we get the first formula in the statement of the lemma. (4.2.3) and (4.2.4)are obtained from it using (2.7.12) and (2.7.13). In this section we study the properties of the quantum counterpart J B ⊂ C B [ G ] of the vanishing ideal of the variety N − ZHsM s − , and the properties of a B –submodule J B ⊂ J B which is a quantum counterpart of the vanishing idealof N − sZHN ′− ⊃ N − ZHsM s − .We start with a technical lemma which will allow us to use properties of the vanishing ideals of the varieties N − ZHsM s − and N − sZHN ′− to prove some properties of their quantum counterparts J B and J B . .3. QUANTIZED VANISHING IDEAL OF THE LEVEL SURFACE OF THE MOMENT MAP Lemma 4.3.1.