Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras
aa r X i v : . [ m a t h . R T ] J un A PROOF OF DE CONCINI–KAC–PROCESI CONJECTURE I.REPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY ANDQ-W ALGEBRAS
A. SEVOSTYANOV
Abstract.
Let U ε ( g ) be the standard simply connected version of the Drinfeld–Jumbo quantumgroup at an odd m-th root of unity ε . De Concini, Kac and Procesi observed that isomorphismclasses of irreducible representations of U ε ( g ) are parameterized by the conjugacy classes in theconnected simply connected algebraic group G corresponding to the simple complex Lie algebra g .They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m dim O . We show that if O intersects one of special transversal slices Σ s to the setof conjugacy classes in G then the dimension of every finite–dimensional irreducible representationof U ε ( g ) corresponding to O is divisible by m codim Σ s . In the second part of this paper is shownthat for every conjugacy class O in G one can find a transversal slice Σ s such that O intersectsΣ s and dim O = codim Σ s . This proves the De Concini–Kac–Procesi conjecture. Our result alsoimplies an equivalence between a category of finite–dimensional U ε ( g )–modules and a category offinite–dimensional representations of a q-W algebra which can be regarded as a truncation of thequantized algebra of regular functions on Σ s . To Michael Arsenyevich Semenov-Tian-Shanskyon the occasion of his 65th birthday. Introduction
It is very well known that the number of simple modules for a finite–dimensional algebra over analgebraically closed field is finite. However, often it is very difficult to classify such representations.In some important particular examples even dimensions of simple modules over finite–dimensionalalgebras are not known.One of the important examples of that kind is representation theory of semisimple Lie algebrasover algebraically closed fields of prime characteristic. Let g ′ be the Lie algebra of a semisimplealgebraic group G ′ over an algebraically closed field k of characteristic p >
0. Let x x [ p ] be the p -th power map of g ′ into itself. The structure of the enveloping algebra of g ′ is quite different fromthe 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 generated by the centralelements 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 explains why all simple g ′ –modules are finite–dimensional). The Kac–Weisfeiler conjectureformulated in [23] and proved in [32] says that if the G ′ –coadjoint orbit of θ has dimension d then p d divides the dimension of every finite–dimensional U θ –module.One can identify θ with an element of g ′ via the Killing form and reduce the proof of the Kac–Weisfeiler conjecture to the case of nilpotent θ . In that case Premet defines a subalgebra U θ ( m θ ) ⊂ U θ generated by a Lie subalgebra m θ ⊂ g ′ such that U θ ( m θ ) has dimension p d and every finite–dimensional U θ –module is U θ ( m θ )–free. Verification of the latter fact uses the theory of support Key words and phrases.
Quantum group. varieties (see [18, 19, 20, 33]). Namely, according to the theory of support varieties, in order toprove that a U θ –module is U θ ( m θ )–free one should check that 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[31]. A proof of the conjecture for p > h , where h is the Coxeter number of the corresponding rootsystem, using localization of D –modules is presented in [3].Another important example of finite–dimensional algebras is related to the theory of quantumgroups at roots of unity. Let g be a complex finite–dimensional semisimple Lie algebra. A remarkableproperty of the standard Drinfeld-Jimbo quantum group U ε ( g ) associated to g , where ε is a primitive m -th root of unity, is that its center contains a huge commutative subalgebra isomorphic to thealgebra Z G of regular functions on (a finite covering of a big cell in) a complex algebraic group G with Lie algebra g . In this paper we consider the simply connected version of U ε ( g ) and the casewhen m is odd. In that case G is the connected, simply connected algebraic group corresponding to g . Consider finite–dimensional representations of U ε ( g ), on which Z G acts according to non–trivialcharacters η g given by evaluation of regular functions at various points g ∈ G . Note that allirreducible representations of U ε ( g ) are of that kind, and every such representation is a representationof the algebra U η g = U ε ( g ) /U ε ( g )Ker η g for some η g . In [12] De Concini, Kac and Procesi showedthat if g and g are two conjugate elements of G then the algebras U η g and U η g are isomorphic.Moreover in [12] De Concini, Kac and Procesi formulated the following conjecture. De Concini–Kac–Procesi conjecture.
The dimension of any finite–dimensional representa-tion of the algebra U η g is divisible by 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 semisimpleLie algebras over fields of prime characteristic.As it is shown in [13] it suffices to verify the De Concini–Kac–Procesi conjecture in case ofexceptional elements g ∈ G (an element g ∈ G is called exceptional if the centralizer in G of itssemisimple part has a finite center). However, the De Concini–Kac–Procesi conjecture is related tothe geometry of the group G which is much more complicated 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 regularelements (see [14]), for the subregular unipotent conjugacy classes in type A n when m is a power ofa prime number (see [5]), for all conjugacy classes in A n when m is a prime number (see [7]), for theconjugacy classes O g of g ∈ SL n when the conjugacy class of the unipotent part of g is spherical(see [6]), and for spherical conjugacy classes (see [4]). In [26] a proof of the De Concini–Kac–Procesiusing localization of quantum D –modules is outlined in case of unipotent conjugacy classes. Incontract to many papers quoted above the strategy of the proof of the De Concini–Kac–Procesiconjecture developed in this paper does not use the reduction to the case of exceptional elements,and all conjugacy classes are treated uniformly.In this paper following Premet’s philosophy we construct certain subalgebras U η g ( m − ) in U η g overwhich U η g –modules are free, at least for some g ∈ G . Since the De Concini–Kac–Procesi conjectureis related to the structure of conjugacy classes in G it is natural to look at transversal slices to theset of conjugacy classes. It turns out that the definition of the subalgebras U η g ( m − ) is related to theexistence of some special transversal slices Σ s to the set of conjugacy classes in G . These slices Σ s associated to (conjugacy classes of) elements s in the Weyl group of g were introduced by the authorin [37]. 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[22] within a different framework. EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 3
A remarkable property of a slice Σ s is that if g is conjugate to an element in Σ s then U η g has asubalgebra of dimension m codim Σ s with a non–trivial character. If g ∈ Σ s (in fact g may belongto a larger variety) then the corresponding subalgebra U η g ( m − ) can be explicitly described in termsof quantum group analogues of root vectors. There are also analogues of subalgebras U η g ( m − ) in U q ( g ) in case of generic q (see [38]).In Section 9 we prove, in particular, that if g ∈ Σ s then every finite–dimensional U η g –module isfree over a subalgebra e U η g ( m − ) isomorphic to U η g ( m − ). Thus the dimension of every such moduleis divisible by m codim Σ s , and if the conjugacy class of g intersects Σ s strictly transversally in thesense that codim Σ s = dim O g , this proves the De Concini–Kac–Procesi conjecture. Thus the DeConcini–Kac–Procesi conjecture is reduced to constructing appropriate transversal slices Σ s suchthat dim O g = codim Σ s for conjugacy classes O g of exceptional elements in G . In [39], Theorem5.2 it is shown that for every conjugacy class O in G one can find a transversal slice Σ s such that O intersects Σ s and dim O = codim Σ s . Thus the De Concini–Kac–Procesi conjecture is completelyproved.In Section 9 it is also shown that the rank of every finite–dimensional U η g –module V over e U η g ( m − )is equal to the dimension of the space V χ of the so-called Whittaker vectors in V which consists ofelements v ∈ V such that xv = χ ( x ) v , x ∈ e U η g ( m − ), and χ is a non–trivial character of e U η g ( m − ).Whittaker vectors are studied in detail in Section 8.The proof of the main statement of Section 9 is reduced to the fact that for certain g every finite–dimensional U η g –module V is free over every subalgebra U η g ( f ) in U η g ( m − ) generated by a quantumanalogue f of a root vector in a Lie subalgebra m − ⊂ g . The support variety technique can not betransferred to the case of quantum groups straightforwardly. The notion of the support variety isstill available in case of quantum groups (see [16, 21, 29]). But in practical applications it is muchless efficient since in case of quantum groups there is no any underlying linear space. However, onecan show that V is free over U η g ( m − ) using a complicated induction over appropriately ordered setof root vectors in m − . In case of restricted representations of a small quantum group this was donein [16]. The situation in [16] is rather similar to the case of the trivial character η correspondingto the identity element 1 ∈ G . In the case considered in this paper the induction is even morecomplicated because the algebra U η g ( m − ) has the Jacobson radical J (see Section 8), and thequotient U η g ( m − ) / J is a non–trivial semisimple algebra. This shows a major difference betweenLie algebras and quantum groups: in case of Lie algebras g ′ over fields of prime characteristic thealgebras U θ ( m θ ) are local while in the quantum group case the algebras U η g ( m − ), which play therole of U θ ( m θ ), are not local.Slices Σ s also appear in Section 10 in a different incarnation. Namely, we show that for g conjugateto an element in Σ s the category of finite–dimensional U η g –modules is equivalent to a category offinite–dimensional modules over an algebra W sε ( G ) which can be regarded as a noncommutativedeformation of a truncated version of the algebra of regular functions on Σ s . In case of generic ε such algebras, called q-W algebras, were introduced and studied in [38]. In fact U η g is the algebraof matrices of size m codim Σ s over the algebra W sε ( G ) which has dimension m dim Σ s . In case of Liealgebras over fields of prime characteristic similar results were obtained in [34].The proofs of statements in Sections 8, 9 and 10 require some preliminary results which arepresented in Sections 2–7. Acknowledgements
The author is grateful to Giovanna Carnovale, Iulian Ion Simion, Lewis Topley and to the membersof representation theory seminars at the Universities of Bologna and Padua for careful reading ofthe manuscript.
A. SEVOSTYANOV Notation
Fix the notation used throughout of the text. Let G be a connected simply connected finite–dimensional complex simple Lie group, g its Lie algebra. Fix a Cartan subalgebra h ⊂ g andlet ∆ be the set of roots of ( g , h ). Let α i , i = 1 , . . . l, l = rank g be a system of simple roots,∆ + = { β , . . . , β N } the set of positive roots, ∆ − = − ∆ + the set of negative roots. Let H , . . . , H l be the set of simple root generators of h .Let a ij be the corresponding Cartan matrix, and let d , . . . , d l be coprime positive integers suchthat 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 ∗ under which α i ∈ h ∗ corresponds to d i H i ∈ h . We denote by α ∨ the element of h thatcorresponds to α ∈ h ∗ under this isomorphism. We shall always identify h and h ∗ by means of theform ( , ). 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 thefundamental reflections 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 same letter. For w ∈ W, g ∈ G we write w ( g ) = wgw − . For any root α ∈ ∆ we also denoteby s α the corresponding reflection.Let b + be the positive Borel subalgebra and b − the opposite Borel subalgebra; let n + = [ b + , b + ]and n − = [ b − , b − ] be their nilradicals. Let H = exp h , N + = exp n + , N − = exp n − , B + = HN + , B − = HN − be the Cartan subgroup, the maximal unipotent subgroups and the Borel sub-groups of G which correspond to the Lie subalgebras h , n + , n − , b + and b − , respectively.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 with respect to the canonical invariant bilinear form. Moreover g α and g − α are non–degenerately paired by this form.Root vectors X α ∈ g α satisfy the following relations:[ X α , X − α ] = ( X α , X − α ) α ∨ . Note also that in this paper we denote by N the set of nonnegative integer numbers, N = { , , . . . } .3. Quantum groups
Let q be an undetermined. The standard simply connected quantum group U q ( g ) associated toa complex finite–dimensional simple Lie algebra g is the algebra over C ( q ) generated by elements L i , L − i , X + i , X − i , i = 1 , . . . , l , and with the following defining relations:(3.1) [ L i , L j ] = 0 , L i L − i = L − i L i = 1 , L i X ± j L − i = q ± δ ij i X ± j ,X + i X − j − X − j X + i = δ ij K i − K − i q i − q − i , where K i = Q lj =1 L a ji j , q i = q d i , EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 5 and the quantum Serre relations:(3.2) 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 (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 − .U q ( g ) is a Hopf algebra with comultiplication defined by∆( L ± i ) = L ± i ⊗ L ± i , ∆( X + i ) = X + i ⊗ K i + 1 ⊗ X + i , ∆( X − i ) = X − i ⊗ K − i ⊗ X − i , antipode defined by S ( L ± i ) = L ∓ i , S ( X + i ) = − X + i K − i , S ( X − i ) = − K i X − i , and counit defined by ε ( L ± i ) = 1 , ε ( X ± i ) = 0 . Now we shall explicitly describe a linear basis for U q ( g ). First following [10] we recall the con-struction of root vectors of U q ( g ) in terms of a braid group action on U q ( g ). Let m ij , i = j beequal to 2 , , , a ij a ji is equal to 0 , , ,
3. The braid group B g associated to g has generators T i , i = 1 , . . . , l , and defining 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.There is an action of the braid group B g by algebra automorphisms of U q ( g ) defined on thestandard generators as follows: T i ( X + i ) = − X − i K i , T i ( X − i ) = − K − i X + i , T i ( L j ) = L j K − δ ij 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,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, where ( X + i ) ( r ) = ( X + i ) r [ r ] q i ! , ( X − i ) ( r ) = ( X − i ) r [ r ] q i ! , r ≥ , i = 1 , . . . , l. Recall that an ordering of a set of positive roots ∆ + is called normal if all simple roots are writtenin an arbitrary order, and for any three roots α, β, γ such that γ = α + β we have either α < γ < β or β < γ < α .Any two normal orderings in ∆ + can be reduced to each other by the so–called elementarytranspositions (see [41], Theorem 1). The elementary transpositions for rank 2 root systems are A. SEVOSTYANOV inversions of the following normal orderings (or the inverse normal orderings):(3.3) 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 where it is assumed that ( α, α ) ≥ ( β, β ). Moreover, any normal ordering in a rank 2 root system isone of orderings (3.3) or one of the inverse orderings.In general an elementary inversion of a normal ordering in a set of positive roots ∆ + is theinversion of an ordered segment of form (3.3) (or of a segment with the inverse ordering) in theordered set ∆ + , where α − β ∆.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 one to one correspondence between normal orderings of ∆ + and reduced decompositions of w (see [42]).Now fix a reduced decomposition w = s i . . . s i D of the longest element w of the Weyl group W of g and define the corresponding root vectors in U q ( g ) by(3.4) X ± β k = T i . . . T i k − X ± i k . Note that one can construct root vectors in the Lie algebra g in a similar way. Namely, if X ± α i are simple root vectors of g then one can introduce an action of the braid group B g by algebraautomorphisms 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,T i ( X − α j ) = ( − a ij ( − a ij )! ad − a ij X − αi X − α j , i = j. Now the root vectors X ± β k ∈ g ± β k of g can be defined by(3.5) X ± β k = T i . . . T i k − X ± α ik . The root vectors X − β satisfy the following relations:(3.6) X − α X − β − q ( α,β ) X − β X − α = X α<δ <...<δ n <β C ( k , . . . , k n )( X − δ n ) ( k n ) ( X − δ n − ) ( k n − ) . . . ( X − δ ) ( k ) , where α < β , the sum is taken over tuples of roots δ , . . . , δ n such that α < δ < . . . < δ n < β , andover k i ∈ N , for α ∈ ∆ + we put ( X ± α ) ( k ) = ( X ± α ) k [ k ] qα ! , k ∈ N , q α = q d i if the positive root α is Weylgroup conjugate to the simple root α i , C ( k , . . . , k n ) ∈ C [ q, q − ], and for each term in the right handside P ni =1 k i δ i = α + β .Let U q ( n + ) , U q ( n − ) and U q ( h ) be the subalgebras of U q ( g ) generated by the X + i , by the X − i andby the L ± i , respectively. EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 7
Now using the root vectors X ± β we can construct a basis of U q ( g ). Define for r = ( r , . . . , r D ) ∈ N D , ( X + ) ( r ) = ( X + β ) ( r ) . . . ( X + β D ) ( r D ) , ( X − ) ( r ) = ( X − β D ) ( r D ) . . . ( X − β ) ( r ) , and for s = ( s , . . . s l ) ∈ Z l , L s = L s . . . L s l l . Proposition 3.1. ( [24] , Proposition 3.3)
The elements ( X + ) ( r ) , ( X − ) ( t ) and L s , for r , t ∈ N D , s ∈ Z l , form linear bases of U q ( n + ) , U q ( n − ) and U q ( h ) , respectively, and the products ( X + ) ( r ) L s ( X − ) ( t ) form a basis of U q ( g ) . In particular, multiplication defines an isomorphism of vector spaces: U q ( n − ) ⊗ U q ( h ) ⊗ U q ( n + ) → U q ( g ) . Let U A ( g ) be the subalgebra in U q ( g ) over the ring A = C [ q, q − ] generated over A by the elements L ± i , K i − K − i q i − q − i , X ± i , i = 1 , . . . , l . The most important for us is the specialization U ε ( g ) of U A ( g ), U ε ( g ) = U A ( g ) / ( q − ε ) U A ( g ), ε ∈ C ∗ . U A ( g ) and U ε ( g ) are Hopf algebras with the comultiplicationinduced from U q ( g ). If in addition ε d i = 1 for i = 1 , . . . , l then U ε ( g ) is generated over C by L ± i , X ± i , i = 1 , . . . , l subject to relations (3.1) and (3.2) where q = ε . In particular, in that case U ε ( g ) is graded by the root lattice as defining relations (3.1) and (3.2) are homogeneous. We alsohave the following obvious consequence of Proposition 3.1. Proposition 3.2.
Let U ε ( n + ) , U ε ( n − ) and U ε ( h ) be the subalgebras of U ε ( g ) generated by the X + i ,by the X − i and by the L ± i , respectively. The elements ( X + ) r = ( X + β ) r . . . ( X + β D ) r D , ( X − ) t =( X − β D ) t D . . . ( X − β ) t and L s , for r , t ∈ N D , s ∈ Z l , form linear bases of U ε ( n + ) , U ε ( n − ) and U ε ( h ) ,respectively, and the products ( X + ) r L s ( X − ) t form a basis of U ε ( g ) . In particular, multiplicationdefines an isomorphism of vector spaces: U ε ( n − ) ⊗ U ε ( h ) ⊗ U ε ( n + ) → U ε ( g ) . The root vectors X − β satisfy the following relations in U ε ( g ) : (3.7) X − α X − β − ε ( α,β ) X − β X − α = X α<δ <...<δ n <β C ( k , . . . , k n )( X − δ n ) ( k n ) ( X − δ n − ) ( k n − ) . . . ( X − δ ) ( k ) , where α < β , the sum is taken over tuples of roots δ , . . . , δ n such that α < δ < . . . < δ n < β , andover k i ∈ N , C ( k , . . . , k n ) ∈ C , and for each term in the right hand side P ni =1 k i δ i = α + β . Quantum groups at roots of unity
Let m be a an odd positive integer number, and m > d i is coprime to all d i for all i , ε a primitive m -th root of unity. In this section, following [10], Section 9.2, we recall some results on the structureof the algebra U ε ( g ). We keep the notation introduced in Section 2.Let Z ε be the center of U ε ( g ). Proposition 4.1. ( [11] , Corollary 3.3, [12] , Theorems 3.5, 7.6 and Proposition 4.5)
Fixthe normal ordering in the positive root system ∆ + corresponding a reduced decomposition w = s i . . . s i D of the longest element w of the Weyl group W of g and let X ± α be the correspondingroot vectors in U ε ( g ) , and X α the corresponding root vectors in g . Let x − α = ( ε α − ε − α ) m ( X − α ) m , x + α = ( ε α − ε − α ) m T ( X − α ) m , where T = T i . . . T i D , α ∈ ∆ + and l i = L mi , i = 1 , . . . , l be elementsof U ε ( g ) .Then the following statements are true.(i) The elements x ± α , α ∈ ∆ + , l i , i = 1 , . . . , l lie in Z ε . A. SEVOSTYANOV (ii) Let Z ( Z ± and Z ) be the subalgebras of Z ε generated by the x ± α and the l ± i (respectively bythe x ± α and by the l ± i ). Then Z ± ⊂ U ε ( n ± ) , Z ⊂ U ε ( h ) , Z ± is the polynomial algebra with gener-ators x ± α , Z is the algebra of Laurent polynomials in the l i , Z ± = U ε ( n ± ) T Z , and multiplicationdefines an isomorphism of algebras Z − ⊗ Z ⊗ Z +0 → Z . The subalgebra Z is independent of the choice of the reduced decomposition w = s i . . . s i D .(iii) U ε ( g ) is a free Z –module with basis the set of monomials ( X + ) r L s ( X − ) t in the statementof Proposition 3.2 for which ≤ r k , t k , s i < m for i = 1 , . . . , l , k = 1 , . . . , D .(iv) Spec( Z ) = C D × ( C ∗ ) l is a complex affine space of dimension equal to dim g , Spec( Z ε ) isa normal affine variety and the map τ : Spec( Z ε ) → Spec( Z ) induced by the inclusion Z ֒ → Z ε is a finite map of degree m l .(v) The subalgebra Z is preserved by the action of the braid group automorphisms T i .(vi) Let G be the connected simply connected Lie group corresponding to the Lie algebra g and G ∗ the solvable algebraic subgroup in G × G which consists of elements of the form ( L ′ + , L ′− ) ∈ G × G , ( L ′ + , L ′− ) = ( t, t − )( n ′ + , n ′− ) , n ′± ∈ N ± , t ∈ H. Then
Spec( Z ) can be naturally identified with the maximal torus H in G , and the map e π : Spec( Z ) = Spec( Z +0 ) × Spec( Z ) × Spec( Z − ) → G ∗ , e π ( u + , t, u − ) = ( t X + ( u + ) , t − X − ( u − ) − ) , u ± ∈ Spec( Z ± ) , t ∈ Spec( Z ) , X ± : Spec( Z ± ) → N ± , X − = exp( x − β D X − β D ) exp( x − β D − X − β D − ) . . . exp( x − β X − β ) , X + = exp( x + β D T ( X − β D )) exp( x + β D − T ( X − β D − )) . . . exp( x + β T ( X − β )) , where x ± β i should be regarded as complex-valued functions on Spec( Z ) , is an isomorphism of varietiesindependent of the choice of reduced decomposition of w . Remark 4.1.
In fact
Spec( Z ) carries a natural structure of a Poisson–Lie group, and the map e π is an isomorphism of algebraic Poisson–Lie groups if G ∗ is regarded as the dual Poisson–Lie groupto the Poisson–Lie group G equipped with the standard Sklyanin bracket (see [12] , Theorem 7.6).We shall not need this fact in this paper. Let K : Spec( Z ) → H be the map defined by K ( h ) = h , h ∈ H . Proposition 4.2. ( [12] , Corollary 4.7)
Let G = N − HN + be the big cell in G . Then the map π = X − KX + : Spec( Z ) → G is independent of the choice of reduced decomposition of w , and is an unramified covering of degree l . Define derivations x ± i of U A ( g ) by(4.1) x + i ( u ) = (cid:20) ( X + i ) m [ m ] q i ! , u (cid:21) , x − i ( u ) = T x + i T − ( u ) , i = 1 , . . . , l, u ∈ U A ( g ) . Let b Z be the algebra of formal power series in the x ± α , α ∈ ∆ + , and the l ± i , i = 1 , . . . , l , whichdefine holomorphic functions on Spec( Z ) = C D × ( C ∗ ) l . Let b U ε ( g ) = U ε ( g ) ⊗ Z b Z , b Z ε = Z ε ⊗ Z b Z . EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 9
Proposition 4.3. ( [11] , Propositions 3.4, 3.5, [12] , Proposition 6.1, Theorem 6.6) (i)On specializing to q = ε , (4.1) induces a well–defined derivation x ± i of U ε ( g ) .(ii)The series exp( tx ± i ) = ∞ X k =0 t k k ! ( x ± i ) k converge for all t ∈ C to a well–defined automorphisms of the algebra b U ε ( g ) .(iii)Let G be the group of automorphisms generated by the one–parameter groups exp( tx ± i ) , i =1 , . . . , l . The action of G on b U ε ( g ) preserves the subalgebras b Z ε and b Z , and hence G acts by holo-morphic automorphisms on the complex algebraic varieties Spec( Z ε ) and Spec( Z ) .(iv)Let O be a conjugacy class in G . The intersection O = O T G is a smooth connected variety,and the variety π − ( O ) is a G –orbit in Spec( Z ) .(v)If P is a G –orbit in Spec( Z ) then the connected components of τ − ( P ) are G –orbits in Spec( Z ε ) . Given a homomorphism η : Z → C , let U η ( g ) = U ε ( g ) /I η , where I η is the ideal in U ε ( g ) generated by elements z − η ( z ), z ∈ Z . By part (iii) of Proposition4.1 U η ( g ) is an algebra of dimension m dim g with linear basis the set of monomials ( X + ) r L s ( X − ) t in the statement of Proposition 3.2 for which 0 ≤ r k , t k , s i < m for i = 1 , . . . , l , k = 1 , . . . , D .If V is an irreducible representation of U ε ( g ) then by the Schur lemma zv = θ ( z ) v for all v ∈ V and z ∈ Z ε and some character θ : Z ε → C . Therefore we get a natural map X : Rep( U ε ( g )) → Spec( Z ε ) , where Rep( U ε ( g )) is the set of equivalence classes of irreducible representations of U ε ( g ), and V is infact a representation of the algebra U η ( g ) for η = τ ( θ ) = τ X ( V ). We shall identify this representationwith V . Observe that every irreducible representation in Rep( U ε ( g )) is a representation of U η ( g ) forsome η ∈ Spec( Z ).If e g ∈ G then for any η ∈ Spec( Z ) we have e gη ∈ Spec( Z ) by part (iii) of Proposition 4.3, andby part (ii) of the same proposition e g induces an isomorphism of algebras, e g : U η ( g ) → U e gη ( g ) . This establishes a bijection between the sets Rep( U η ( g )) and Rep( U e gη ( g )) of equivalence classes ofirreducible representations of U η ( g ) and U e gη ( g ),(4.2) e g : Rep( U η ( g )) → Rep( U e gη ( g )) . For every finite-dimensional representation V of U η ( g ), and e g ∈ G we denote by V e g the corre-sponding representation of U e gη ( g ).For any element g ∈ G let g s , g u ∈ G be the semisimple and the unipotent part of g so that g = g s g u . Recall that g is called exceptional if the centralizer of g s in G has a finite center.Let ϕ = πτ X : Rep( U ε ( g )) → G be the composition of the three maps π , τ and X defined above.An irreducible representation V of U ε ( g ) is called exceptional if ϕ ( V ) ∈ G ⊂ G is an exceptionalelement.Observe that the conjugacy class of every non–exceptional element contains an element g ∈ G such that(4.3) g s ∈ H, g u ∈ N − , (4.4) the Lie algebra h g of the center of the centralizer of g s in G is non − − trivial , and(4.5) ∆ ′ = { α ∈ ∆ : α | h g = 0 } = Z Γ ′ ∩ ∆ , where Γ ′ ⊂ Γ is a proper subset of the set of simple positive roots Γ.Therefore if V is a non–exceptional irreducible representation of U ε ( g ) then V can be regarded asa representation of the algebra U η ( g ) for η = τ X ( V ), and by part (iv) of Proposition 4.3 there existsan element e g ∈ G such that π ( e gη ) satisfies properties (4.3)-(4.5), and by (4.2) V e g can be regardedas a representation of the algebra U e gη ( g ).Replacing V with V e g we may assume that V is an irreducible representation of the algebra U η ( g )such that g = π ( η ) satisfies (4.3)-(4.5).Let U ′ ε ( g ) be the subalgebra of U ε ( g ) generated by U ε ( h ) and all the elements X ± i such that α i ∈ Γ ′ . Denote by U ′ η ( g ) the quotient of U ′ ε ( g ) by the ideal generated by elements z − η ( z ), z ∈ Z ∩ U ′ ε ( g ). Now let U gε ( g ) = U ′ ε ( g ) U ε ( n + ) and U gη ( g ) be the quotient of U gε ( g ) by the idealgenerated by elements z − η ( z ), z ∈ Z ∩ U gε ( g ). The algebras U gε ( g ) and U gη ( g ) can be regardedas quantum analogues of the parabolic subalgebras associated to the subset Γ ′ of simple roots. Letalso U ′′ η ( g ) be the subalgebra of U ′ η ( g ) generated by all the elements X ± i and L ± i such that α i ∈ Γ ′ . U ′′ η ( g ) can be regarded as the semisimple part of the Levi factor U ′ η ( g ).The following fundamental proposition states that V is in fact induced from a representation ofthe algebra U gη ( g ). Proposition 4.4. ( [12] , Theorem 6.8, [13] , §
8, Theorem) (i)The U η ( g ) –module V contains a unique irreducible U gη ( g ) –submodule V ′ which remains irre-ducible when restricted to U ′′ η ( g ) .(ii)The U η ( g ) –module V is induced from the U gη ( g ) –module V ′ , V = U η ( g ) ⊗ U gη ( g ) V ′ , with the left action defined by left multiplication on U η ( g ) . In particular, dim V = m t/ dim V ′ ,where t = | ∆ \ ∆ ′ | .(iii)The map V V ′ establishes a bijection Rep( U η ( g )) → Rep( U ′′ η ( g )) , and V ′ can be regardedas an exceptional representation of the algebra U ε ( g ′ ) , where g ′ is the Lie subalgebra of g generatedby the Chevalley generators corresponding to α i ∈ Γ ′ . Realizations of quantum groups associated to Weyl group elements
Some important ingredients that will be used in the proof of the main statement in Section 9 arecertain subalgebras of the quantum group. These subalgebras are defined in terms of realizations ofthe algebra U ε ( g ) associated to Weyl group elements. We introduce these realizations in this section.A similar construction in case of quantum groups U q ( g ) with generic q was introduced in [38].Let s be an element of the Weyl group W of the pair ( g , h ), and h ′ the orthogonal complementin h , with respect to the Killing 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 canonical bilinearform on g . The restriction of the natural action of s on h ∗ to the subspace h ′∗ has no fixed points.Therefore one can define the Cayley transform s − s P h ′∗ of the restriction of s to h ′∗ , where P h ′∗ isthe orthogonal projection operator onto h ′∗ in h ∗ , with respect to the Killing form.Recall also that in the classification theory of conjugacy classes in the Weyl group W of thecomplex simple Lie algebra g the so-called primitive (or semi–Coxeter in another terminology) el-ements play a primary role. The primitive elements w ∈ W are characterized by the propertydet(1 − w ) = det a , where a is the Cartan matrix of g . According to the results of [9] the element EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 11 s of the Weyl group of the pair ( g , h ) is primitive in the Weyl group W ′ of a regular semisimple Liesubalgebra g ′ ⊂ g , rank g ′ = dim h ′ , of the form g ′ = h ′ + X α ∈ ∆ ′ g α , where ∆ ′ is a root subsystem of the root system ∆ of g , g α is the root subspace of g correspondingto root α .Moreover, by Theorem C in [9] s can be represented as a product of two involutions,(5.1) s = s s , 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 ′ arepositive and mutually orthogonal, and the roots γ , . . . γ l ′ form a linear basis of h ′∗ , in particular l ′ is the rank of g ′ . The matrix elements of the Cayley transform of the restriction of s to h ′∗ withrespect the basis γ , . . . , γ l ′ can be computed as follows. Lemma 5.1. ( [38] , Lemma 6.2)
Let P h ′∗ be the orthogonal projection operator onto h ′∗ in h ∗ , withrespect to the Killing form. Then the matrix elements of the operator s − s P h ′∗ in the basis γ , . . . , γ l ′ are of the form: (5.2) (cid:18) s − s P h ′∗ γ i , γ j (cid:19) = ε ij ( γ i , γ j ) , where ε ij = − i < j i = j i > j . Let γ ∗ i , i = 1 , . . . , l ′ be the basis of h ′∗ dual to γ i , i = 1 , . . . , l ′ with respect to the restriction ofthe bilinear form ( · , · ) to h ′∗ . Since the numbers ( γ i , γ j ) are integer each element γ ∗ i has the form γ ∗ i = P l ′ j =1 m ij γ j , where m ij ∈ Q . Therefore by the previous lemma the numbers p ij = 1 d j (cid:18) s − s P h ′∗ α i , α j (cid:19) =(5.3) = 1 d j l ′ X k,l,p,q =1 ( γ k , α i )( γ l , α j ) (cid:18) s − s P h ′∗ γ p , γ q (cid:19) m kp m lq , i, j = 1 , . . . , l are rational. Let d be a positive integer such that p ij ∈ d Z for any i < j (or i > j ), i, j = 1 , . . . , l .Now we suggest a new realization of the quantum group U ε ( g ) associated to s ∈ W . Let n be apositive integer number, n ∈ N , n >
0. Assume that ε d i = 1. Let U sε ( g ) be the associative algebra over C generated by elements e i , f i , L ± i , i = 1 , . . . l subject to the relations:(5.4) [ L i , L j ] = 0 , L i L − i = L − i L i = 1 , L i e j L − i = ε δ ij i e j , L i f j L − i = ε − δ ij i f j , ε i = ε d i ,e i f j − ε c ij f j e i = δ i,j K i − K − i ε i − ε − i , c ij = nd (cid:16) s − s P h ′∗ α i , α j (cid:17) , where K i = Q lj =1 L a ji j , P − a ij r =0 ( − r ε rc ij (cid:20) − a ij r (cid:21) ε i ( e i ) − a ij − r e j ( e i ) r = 0 , i = j, P − a ij r =0 ( − r ε rc ij (cid:20) − a ij r (cid:21) ε i ( f i ) − a ij − r f j ( f i ) r = 0 , i = j. Theorem 5.2.
Assume that ε d i = 1 . For every solution n ij ∈ Z , i, j = 1 , . . . , l of equations (5.5) d j n ij − d i n ji = c ij there exists an algebra isomorphism ψ { n ij } : U sε ( g ) → U ε ( g ) defined by the formulas: ψ { n ij } ( e i ) = X + i Q lp =1 L n ip p ,ψ { n ij } ( f i ) = Q lp =1 L − n ip p X − i ,ψ { n ij } ( L ± i ) = L ± i . The proof of this theorem is similar to the proof of Theorem 4.1 in [36].
Remark 5.2.
The general solution of equation (5.5) is given by (5.6) n ij = 12 d j ( c ij + s ij ) , where s ij = s ji . If p ij ∈ d Z for any i < j , we put s ij = c ij i < j i = j − c ij i > j . Then n ij = d j c ij i < j i = j i > j . By the choice of c ij and d we have d j c ij = ndd j (cid:16) s − s P h ′∗ α i , α j (cid:17) = ndp ij ∈ n Z for i < j , i, j = 1 , . . . , l .Therefore n ij ∈ Z for any i, j = 1 , . . . , l , and integer valued solutions to equations (5.5) exist if p ij ∈ d Z for any i < j . A similar consideration shows that if p ij ∈ d Z for any i > j integer valuedsolutions to equations (5.5) exist as well. We call the algebra U sε ( g ) the realization of the quantum group U ε ( g ) corresponding to the element s ∈ W . Remark 5.3.
Let n ij ∈ Z be a solution of the homogeneous system that corresponds to (5.5), (5.7) d i n ji − d j n ij = 0 . EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 13
Then the map defined by (5.8) X + i X + i Q lp =1 L n ip p ,X − i Q lp =1 L − n ip p X − i ,L ± i L ± i is an automorphism of U ε ( g ) . Therefore for given element s ∈ W the isomorphism ψ { n ij } is defineduniquely up to automorphisms (5.8) of U ε ( g ) . Now we shall study the algebraic structure of U sε ( g ). Denote by U sε ( n ± ) the subalgebra in U sε ( g )generated by e i ( f i ) , i = 1 , . . . l . Let U sε ( h ) be the subalgebra in U sε ( g ) generated by L ± i , i = 1 , . . . , l .We shall construct a Poincar´e–Birkhoff-Witt basis for U sε ( g ). Proposition 5.3. (i) For any normal ordering of the root system ∆ + and for any integer valued solu-tion of equation (5.5) the elements e β = ψ − { n ij } ( X + β Q li,j =1 L c i n ij j ) and f β = ψ − { n ij } ( Q li,j =1 L − c i n ij j X − β ) , β = P li =1 c i α i ∈ ∆ + lie in the subalgebras U sε ( n + ) and U sε ( n − ) , respectively. The elements f β , β ∈ ∆ + satisfy the following commutation relations (5.9) f α f β − ε ( α,β )+ nd ( s − s P h ′∗ α,β ) f β f α = X α<δ <...<δ n <β C ′ ( k , . . . , k n ) f k n δ n f k n − δ n − . . . f k δ , α < β, where C ′ ( k , . . . , k n ) ∈ C .(ii) Moreover, the elements ( e ) r = ( e β ) r . . . ( e β D ) r D , ( f ) t = ( f β D ) t D . . . ( f β ) t and L s = L s . . . L s l l for r , t ∈ N D , s ∈ Z l form bases of U sε ( n + ) , U sε ( n − ) and U sε ( h ) , and the products ( f ) t L s ( e ) r form a basis of U sε ( g ) . In particular, multiplication defines an isomorphism of vectorspaces, U sε ( n − ) ⊗ U sε ( h ) ⊗ U sε ( n + ) → U sε ( g ) . (iii) The subalgebra ψ − { n ij } ( Z ) ⊂ U sε ( g ) is the tensor product of the polynomial algebra withgenerators e mα , f mα , α ∈ ∆ + and of the algebra of Laurent polynomials in l i , i = 1 , . . . , l .(iv) U sε ( g ) is a free ψ − { n ij } ( Z ) –module with basis the set of monomials ( f ) r L s ( e ) t for which ≤ r k , t k , s i < m for i = 1 , . . . , l , k = 1 , . . . , D . The proof of this proposition is similar to the proof of Proposition 4.2 in [38].6.
Nilpotent subalgebras and quantum groups
In this section we define the subalgebras of U ε ( g ) which resemble nilpotent subalgebras in g andpossess non–trivial characters. We start by recalling the definition of certain normal orderings ofroot systems associated to Weyl group elements (see [38], Section 5 for more details). The definitionof subalgebras of U ε ( g ) having non–trivial characters will be given in terms of root vectors associatedto such normal orderings.Let s be an element of the Weyl group W of the pair ( g , h ) and h R the real form of h , the reallinear span of simple coroots in h . The set of roots ∆ is a subset of the dual space h ∗ R . Denote by h ′ R ⊂ h R the subspace corresponding to h ′ ⊂ h .The Weyl group element s naturally acts on h R as an orthogonal transformation with respect tothe scalar product induced by the Killing form of g . Now we recall some results of [8], Sect. 10.4 onthe spectral decomposition for the action of s on h R .Let f , . . . , f l ′ be the vectors of unit length in the directions of γ , . . . γ l ′ , and b f , . . . , b f l ′ the basisof 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 real matrix, and hence it is diagonalizable and has real eigenvalues. The following proposition gives a recipe for constructing a spectral decomposition for the actionof the orthogonal transformation s on h R . Proposition 6.1.
Let λ = 0 , ± be a (real) eigenvalue of the symmetric matrix I − M , and u ∈ R l ′ acorresponding non–zero real eigenvector with components u i , i = 1 , . . . , l ′ . Let a λ , b λ ∈ h R be definedby (6.1) a u = n X k =1 u i b f i , b u = l ′ X k = n +1 u i b f i . Then the angle θ between a u and b u is given by cos θ = λ .The plane h λ ⊂ h R spanned by a u and b u is invariant with respect to the involutions s , , s actson h λ as the reflection in the line spanned by b u , and s acts on h λ as the reflection in the linespanned by a u . The orthogonal transformation s = s s acts on h λ as a rotation through an angle θ .In particular, if λ = 0 , ± is an eigenvalue of I − M then − λ is also an eigenvalue of I − M ,and if λ = µ are two positive eigenvalues of I − M , λ, µ = 1 then the planes h λ and h µ are mutuallyorthogonal.Moreover, let λ = 0 , ± be an eigenvalue of I − M of multiplicity greater than , and u k ∈ R l ′ , k = 1 , . . . , mult λ a basis of the eigenspace corresponding to λ . If the basis u k is orthonormal withrespect to the standard scalar product on R l ′ then the corresponding planes h kλ defined with the helpof u k , k = 1 , . . . , mult λ are mutually orthogonal.Proof. All statements of this proposition, except for the last part, are proved by repeating thearguments given in the proofs of Lemma 10.4.2, Proposition 10.4.3 in [8] and using the spectraltheory of orthogonal transformations.For the last statement one has to use some calculations from the proof of Lemma 10.4.3 in [8].More precisely, by definition the matrix M can be written in a block form,(6.2) M = (cid:18) I n AA ⊤ I l ′ − n (cid:19) , where A ia a n × ( l ′ − n ) matrix, A ⊤ is the transpose to A , I n and I l ′ − n are the unit matrixes ofsizes n and l ′ − n . M − is also symmetric and has a similar block form,(6.3) M − = (cid:18) B CC ⊤ D (cid:19) , B = B ⊤ , D = D ⊤ , 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,(6.4) u = (cid:18) e u ee u (cid:19) . 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 decomposition (6.4).If u is a non–zero eigenvector of I − M corresponding to an eigenvalue λ = 0 , ± I − M ) u = λu gives(6.5) − A ee u = λ e u, − A ⊤ e u = λ ee u. Since M − M = I one has(6.6) BA + C = 0 , C ⊤ + DA ⊤ = 0 . EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 15
Multiplying the first and the second equations in (6.5) from the left by B and D , respectively, andusing (6.6) we obtain that(6.7) C ee u = λB e u, C ⊤ e u = λD ee u. Now if u , are two non–zero eigenvectors of I − M corresponding to an eigenvalue λ = 0 , ± 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 , where · stands for the standard scalar product in R l ′ .From (6.7) and the last formula we also obtain that(6.9) ( 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 ) . Similarly,(6.10) ( a u , b u ) = λ ( a u , a u ) = e u · C ee u , ( b u , a u ) = λ ( a u , a u ) = ee u · C ⊤ e u Now (6.8), (6.9), (6.10) and the identity 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 u , are mutually orthogonal a u and a u are also mutually orthogonal, and from (6.9) and(6.10) we obtain that b u and b u , a u and b u , a u and b u are mutually orthogonal. Therefore theplanes spanned by a u , b u and by a u , b u are mutually orthogonal. This completes the proof. (cid:3) Let h − be the subspace of h R on which s acts by multiplication by −
1. One can choose one–dimensional s , –invariant subspaces in h − such that h − is the orthogonal direct sum of thosesubspaces. Indeed, there is an orthogonal vector space decomposition h − = ( h − ∩ h − ) ⊕ ( h − ∩ h − ),and h − ∩ h , − = h − ∩ h , , where h − and h , − are the subspaces of h R on which s and s , ,respectively, act by multiplication by −
1, and h , are the subspaces of h R fixed by the action of s , . This orthogonal vector space decomposition is a consequence of the relation s x = − s x whichobviously holds for any x ∈ h − . The above mentioned identity implies that h − ∩ h , − = h − ∩ h , .These identities and the obvious orthogonal decompositions h − = ( h − ∩ h , − ) ⊕ ( h − ∩ h , ) imply h − = ( h − ∩ h − ) ⊕ ( h − ∩ h − ). Thus the one–dimensional subspaces of h − ∩ h − or of h − ∩ h − are also invariant with respect to the involutions s and s . The required decomposition of h − is adecomposition into an orthogonal direct sum of such one–dimensional subspaces.Using the previous proposition and the results of the discussion above we can decompose h R intoa direct orthogonal sum of s –invariant subspaces,(6.11) h R = K M i =0 h i , where h is the linear subspace of h R fixed by the action of s , each h i is also invariant with respectto both involutions s , in the decomposition s = s s , and each of the subspaces h i ⊂ h R , i =1 , . . . , K , is either 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 on it 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 by multiplication by − s has finite order θ i = πm 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 –invariantsubspace h i with respect to the natural pairing between h R and h ∗ R .Now we consider certain s –invariant subsets of roots ∆ i , i = 0 , . . . , K , defined as follows(6.12) ∆ i = { α ∈ ∆ : h j ( α ) = 0 , j > i, h i ( α ) = 0 } , where we formally assume that h K +1 = 0. Note that for some indexes i the subsets ∆ i are empty,and that the definition of these subsets depends on the order of terms in direct sum (6.11).Now consider the nonempty s –invariant subsets of roots ∆ i k , k = 0 , . . . , T . For convenience weassume that indexes i k are labeled in such a way that i j < i k if and only if j < k . According to thisdefinition ∆ = { α ∈ ∆ : sα = α } is the set of roots fixed by the action of s . Observe also that theroot system ∆ is the disjoint union of the subsets ∆ i k ,∆ = T [ k =0 ∆ i k . Now assume that(6.13) | h i k ( α ) | > | X p ≤ j
For every element w ∈ W one can introduce the set ∆ w = { α ∈ ∆ + : w ( α ) ∈ − ∆ + } , and thenumber of the elements in the set ∆ w is equal to the length l ( w ) of the element w with respect tothe system Γ of simple roots in ∆ + .Now recall that s can be represented as a product of two involutions,(6.17) s = s s , 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 ′ arepositive and mutually orthogonal, and the roots γ , . . . γ l ′ form a linear basis of h ′ . Proposition 6.2. ( [38] , Proposition 5.1)
Let s ∈ W be an element of the Weyl group W of thepair ( g , h ) , ∆ the root system of the pair ( g , h ) and ∆ + the system of positive roots defined with thehelp of element (6.14), ∆ + = { α ∈ ∆ | ¯ h ( α ) > } .Then the decomposition s = s s is reduced in the sense that l ( s ) = l ( s ) + l ( s ) , where l ( · ) is thelength function in W with respect to the system of simple roots in ∆ + , and ∆ s = ∆ s S s (∆ s ) , ∆ s − = ∆ s S s (∆ s ) (disjoint unions). Moreover, there is a normal ordering of the root system ∆ + 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 ) , . . . , (6.18) 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 ) } = ∆ s , { 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 ( α ) = − α } , { 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 ) } = ∆ s , { 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 ( α ) = − α } , { β , . . . , β D } = (∆ ) + = { α ∈ ∆ + | s ( α ) = α } , where s , s are the involutions entering decomposition (5.1), s = s γ . . . s γ n , s = s γ n +1 . . . s γ l ′ , theroots in each of the sets γ , . . . , γ n and γ n +1 , . . . , γ l ′ are positive and mutually orthogonal.The length of the ordered segment ∆ m + ⊂ ∆ in normal ordering (6.18), ∆ 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 , (6.19) 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 (6.20) D − ( l ( s ) − l ′ D ) , where D is the number of roots in ∆ + , l ( s ) is the length of s and D is the number of positive rootsfixed by the action of s .Moreover, for any two roots α, β ∈ ∆ m + such that α < β the sum α + β can not be representedas a linear combination P qk =1 c k γ i k , where c k ∈ N and α < γ i < . . . < γ i q < β . Remark 6.4.
In case when s = s is an involution the last root in the segment ∆ m + is the rootpreceding β in normal ordering (6.18). We call the system of positive roots ∆ + ordered as in (6.18) the normally ordered system ofpositive roots associated to (the conjugacy class of) the Weyl group element s ∈ W . We shall alsoneed the circular ordering in the root system ∆ corresponding to normal ordering (6.18) of thepositive root system ∆ + .Let β , β , . . . , β D be a normal ordering of a positive root system ∆ + . Then following [25] onecan introduce the corresponding circular normal ordering of the root system ∆ where the roots in∆ are located on a circle in the following way β β r r r r r β D − β − β rrrrr - β D ◗◗s◗◗❦ Fig.1Let α, β ∈ ∆. One says that the segment [ α, β ] of the circle is minimal if it does not containthe opposite roots − α and − β and the root β follows after α on the circle above, the circle beingoriented clockwise. In that case one also says that α < β in the sense of the circular normal ordering,(6.21) α < β ⇔ the segment [ α, β ] of the circle is minimal . Later we shall need the following property of minimal segments which is a direct consequence ofProposition 3.3 in [24].
Lemma 6.3.
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 in the sense of the circular normal ordering. EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 19
Now we can define the subalgebras of U ε ( g ) which resemble nilpotent subalgebras in g and possessnon–trivial characters. Theorem 6.4.
Let s ∈ W be an element of the Weyl group W of the pair ( g , h ) , ∆ the root systemof the pair ( g , h ) . Fix a decomposition (5.1) of s and let ∆ + be a system of positive roots associatedto s . Assume that ε d i = 1 and that ε nd − = 1 , where d and n are introduced in Section 5. Let U sε ( g ) be the realization of the quantum group U ε ( g ) associated to s . Let f β ∈ U sε ( n − ) , β ∈ ∆ + bethe root vectors associated to the corresponding normal ordering (6.18) of ∆ + .Then elements f β ∈ U sε ( n − ) , β ∈ ∆ m + , where ∆ m + ⊂ ∆ is ordered segment (6.19), generate asubalgebra U sε ( m − ) ⊂ U sε ( g ) . The elements f r = f r D β D . . . f r β , r i ∈ N , i = 1 , . . . D and r i can bestrictly positive only if β i ∈ ∆ m + , form a linear basis of U sε ( m − ) .Moreover the map χ s : U sε ( m − ) → C defined on generators by (6.22) χ s ( f β ) = (cid:26) β
6∈ { γ , . . . , γ l ′ } c i β = γ i , c i ∈ C is a character of U sε ( m − ) .Proof. The first statement of the theorem follows straightforwardly from commutation relations (5.9)and Proposition 5.3.In order to prove that the map χ s : U sε ( m − ) → C defined by (6.22) is a character of U sε ( m − ) weshow that all relations (5.9) for f α , f β with α, β ∈ ∆ m + , which are obviously defining relations inthe subalgebra U sε ( m − ), belong to the kernel of χ s . By definition the only generators of U sε ( m − ) onwhich χ s may not vanish are f γ i , i = 1 , . . . , l ′ . By the last statement in Proposition 6.2 for any tworoots α, β ∈ ∆ m + such that α < β the sum α + β can not 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 . Hence for any two roots α, β ∈ ∆ m + suchthat α < β the value of the map χ s on the r.h.s. of the corresponding commutation relation (5.9) isequal to zero.Therefore it suffices to prove that χ s ( 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. Since ε nd − = 1 and ( s − s P h ′∗ γ i , γ j ) are integer numbers for any i, j = 1 , . . . , l ′ , the last identityalways holds provided ( γ i , γ j ) + ( s − s P ∗ h ′ γ i , γ j ) = 0 for i < j . As we saw in Lemma 5.1 this is indeedthe case. This completes the proof. (cid:3) Some facts about the geometry of the conjugation action
In this section we collect some results on the geometry of the conjugation action that will beused later. We keep the notation of the previous section. Let r ∈ End g be a linear operator on g satisfying the classical modified Yang–Baxter equation,[ rX, rY ] − r ([ rX, Y ] + [ X, rY ]) = − [ X, Y ] , X, Y ∈ g . One can check that if we define operators r ± ∈ End g by r ± = 12 ( r ± id )then the linear subspace g ∗ ⊂ g ⊕ g , g ∗ = { ( X + , X − ) , X ± = r ± X, X ∈ g } is a Lie subalgebra in g ⊕ g (see, for instance, [35]). We denote by G ∗ the corresponding subgroup in G × G .Let r , r s ∈ End g be the linear operators on g defined by r = P + − P − , r s = P + − P − + 1 + s − s P h ′ , where P + , P − and P h ′ are the projection operators onto n + , n − and h ′ in the direct sum(7.1) g = n + + h ′ + h ′⊥ + n − , and h ′⊥ is the orthogonal complement to h ′ in h with respect to the Killing form. One can checkthat both r and r s satisfy the classical modified Yang–Baxter equation. Therefore one can definethe corresponding subgroups G ∗ , G ∗ s ⊂ G × G .Note also that r s + = P + + 11 − s P h ′ + 12 P h ′⊥ , r s − = − P − + s − s P h ′ − P h ′⊥ , where P h ′⊥ is the projection operator onto h ′⊥ in direct sum (7.1). Hence every element ( L + , L − ) ∈ G ∗ s may be uniquely written as(7.2) ( L + , L − ) = ( h + , h − )( n + , n − ) , where n ± ∈ N ± , h + = exp (( − s P h ′ + P h ′⊥ ) x ) , h − = exp (( s − s P h ′ − P h ′⊥ ) x ) , x ∈ h . In particular, G ∗ s is a solvable algebraic subgroup in G × G .Similarly we have r = P + + 12 P h , r − = − P − − P h , P h = P h ′ + P h ′⊥ , and hence every element ( L ′ + , L ′− ) ∈ G ∗ may be uniquely written as( L ′ + , L ′− ) = ( h ′ + , h ′− )( n ′ + , n ′− ) , n ′± ∈ N ± , h ′ + = exp ( 12 x ′ ) , h ′− = exp ( − x ′ ) , x ′ ∈ h . In particular, G ∗ is also a solvable algebraic subgroup in G × G .We shall need an isomorphism of varieties φ : G ∗ → G ∗ s which is uniquely defined by the require-ment that if φ ( L ′ + , L ′− ) = ( L + , L − ) then(7.3) L = tL ′ t − , L ′ = L ′− ( L ′ + ) − , L = L − L − , t = e Ax ′ , where A ∈ End h is the endomorphism of h defined by(7.4) AH i = 12 nd l X j =1 n ij d i Y j , i = 1 , . . . , l,n ij are solutions to equations (5.5), and Y i = l X j =1 d i ( a − ) ij H j , ( Y i , H j ) = δ ij are the weight–type generators of h (see [38] for more detail).In fact (7.3) is an isomorphism of Poisson manifolds if G ∗ is regarded as the dual Poisson–Liegroup to the Poisson–Lie group G equipped with the standard Sklyanin bracket, and G ∗ s is regardedas the dual Poisson–Lie group to the Poisson–Lie group G equipped with the Sklyanin bracketassociated to the r–matrix r s (see [38], Section 10). We shall not need this fact in this paper.Formula (7.2) and decomposition of N + into a product of one–dimensional subgroups correspond-ing to roots also imply that every element L − may be represented in the form(7.5) L − = exp hP li =1 b i ( s − s P h ′ − P h ′⊥ ) H i i × Q β exp [ b − β X − β ] , b i , b − β ∈ C , where the product over roots is taken in the same order as in (6.18), and the root vectors X − β areconstructed as in (3.5) using the normal ordering of ∆ + opposite to (6.18). EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 21
Let M ± be the subgroups in N ± corresponding to the Lie subalgebras m ± ⊂ n ± which aregenerated by root vectors X ± β , β ∈ ∆ m + . Now define a map µ M + : G ∗ s → M − by(7.6) µ M + ( L + , L − ) = m − , where for L − given by (7.5) m − is defined as follows(7.7) m − = Y β ∈ ∆ m + exp [ b − β X − β ] , and the product over roots is taken in the same order as in the normally ordered segment ∆ m + .By definition µ M + is a morphism of algebraic varieties.Let u be the element defined by(7.8) u = l ′ Y i =1 exp [ t i X − γ i ] ∈ M − , t i ∈ C , where the product over roots is taken in the same order as in the normally ordered segment ∆ m + .Let X α ( t ) = exp( tX α ) ∈ G , t ∈ C be the one–parametric 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 is a representative for the reflection s α corresponding to the root α .Denote by s ∈ G the following representative of the Weyl group element s ∈ W ,(7.9) s = s γ ( t ) . . . s γ l ′ ( t l ′ ) , where the numbers t i are defined in (7.8), and we assume that t i = 0 for any i .Let Z be the subgroup of G generaled by the semi–simple part of the Levi factor L correspondingto the Lie subalgebra l and by the centralizer of s in H . Denote by N the subgroup of G correspondingto the Lie subalgebra n and by N the opposite unipotent subgroup in G with the Lie algebra n . Bydefinition we have that N + ⊂ ZN .Now we formulate the main proposition in which transversal slices to conjugacy classes in G aredescribed. Proposition 7.1. ( [37] , Propositions 2.1 and 2.2)
Let N s = { v ∈ N | svs − ∈ N } . Then theconjugation map (7.10) N × sZN s → N sZN is an isomorphism of varieties. Moreover, the variety Σ s = sZN s is a transversal slice to the set ofconjugacy classes in G . Assume that ∆ + is ordered as in (6.18). We shall also use the corresponding circular normalordering on ∆. Let ∆ m + = { α ∈ ∆ + : α < γ } , ∆ m + = { α ∈ ∆ + : α > γ l ′ } .Let λ : G ∗ s → G be the map defined by, λ ( L + , L − ) = L − L − . Consider the space µ − M + ( u ) which can be explicitly described as follows(7.11) µ − M + ( u ) = { ( h + n + , h − x ux ) | n + ∈ N + , h ± = e r s ± x , x ∈ h , x ∈ M − , x ∈ M − } , where M , − is the subgroup of G generated by the one–parametric subgroups corresponding to theroots from the segment − ∆ , m + . Therefore(7.12) λ ( µ − M + ( u )) = { h − x ux n − h − | n + ∈ N + , h ± = e r s ± x , x ∈ h , x , ∈ M , − } . We shall need the following improved version of Proposition 12.1 in [38].
Proposition 7.2.
Let λ : G ∗ s → G be the map defined by, λ ( L + , L − ) = L − L − . Suppose that the numbers t i introduced in (7.8) are not equal to zero for all i . Then λ ( µ − M + ( u )) is asubvariety in N sZN . All elements of λ ( µ − M + ( u )) are of the form h − n s skh − with h ± = e r ± x , where x ∈ h is arbitrary, and n s and k are some elements of N ′ s = { v ∈ N + | s − vs ∈ N − } ⊂ N and of ZN , respectively. In particular, (7.13) u = m s sm, m s ∈ N ′ s , m ∈ N. The closure λ ( µ − M + ( u )) with respect to Zariski topology is also contained in N sZN .Proof.
First we show that x ux n − belongs to N sZN . Fix the circular normal ordering on ∆associated to normal ordering (6.18) of ∆ + . Observe that the segment which consists of α ∈ ∆ suchthat γ ≤ α < − γ is minimal with respect to the circular normal ordering, and its intersection with∆ − is − ∆ m + . Therefore by Lemma 6.3 we have x X − γ ( t ) . . . X − γ n ( t n ) = x X γ ( t − ) . . . X γ n ( t − n ) X γ n ( − t − n ) . . . X γ ( − t − ) X − γ ( t ) . . . X − γ n ( t n ) == n x ′ X γ n ( − t − n ) . . . X γ ( − t − ) X − γ ( t ) . . . X − γ n ( t n ) , n ∈ N + , x ′ ∈ M − . Using the relations X γ ( − t − ) X − γ ( t ) = X − γ ( − t ) s γ one can rewrite the last identity asfollows x X − γ ( t ) . . . X − γ n ( t n ) = n x ′ X γ n ( − t − n ) . . . X γ ( − t − ) X − γ ( − t ) s γ X − γ ( t ) . . . X − γ n ( t n ) . Now we move X − γ ( − t ) to the left from the product X γ n ( − t − n ) . . . X γ ( − t − ) in the last formula.Since the segment which consists of α ∈ ∆ such that γ ≤ α ≤ − γ is minimal with respect to thecircular normal ordering, and its intersection with ∆ − is ∆ − = { α ∈ ∆ − : α ≤ − γ } , one has byLemma 6.3, using commutation relations between one–parametric subgroups corresponding to rootsand the orthogonality of roots γ and γ x X − γ ( t ) . . . X − γ n ( t n ) = n x ′′ X γ n ( − t − n ) . . . X γ ( − t − ) s γ X γ ( − t − ) X − γ ( t ) . . . X − γ n ( t n ) , where n ∈ N + , x ′′ ∈ M , and M is the subgroup of G generated by the one–parametric subgroupscorresponding to roots from ∆ − .Now we can use the relation X γ ( − t − ) X − γ ( t ) = X − γ ( − t ) s γ and apply similar argumentsto get x X − γ ( t ) . . . X − γ n ( t n ) = n x ′′′ X γ n ( − t − n ) . . . X γ ( − t − ) s γ s γ X γ ( − t − ) X − γ ( t ) . . . X − γ n ( t n ) , where n ∈ N + , x ′′′ ∈ M , and M is the subgroup of G generated by the one–parametric subgroupscorresponding to roots from ∆ − = { α ∈ ∆ − : α ≤ − γ } .We can proceed in a similar way to obtain the following representation(7.14) x X − γ ( t ) . . . X − γ n ( t n ) = n e xs γ . . . s γ n , n ∈ N + , e x ∈ M n , where M n is the subgroup of G generated by the one–parametric subgroups corresponding to rootsfrom ∆ n − = { α ∈ ∆ − : α ≤ − γ n } .By the definition of normal ordering (6.18) one also has s − γ n . . . s − γ M n s γ . . . s γ n ⊂ N , and hence(7.14) can be rewritten in the following form(7.15) x X − γ ( t ) . . . X − γ n ( t n ) = ns γ . . . s γ n n ′ = ns n ′ , n ∈ N + , n ′ ∈ N. If x = 1 the arguments presented above, together with the fact the roots ± γ i , i = 1 , . . . n belongto the root subsystem { α ∈ ∆ : s α = − α } which has trivial intersection with ∆ , yield(7.16) X − γ ( t ) . . . X − γ n ( t n ) = ms m ′ , m, m ′ ∈ N. EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 23
Similarly, taking into account that Z normalizes N , one has(7.17) X − γ n +1 ( t n +1 ) . . . X − γ l ′ ( t l ′ ) x = n ′′ s γ n +1 . . . s γ l ′ n ′ = n ′′ s n ′′′ , n ′′ ∈ N, n ′′′ ∈ Z − Z + N, where Z − = Z T N − , Z + = Z T N + , and(7.18) X − γ n +1 ( t n +1 ) . . . X − γ l ′ ( t l ′ ) = m ′′ s m ′′′ , m ′′ , m ′′′ ∈ N. Combining (7.15) and (7.17) we obtain(7.19) x ux n − = ns gs k, g = n ′ n ′′ ∈ N, k = n ′′′ n − ∈ Z − Z + N, and from (7.16) and (7.18) we also have(7.20) u = ms g ′ s m ′′′ , g ′ = m ′ m ′′ ∈ N. Let M , be the subgroups of G generated by the one–parametric subgroups corresponding tothe roots from the segments ∆ s and ∆ s , respectively, M ′ + the subgroup of G generated by theone–parametric subgroups corresponding to the roots from the segment ∆ + \ (∆ s S ∆ s S (∆ ) + ).By the definition of these subgroups every element g ∈ N has a unique factorization g = g g g ,where g , ∈ M , , and g ∈ M ′ + . Applying this factorization to the element g in (7.19) andrecalling the properties of normal ordering (6.18), we have ( s ) − M s ⊂ N , s M ( s ) − ⊂ N ,( s ) − M ′ + s ⊂ N . Now we derive from (7.19) that(7.21) x ux n − = b ns s k ′ = b nsk ′ . for some b n ∈ N + , k ′ ∈ Z − Z + N .Finally factorizing b n as b n = n s e n , where n s ∈ N ′ s = { v ∈ N + | s − vs ∈ N − } ⊂ N and e n ∈ e N = { v ∈ N + | s − vs ∈ N + } we arrive at(7.22) x ux n − = n s sk ′′ , n s ∈ N ′ s , k ′′ ∈ ZN.
Hence x ux n − ∈ N sZN .Similarly from (7.20) we deduce u = m s sm, m s ∈ N ′ s , m ∈ N. Let H ′ ⊂ H be the connected subgroup corresponding to the Lie subalgebra h ′ ⊂ h , and H ⊂ H the connected subgroup corresponding to the orthogonal complement h of h ′ in h with respect tothe Killing form. Note that h is the space of fixed points for the action of s on h . We obviouslyhave H = H ′ H (direct product of subgroups). From the definition 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 ± = e r s ± x , x ∈ h are obtained in this way.Next observe that the space N sZN is invariant with respect to the following action of H :(7.23) h ◦ L = h − Lh − , h = h + = h h ′ , h − = h − s ( h ′ ) . Indeed, let L = vszw, v, w ∈ N, z ∈ Z be an element of N sZN . Then(7.24) h ◦ L = h − vh − − h − sh − h + zwh − = h − vh − − sh − h + zwh − Since s − h − s = h − h ′ . The r.h.s. of the last equality belongs to N sZN because H normalizes N and Z .Comparing action (7.23) with (7.12) and recalling that x ux n − ∈ N sZN we deduce λ ( µ − M + ( u )) ⊂ N sZN .The variety λ ( µ − M + ( u )) is not closed in G . But following Corollary 2.5 and Proposition 2.10 in[17] we shall show that N sZN is closed in G .Observe that an element g ∈ G belongs to N sZN = N ′ s sZN if and only if s − g ∈ s − N ′ s sZN .The variety s − N ′ s sZN is a subvariety of N ZN . First we prove that
NZN is closed in G . Let V γ i , i = 1 , . . . , l ′ be the irreducible finite-dimensional representation of G with highest weight γ i . Denote by v γ i a nonzero highest weight vector in V γ i and by < · , · > the Hermitian form on V γ i normalized in such a way that < v γ i , v γ i > = 1.We claim that an element g ∈ G belongs to N ZN iff < v γ i , gv γ i > = 1, i = 1 , . . . , l ′ . Indeed,according to the Bruhat decomposition g ∈ B − wB + for some w ∈ W . In this case g can be writtenuniquely in the form g = n − whn + for some n ± ∈ N ± , h ∈ H . Now < v γ i , gv γ i > = γ i ( h )
NZN is closed in G . The variety s − N ′ s sZN is a closed subvariety of NZN as s − N ′ s s isthe closed algebraic subgroup in N generated by the one–parametric subgroups corresponding tothe roots from the set { α ∈ − ∆ + : s ( α ) ∈ ∆ + } . So finally s − N ′ s sZN is closed in G , and hence N sZN = N ′ s sZN is also closed.Therefore the closure λ ( µ − M + ( u )) is contained in N sZN . This completes the proof. (cid:3)
We shall need a short technical lemma which will play the key role in the proof of the mainstatement of this paper.
Lemma 7.3.
Suppose that the numbers t i defined in (7.8) are not equal to zero for all i . Thenfor any η ∈ λ ( µ − M + ( u )) and h ′ ∈ H ′ one can find n ∈ N + such that n ηn − ∈ λ ( µ − M + ( u )) , and n ηn − = h − x ux n − h − , h + = h h ′ , h − = h − s ( h ′ ) for some h ∈ H , n + ∈ N + , x , ∈ M , − .Proof. Recall that u = m s sm, m s ∈ N ′ s , m ∈ N. By (7.21) η can be represented in the form η = h − b nsk ′ h − , b n ∈ N + , k ′ ∈ Z − Z + N, h ± = e r s ± x , x ∈ h . Let h ′ + = ( h − ) H ′ h ′ , h ′− = ( h − − ) H ′ s ( h ′ ), where ( · ) H ′ stands for the H ′ –component of an elementof H in the decomposition H = H H ′ , and define m = h ′− m s h ′− − , m = h ′ + mh ′− , so that(7.25) h ′− uh ′− = m sm , m , ∈ N. Let n = h − m b n − h − − ∈ N + . Then, since N + = Z + N , H normalizes N + , and Z normalizes N ,we have n ηn − = h − m sk ′′ h − = h − m sm m − k ′′ h − = h − m sm kh − , b n ∈ N + , k, k ′′ ∈ Z − Z + N. Now by (7.25) the last formula can be rewritten as n ηn − = h − h ′− uh ′− kh − = h + u e kh − − , e k ∈ Z − Z + N ⊂ M − N + , where h + = ( h + ) H h ′ , h − = ( h − ) H s ( h ′ ), and ( · ) H stands for the H –component of an element of H in the decomposition H = H H ′ . The element in the right hand side of the last identity belongsto λ ( µ − M + ( u )) by definition and is of the form h − x ux n − h − , where x n − = e k ∈ Z − Z + N ⊂ M − N + , x = 1 ∈ M − , h + = h h ′ , h − = h − s ( h ′ ), h = ( h + ) H ∈ H . This completes theproof. (cid:3) EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 25
Consider the restriction of the action of G on itself by conjugations to the subgroup M + . Denoteby π λ : G → G/M + the canonical projection onto the quotient with respect to this action. Thequotient π λ ( λ ( µ − M + ( u ))) can be described as follows. Proposition 7.4. ( [40] , Theorem 6.4)
Suppose that the numbers t i defined in (7.8) are not equalto zero for all i . Then λ ( µ − M + ( u )) is invariant under conjugations by elements of M + , the conjugationaction of M + on λ ( µ − M + ( u )) is free, the quotient π λ ( λ ( µ − M + ( u ))) is a smooth variety isomorphic to sZN s , and λ ( µ − M + ( u )) ≃ M + × π λ ( λ ( µ − M + ( u ))) ≃ M + × sZN s . Whittaker vectors
In this section we introduce the notion of Whittaker vectors for modules over quantum groupsat roots of unity and prove an analogue of Engel theorem for them. We start by studying someproperties of quantum groups at roots of unity.From now on we fix an element s ∈ W . Let ∆ + be a system of positive roots associated to s .We also fix positive integer d such that p ij ∈ d Z for any i < j (or i > j ), i, j = 1 , . . . , l , where thenumbers p ij are defined by formula (5.3). We shall always assume that m > d i is coprime to all d i , i = 1 , . . . , l and that there exists a positive integer n such that ε nd − = 1. We fix an integer valuedsolution n ij to equations (5.5) and identify the algebra U s − ε ( g ) associated to the Weyl group element s − with U ε ( g ) using Theorem 5.2 and the solution − n ij − δ ij to equations (5.5) (a motivation foradding the extra term − δ ij to n ij will be given later; as it was explained in Remark 5.3 this termis a solution to homogeneous equations (5.7) and corresponds to an automorphism of U ε ( g )). Usingthis identification U s − ε ( m − ) can be regarded as a subalgebra in U ε ( g ). Therefore for every character η : Z → C one can define the corresponding subalgebra in U η ( g ). We denote this subalgebra by U η ( m − ).First we study some properties of the finite dimensional algebras U η ( g ) and U η ( m − ). We remindthat a finite dimensional algebra is called Frobenius if its left regular representation is isomorphicto the dual of the right regular representation. Thus any free module over a Frobenius algebra isalso injective and projective. Proposition 8.5.
For any character η : Z → C the algebra U η ( g ) and its subalgebra U η ( m − ) areFrobenius algebras.Proof. The proof of this proposition is parallel to the proof of a similar statement for Lie algebrasover fields of prime characteristic (see Proposition 1.2 in [20]) and for the restricted form of thequantum group in [27]. We shall only briefly outline the main steps of the proof for U η ( g ). Theproof for U η ( m − ) is similar.The key ingredient of the proof is the De Concini-Kac filtration on U ε ( g ) ≃ U s − ε ( g ) which isdefined as follows (see [11]). For r , t ∈ N D introduce the element u r , t ,t = e r tf t , t ∈ U ε ( h ), wherewe use the notation of Theorem 5.2 and Proposition 5.3. Here the generators f α , e α , α ∈ ∆ + andthe ordered products of them are defined with the help of the normal ordering of ∆ + opposite to(6.18). Define also the height of the element u r , t ,t as follows ht ( u r , t ,t ) = P Di =1 ( t i + r i )ht β i ∈ N ,where ht β i is the height of the root β i . Introduce also the degree of u r , t ,t by d ( u r , t ,t ) = ( r , . . . , r D , t D , . . . , t , ht ( u r , t ,t )) ∈ N D +1 . Equip N D +1 with the total lexicographic order and for k ∈ N D +1 denote by ( U ε ( g )) k the spanof elements u r , t ,t with d ( u r , t ,t ) ≤ k in U ε ( g ). Then Proposition 1.7 in [11] implies that ( U ε ( g )) k isa filtration of U ε ( g ) such that the associated graded algebra is the associative algebra over C with generators e α , f α , α ∈ ∆ + , L ± i , i = 1 , . . . l subject to the relations(8.26) L i L j = L j L i , L i L − i = L − i L i = 1 , L i e α L − i = ε α ( Y i ) e α , L i f α L − i = ε − α ( Y i ) f α ,e α f β = ε − nd ( s − s P h ′∗ α,β ) f β e α ,e α e β = ε ( α,β ) − nd ( s − s P h ′∗ α,β ) e β e α , α > β,f α f β = ε ( α,β ) − nd ( s − s P h ′∗ α,β ) f β f α , α > β, where Y i = P lj =1 d i ( a − ) ij H j are the weight–type generators of h . Such algebras are called semi–commutative.By Theorem 61.3 in [15] it suffices to show that there is a non–degenerate bilinear form B η : U η ( g ) × U η ( g ) → C which is associative in the sense that B η ( ab, c ) = B η ( a, bc ) , a, b, c ∈ U η ( g ) . Consider the free Z –basis of U ε ( g ) introduced in part (iv) of Proposition 5.3. This basis consistsof the monomials x I = ( f ) r L s ( e ) t , I = ( r , . . . , r D , s , . . . , s l , t , . . . , t D ) for which 0 ≤ r k , t k , s i < m for i = 1 , . . . , l , k = 1 , . . . , D . Set I ′ = ( m − − r , . . . , m − − r D , m − − s , . . . , m − − s l , m − − t , . . . , m − − t D ) and P = ( m − , . . . , m − U ε ( g ) → Z be the Z –linear map defined on the basis x I of monomials byΦ( x I ) = (cid:26) I = P . Let x = P I c I x I , c I ∈ Z be an element of U ε ( g ), and c K = 0 a coefficient such that d ( x K ) ismaximal possible with c K = 0 in the sum defining x .Using the definition of the De Concini–Kac filtration and commutation relations (8.26) one cancheck that Φ( xx K ′ ) = a x c K , where a x is a nonzero complex number (see [27], proof of Theorem 2.2,Assertion I for details).Therefore the bilinear form B η : U η ( g ) × U η ( g ) → C associated to the associative Z –bilinearpairing B : U ε ( g ) ⊗ Z U ε ( g ) → Z , B ( x, y ) = Φ( xy ) is non–degenerate and associative. Thiscompletes the proof. (cid:3) In order to define Whittaker vectors for quantum groups at roots of unity we shall need someauxiliary notions that we are going to discuss now.Consider the isomorphism of varieties φ ◦ e π : Spec( Z ) → G ∗ s constructed with the help of the normal ordering β , . . . , β D of the positive root system ∆ + oppositeto (6.18) and with the help of the solution n ij of equations (5.5). We shall need some property ofelements η ∈ Spec( Z ) such that φ ◦ e π ( η ) ∈ µ − M + ( u ). To describe this property we observe that astraightforward calculation using the explicit form of the isomorphism ψ {− n ij − δ ij } shows that the n − –component Y − of the map φ ◦ e π in the image G ∗ s with respect to factorization (7.2) has the form Y − : Spec( Z ) → N − , (8.27) Y − = exp( y − β D X − β D ) exp( y − β D − X − β D − ) . . . exp( y − β X − β ) , where y − α = k α f mα , for some k α ∈ C , k α = 0, and y − α should be regarded as complex-valued functionson Spec( Z ), the elements f α ∈ U s − ε ( m − ) are constructed using the normal ordering opposite to EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 27 (6.18), so the order of terms corresponding to roots in the product (8.27) coincides with the orderof roots in normal ordering (6.18).The following property of elements η ∈ Spec( Z ) , φ ◦ e π ( η ) ∈ µ − M + ( u ) is a direct consequence offormula (8.27) and of the definition of the variety µ − M + ( u ) in terms of the map µ M + (see formulas(7.6), (7.7), (7.8) and (7.11)). Lemma 8.6.
Let η be an element of Spec( Z ) . Assume that φ ◦ e π ( η ) ∈ µ − M + ( u ) . Then for β ∈ ∆ m + we have (8.28) η ( f mβ ) = (cid:26) a i = t i k γi β = γ i , i = 1 , . . . , l ′ β
6∈ { γ , . . . , γ l ′ } . Finally consider the subalgebra U η ( h ) ⊂ U η ( g ) generated by L , . . . , L l . Since η ( L mi ) = 0, i = 1 , . . . , l the elements L , . . . , L l act on any finite–dimensional U η ( g )–module V as mutually com-muting semisimple automorphisms. Therefore if by a weight we mean an l –tuple ω = ( ω , . . . , ω l ) ∈ ( C ∗ ) l , the space V has a weight decomposition with respect to the action of U η ( h ), V = M ω ∈ ( C ∗ ) l V ω , where V ω = { v ∈ V, L i v = ω i v, ω i ∈ C ∗ , i = 1 , . . . , l } is the weight space corresponding to weight ω .Observe that by Proposition 6.2 for any two roots α, β ∈ ∆ m + such that α < β the sum α + β 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 ,and hence from commutation relations (5.9) one can deduce that the elements f β ∈ U η ( m − ), β ∈ Θ,where Θ = { α ∈ ∆ m + : α
6∈ { γ , . . . , γ l ′ }} , generate an ideal J in U η ( m − ). Lemma 8.7.
Let η be an element of Spec( Z ) . Assume that t i = 0 , i = 1 , . . . , l ′ in formula (7.8)and φ ◦ e π ( η ) ∈ µ − M + ( u ) , so that f mγ i = η ( f mγ i ) = a i = 0 in U η ( m − ) for i = 1 , . . . , l ′ and f mβ = 0 in U η ( m − ) for β ∈ ∆ m + , β
6∈ { γ , . . . , γ l ′ } . Then the ideal J is the Jacobson radical of U η ( m − ) and U η ( m − ) / J is isomorphic to the truncated polynomial algebra C [ f γ , . . . , f γ l ′ ] / { f mγ i = a i } i =1 ,...,l ′ .Proof. First we show that J is nilpotent.Let δ < δ < . . . < δ b be the roots in the segment ∆ m + . By commutation relations (5.9) theelements(8.29) x k ,...,k b = f k δ f k δ . . . f k b δ b for k i ∈ N , k i < m , and k i > δ i
6∈ { γ , . . . , γ l ′ } , form a linear basis of J . Denoteby I the set of b –tuples ( k , . . . , k b ) described above.By commutation relations (5.9), for any given 1 ≤ k < b the elements x k ,...,k b , ( k , . . . , k b ) ∈ I with k i = 0 for i ≤ k , if there are any such elements, generate an ideal J k in J . These elements alsoform a linear basis of J k . Note that for the last few k the sets of elements x k ,...,k b , ( k , . . . , k b ) ∈ I with k i = 0 for i ≤ k may be empty indeed, and these are all possibilities when those sets are empty.Note that J is finite–dimensional. In particular, it is Artinian. Hence in order to prove that J is nilpotent it suffices to show that it is a nil–ideal, i.e. each element of J is nilpotent.Let x be an element of J , x = v X i =1 x i , where each x v is the product of a complex number and of an element of the linear basis of J described above. Then for any positive integer Ex E = X n + ... + n v = E x n i . . . x n v i v . In particular, for each term in the sum above corresponding to a tuple n , . . . , n v there is n j in thattuple such that n j ≥ [ Ev ]. Therefore in order to prove that J is nilpotent it suffices to show thateach element of basis (8.29) is nilpotent. The same observation obviously applies to each of theideals J k and to their bases. We shall show that J is nilpotent using induction over k and startingfrom the maximal possible k when J k is not empty.Let K be maximal possible such that J K is not empty. If y = x k ,...,k b is an element of thelinear basis of J K introduced above then y must be of the form y = f pβ f k δ i . . . f k r δ ir , δ i , . . . , δ i r ∈{ γ n +1 , . . . , γ l ′ } , β < δ i , β ∈ Θ is the last root preceding γ l ′ in normal ordering (6.18). Here it isassumed that f k δ . . . f k r δ r = 1 if the set { γ n +1 , . . . , γ l ′ } is empty.Commutation relations (5.9) and the relation f mβ = 0 imply that for some C ∈ C y m = Cf mpβ f mk δ i . . . f mk r δ ir = 0 . We deduce that J K is nilpotent.Now assume that J k is nilpotent for some k ≤ K . We show that J k − is nilpotent. Let z = f k δ j f k δ j +1 . . . f k b δ b be an element of the linear basis of J k − . Recall that by Proposition 6.2 for any tworoots α, β ∈ ∆ m + such that α < β the sum α + β can not 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 . Therefore using commutation relations(5.9) we obtain(8.30) z F = k +( k − F − X p =0 f pδ j z p + k F X p = k +( k − F − f pδ j w p , where each z p ∈ J k is a sum of terms each of which is a product of at least F − J k .If f k δ j +1 . . . f k b δ b ∈ J k then each w p ∈ J k is a sum of terms each of which is a product of at least F elements of J k . If f k δ j +1 . . . f k b δ b
6∈ J k then δ j ∈ Θ. Thus, by the induction assumption, for F largeenough one has z p = 0 for all p . If f k δ j +1 . . . f k b δ b ∈ J k one has w p = 0 for all p as well. Otherwise δ j ∈ Θ and one can choose F large enough so that f k +( k − F − δ j = 0. In all cases the right handside of equation (8.30) vanishes if F is large enough. Hence z is nilpotent, and J k − is nilpotent aswell.We deduce that the ideal J is nilpotent. Hence J is contained in the Jacobson radical of U η ( m − ).Using commutation relations (5.9) we also have (see the proof of Theorem 6.4) f γ i f γ j − f γ j f γ i ∈ J . Therefore the quotient algebra U η ( m − ) / J is isomorphic to the truncated polynomial algebra C [ f γ , . . . , f γ l ′ ] / { f mγ i = a i } i =1 ,...,l ′ which is semisimple. Therefore J coincides with the Jacobson radical of U η ( m − ). (cid:3) Next, commutation relations (5.9) and part (iv) of Proposition 5.3 also imply the following lemma.
Lemma 8.8.
Let β , . . . , β D be the normal ordering of ∆ + opposite to (6.18). Then for any character η : Z → C the elements f r D β D . . . f r β f n γ . . . f n l ′ γ l ′ , where r i , n j ∈ N , ≤ r i , n j ≤ m − , i = 1 , . . . D , j = 1 , . . . , l ′ , and r i can be strictly positive only if β i ∈ ∆ m + , β i
6∈ { γ , . . . , γ l ′ } , form a linear basisof U η ( m − ) . EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 29
The elements f r D β D . . . f r β f n γ . . . f n l ′ γ l ′ r i , n j ∈ N , ≤ r i , n j ≤ m − , i = 1 , . . . D , j = 1 , . . . , l ′ , r i can be strictly positive only if β i ∈ ∆ m + , β i
6∈ { γ , . . . , γ l ′ } , and at least one r i is strictly positive,form a linear basis of J . In Theorem 6.4 we constructed some characters of the algebra U sε ( m − ). Similarly one can definecharacters of the algebra U s − ε ( m − ). Now we show that the algebra U η ( m − ) has a finite number ofirreducible representation which are one–dimensional, and all those representations can be obtainedfrom each other by twisting with the help of automorphisms of U η ( m − ). Proposition 8.9.
Let η be an element of Spec( Z ) . Assume that t i = 0 , i = 1 , . . . , l ′ in formula(7.8) and φ ◦ e π ( η ) ∈ µ − M + ( u ) , so that η ( f mγ i ) = a i = 0 , i = 1 , . . . , l ′ . Then all non–zero irreduciblerepresentations of the algebra U η ( m − ) are one–dimensional and have the form (8.31) χ ( f β ) = (cid:26) β
6∈ { γ , . . . , γ l ′ } c i β = γ i , i = 1 , . . . , l ′ , where complex numbers c i satisfy the conditions c mi = a i , i = 1 , . . . , l ′ . Moreover, all non–zeroirreducible representations of U η ( m − ) can be obtained from each other by twisting with the help ofautomorphisms of U η ( m − ) .Proof. Let V be a non–zero finite–dimensional irreducible U η ( m − )–module. By Corollary 54.13 in[15] elements of the ideal J ⊂ U η ( m − ) act by zero transformations on V . Hence V is in fact anirreducible representation of the algebra U η ( m − ) / J which is isomorphic to the truncated polynomialalgebra C [ f γ , . . . , f γ l ′ ] / { f mγ i = a i } i =1 ,...,l ′ . The last algebra is commutative and all its complex irreducible representations are one–dimensional.Therefore V is one–dimensional, and if v is a nonzero element of V then f γ i v = c i v , for some c i ∈ C , i = 1 , . . . , l ′ . Note that η ( f mγ i ) = a i = 0, i = 1 , . . . , l ′ and hence c mi = a i = 0, i = 1 , . . . , l ′ . Inparticular, the elements f γ i act on V by semisimple automorphisms.If we denote by χ : U η ( m − ) → C the character of U η ( m − ) such that χ ( f β ) = (cid:26) β
6∈ { γ , . . . , γ l ′ } c i β = γ i and by C χ the corresponding one–dimensional representation of U η ( m − ) then we have V = C χ .Now we have to prove that the representations C χ for different characters χ are obtained fromeach other by twisting with the help of automorphisms of U η ( m − ).Since c mi = a i , i = 1 , . . . , l ′ , for given η such that φ ◦ e π ( η ) ∈ µ − M + ( u ) there are only finitely manypossible characters χ . If χ and χ ′ are two such characters, χ ( f γ i ) = c i , i = 1 , . . . , l ′ and χ ′ ( f γ i ) = c ′ i , i = 1 , . . . , l ′ then the relations c mi = c ′ im = a i , i = 1 , . . . , l ′ imply that c ′ i = ε m i c i , 0 ≤ m i ≤ m − m i ∈ Z , i = 1 , . . . , l ′ .Now observe that for any h ∈ h the map defined by f α ε α ( h ) f α , α ∈ ∆ m + is an automorphismof the algebra U s − ε ( m − ) generated by elements f α , α ∈ ∆ m + with defining relations (5.9). Here theprincipal branch of the analytic function ε z is used to define ε α ( h ) , so that ε α ( h ) ε β ( h ) = ε ( α + β )( h ) for any α, β ∈ ∆ m + . If in addition ε mγ i ( h ) = 1, i = 1 , . . . , l ′ the above defined map gives rise toan automorphism ς of U η ( m − ). Indeed in that case ( ε γ i ( h ) f γ i ) m = f mγ i , i = 1 , . . . , l ′ and all theremaining defining relations f mγ i = η ( f mγ i ) = a i = 0, i = 1 , . . . , l ′ , f mβ = η ( f mβ ) = 0, β ∈ ∆ m + , β
6∈ { γ , . . . , γ l ′ } of the algebra U η ( m − ) are preserved by the action of the above defined map ς .Now fix h ∈ h such that γ i ( h ) = m i , i = 1 , . . . , l ′ . Obviously we have ε mm i = 1, i = 1 , . . . , l ′ . Weclaim that the representation C χ twisted by the corresponding automorphism ς coincides with C χ ′ .Indeed, we obtain χ ( ςf γ i ) = χ ( ε m i f γ i ) = ε m i c i = c ′ i , i = 1 , . . . , l ′ . This completes the proof of the proposition. (cid:3)
Let V be a U η ( g )–module, where η is an element of Spec( Z ) such that φ ◦ e π ( η ) ∈ µ − M + ( u ).Assume that t i = 0, i = 1 , . . . , l ′ in formula (7.8). Let χ : U η ( m − ) → C be a character defined inthe previous proposition, C χ the corresponding one–dimensional U η ( m − )–module. Then the space V χ = Hom U η ( m − ) ( C χ , V ) is called the space of Whittaker vectors of V . Elements of V χ are calledWhittaker vectors.The following proposition is an analogue of Engel theorem for quantum groups at roots of unity. Proposition 8.10.
Assume that t i = 0 , i = 1 , . . . , l ′ in formula (7.8). Suppose also that forany integers p j ∈ Z , j = 1 , . . . , l the system of equations Y j ( P l ′ i =1 m i γ i ) = mp j , j = 1 , . . . , l forunknowns m i ∈ {− m + 1 , . . . , m − } , i = 1 , . . . , l ′ has no nontrivial solutions. Let η be an element of Spec( Z ) such that φ ◦ e π ( η ) ∈ µ − M + ( u ) . Let χ : U η ( m − ) → C be any character defined in the previousproposition. Then any non–zero finite–dimensional U η ( g ) –module contains a non–zero Whittakervector.Proof. Consider the subalgebra U η ( m − + h ) in U η ( g ) generated by the elements of U η ( m − ) and by L ± i , i = 1 , . . . , l . Let I be the ideal in U η ( m − + h ) generated by J . Lemma 8.11.
Assume that t i = 0 , i = 1 , . . . , l ′ in formula (7.8). Suppose also that for any integers p j ∈ Z , j = 1 , . . . , l the system of equations Y j ( P l ′ i =1 m i γ i ) = mp j , j = 1 , . . . , l for unknowns m i ∈ {− m + 1 , . . . , m − } , i = 1 , . . . , l ′ has no nontrivial solutions.Let η be an element of Spec( Z ) such that φ ◦ e π ( η ) ∈ µ − M + ( u ) . Let V be a non–zero finite–dimensional U η ( m − + h ) / I –module. Then V is free over the subalgebra F of U η ( m − + h ) / I generatedby the classes of the elements f γ i , i = 1 , . . . , l ′ in U η ( m − + h ) / I , and one can choose a weight F –basis in V . Fix numbers c i , i = 1 , . . . , l ′ such that c mi = a i , i = 1 , . . . , l ′ , where a i are defined by(8.28). Then the rank of V over F is equal to the dimension of the subspace of V which consistsof elements v such that f γ i v = c i v , i = 1 , . . . , l ′ Proof.
Denote the classes of f γ i , i = 1 , . . . , l ′ and of L ± i , i = 1 , . . . , l in U η ( m − + h ) / I by the sameletters. Then U η ( m − + h ) / I has generators f γ i , i = 1 , . . . , l ′ and L ± i , i = 1 , . . . , l , and relations L i L − i = 1 , L i L j = L j L i , L mi = η ( L i ) , f γ i f γ j = f γ j f γ i , f mγ i = a i , L i f γ j = ε Y i ( γ j ) f γ j L i . From the relations L mi = η ( L i ) = 0 and f mγ i = a i we obtain that the elements f γ , . . . , f γ l ′ and L , . . . , L l act on V by semisimple automorphisms. In particular, V has a weight space decompo-sition for the action of the commutative subalgebra generated by the L i . If v ∈ V is a non–zerovector of weight ω then(8.32) L j f n γ . . . f n l ′ γ l ′ v = ε Y j ( P l ′ i =1 n i γ i ) ω j f n γ . . . f n l ′ γ l ′ v. Since for any integers p j ∈ Z , j = 1 , . . . , l the system of equations Y j ( P l ′ i =1 m i γ i ) = mp j , j = 1 , . . . , l for unknowns m i ∈ {− m + 1 , . . . , m − } , i = 1 , . . . , l ′ has no nontrivial solutions, and the elements f γ i act on V by semisimple automorphisms, (8.32) implies that the non–zero vectors f n γ . . . f n l ′ γ l ′ v have different weights for different l ′ –tuples ( n , . . . , n l ′ ), 0 ≤ n i ≤ m −
1, and hence they are linearlyindependent in V .This implies that one can choose linearly independent weight vectors v k ∈ V , k = 1 , . . . , M suchthat V = M M k =1 V k (direct sum of F − modules) , where V k is the free F –module with the linear basis f n γ . . . f n l ′ γ l ′ v k , for 0 ≤ n i ≤ m − i = 1 , . . . , l ′ . EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 31
One check directly that the vectors l ′ Y i =1 m − X j =0 c m +1 − ji ε − ( j +1) q i f jγ i v k , q i = 0 , , , . . . , m − V k , and the vector(8.33) w k = l ′ Y i =1 m − X j =0 c m +1 − ji f jγ i v k is the only vector in V k satisfying the conditions f γ i w k = c i w k , i = 1 , . . . , l ′ . Thus the rank M of V over F is equal to the dimension of the subspace of such vectors. (cid:3) Arguments similar to those used in the proof of Lemma 8.7 together with the commutationrelations in the first line of (5.4) show that the ideal I is nilpotent. Hence elements of the ideal I ⊂ U η ( m − + h ) act by nilpotent transformations on V . Therefore from Engel theorem one candeduce that the subspace V I = { v ∈ V | xv = 0 ∀ x ∈ I} , V I ⊂ V , is non–zero. Now the statementof Proposition 8.10 follows from Lemma 8.11 applied to the U η ( m − + h ) / I –module V I and from thedefinition of Whittaker vectors.. (cid:3) Remark 8.5.
The fact that any non–zero finite–dimensional U η ( g ) –module V contains a non–zero Whittaker vector with respect to some character χ : U η ( m − ) → C can be proved in a muchsimpler way. Indeed, V can be regarded as a finite–dimensional U η ( m − ) –module. Since V is finite–dimensional it has a composition series. In particular, V contains an irreducible U η ( m − ) –submodulewhich must be isomorphic to C χ for some character χ : U η ( m − ) → C by Proposition 8.9. Some properties of finite–dimensional modules over quantum groups at roots ofunity
This section is central in the paper. We shall prove that finite–dimensional modules over quantumgroups at roots of unity are free over certain subalgebras. More precisely, we have the followingtheorem.
Theorem 9.12.
Let ζ be an element of Spec( Z ) . Assume that for any integers p j ∈ Z , j = 1 , . . . , l the system of equations Y j ( P l ′ i =1 m i γ i ) = mp j , j = 1 , . . . , l for unknowns m i ∈ {− m + 1 , . . . , m − } , i = 1 , . . . , l ′ has no nontrivial solutions. Suppose that t i = 0 , i = 1 , . . . , l ′ in formula (7.8) and thatthere exists a quantum coadjoint transformation e g ′ such that φ ◦ e π ( η ) ∈ µ − M + ( u ) , where η = e g ′ ζ .Then there exists a quantum coadjoint transformation e g ∈ G such that φ ◦ e π ( e gη ) ∈ µ − M + ( u ) and anynon–zero finite–dimensional U e gη ( g ) –module V is free over U e gη ( m − ) of rank equal to the dimensionof the space of Whittaker vectors V χ , where χ is a character of U e gη ( m − ) , and hence any non–zerofinite–dimensional U ζ ( g ) –module is free over e g ′− e g − U e gη ( m − ) .Proof. Let ζ be an element of Spec( Z ) satisfying the conditions imposed in the formulation ofTheorem 9.12, η = e g ′ ζ and e g ∈ G an arbitrary quantum coadjoint transformation such that φ ◦ e π ( e gη ) ∈ µ − M + ( u ). Let V be a finite–dimensional non–zero U e gη ( g )–module.In the proof we shall use the notation of Lemma 8.11. As we already observed in the proof ofProposition 8.10 the ideal I is nilpotent. Therefore from Engel theorem one can deduce that thesubspace V I = { v ∈ V | xv = 0 ∀ x ∈ I} , V I ⊂ V , is non–zero.By Lemma 8.11 V I is free over the algebra F with a weight basis v k , k = 1 , . . . , M . As in Lemma8.11 we denote by V k I the free F –submodule in V I generated by v k . Since the elements f γ i act on V I by semisimple automorphisms the m -th powers of which aremultiplications by non–zero numbers we can assume that if V k I and V k ′ I contain vectors of the sameweight then the weight of v k is equal to the weight of v k ′ .Let V ′I be the linear space with the linear basis v k ∈ V , k = 1 , . . . , M . Fix a linear basis Υ of V which consists of weight vectors and contains all elements f n γ . . . f n l ′ γ l ′ v k , for 0 ≤ n i ≤ m − i = 1 , . . . , l ′ , k = 1 , . . . , M . Let ρ : V → V ′I be the linear projection such that ρv = 0 for v ∈ Υ, v = v k for some k . Obviously ρ sends weight vectors to weight vectors.Consider the left U e gη ( m − )–module Hom C ( U e gη ( m − ) , V ′I ) with the left U e gη ( m − )–action inducedby multiplication in U e gη ( m − ) from the right. Note that since by Proposition 8.5 the algebra U e gη ( m − ) is Frobenius and the space V ′I is finite–dimensional we have a U e gη ( m − )–module isomorphismHom C ( U e gη ( m − ) , V ′I ) ≃ U e gη ( m − ) ⊗ V ′I . Therefore Hom C ( U e gη ( m − ) , V ′I ) is a free U e gη ( m − )–module.Now let σ : V → Hom C ( U e gη ( m − ) , V ′I ) be the homomorphism of U e gη ( m − )–modules defined by σ ( v )( x ) = ρ ( xv ), x ∈ U e gη ( m − ), v ∈ V . We claim that σ is an isomorphism when e g is chosen in anappropriate way. Since Hom C ( U e gη ( m − ) , V ′I ) is a free U e gη ( m − )–module this will imply that V is freeover U e gη ( m − ) of rank equal to the dimension of V ′I . By Lemma 8.11 that dimension is equal to thedimension of the space of Whittaker vectors in V .First we show that σ is injective. Indeed, the kernel Ker σ of σ is a U e gη ( m − )–submodule of V ,and hence, by Engel theorem, if Ker σ = { } it must contain a non–zero element v annihilated bythe nilpotent transformations f β , β ∈ ∆ m + , β
6∈ { γ , . . . , γ l ′ } . Thus by definition v ∈ V I . Since V I is free over F with basis v k , v can be uniquely represented as a linear combination of elements f n γ . . . f n l ′ γ l ′ v k , for 0 ≤ n i ≤ m − i = 1 , . . . , l ′ , k = 1 , . . . , M , v = X ≤ n i ≤ m − ,k =1 ,...,M c kn ...n l ′ f n γ . . . f n l ′ γ l ′ v k . Recall that elements f γ i act on V by automorphisms the m -th powers of which are multiplicationsby non–zero numbers. Therefore if c k ′ n ′ ...n ′ l ′ = 0 then the element w = f m − n ′ l ′ γ l ′ . . . f m − n ′ γ v can berepresented in the form w = X ≤ n i ≤ m − ,k =1 ,...,M d kn ...n l ′ f n γ . . . f n l ′ γ l ′ v k , where d k ′ ... = 0.Now we have σ ( w )(1) = ρ ( w ) = X k =1 ,...,M d k ... v k , where at least one coefficient d k ′ ... = 0. Since the elements v k are linearly independent we deducethat σ ( w )(1) = 0, and hence σ ( w ) = 0.On the other hand if v ∈ Ker σ then we also have w = f m − n ′ l ′ γ l ′ . . . f m − n ′ γ v ∈ Ker σ since Ker σ isa U e gη ( m − )–submodule of V . Thus we arrive at a contradiction, and hence σ is injective.Now we prove that σ is surjective. We start with the following lemma. Lemma 9.13.
Let η be an element of Spec( Z ) . Suppose that t i = 0 , i = 1 , . . . , l ′ in formula (7.8)and that φ ◦ e π ( η ) ∈ µ − M + ( u ) . Then there exists a quantum coadjoint transformation e g ∈ G suchthat φ ◦ e π ( e gη ) ∈ µ − M + ( u ) and for any α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } any non–zero finite–dimensional U e gη ( g ) –module V is free over the subalgebra U e gη ( f α ) of U e gη ( g ) generated by the unit and by f α .Proof. Recall that by Lemma 8.6 for α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } we have f mα = 0. Hence U e gη ( f α ) isisomorphic to the truncated polynomial algebra U e gη ( f α ) = C [ f α ] / { f mα = 0 } , and in order to show EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 33 that V is U e gη ( f α )–free it suffices to verify that all Jordan blocks of the nilpotent endomorphismgiven by the action of f α in V have size m .Denote by V f α the kernel of f α in V . Since f mα = 0 the subspace V f α is not trivial. Let w i , i = 1 , . . . , P be a linear basis of V f α . We show that the vectors e kα w i , k = 0 , . . . , m − i = 1 , . . . , P are linearly independent if e g is chosen in an appropriate way.Assume that they are linearly dependant. Then there are non–zero elements w k ∈ V f α , k =0 , . . . , Q , Q ≤ m − Q X k =0 a k e kα w k = 0 , where a k ∈ C and a Q = 0.Now from formula (13), Sect. 9.3 in [10] we deduce the following commutation relations(9.35) f α e kα = e kα f α − [ k ] ε α e k − α ε k − α K α − ε − kα K − α ε α − ε − α , where if α = s i . . . s i p − α i p then K α = T i . . . T i p − K i p .Applying f α to relation (9.34), using commutation relations (9.35) and the fact that f α v = 0 forany v ∈ V f α we obtain that(9.36) Q X k =1 a k e k − α [ k ] ε α ε k − α K α − ε − kα K − α ε α − ε − α w k = 0 . Now observe that commutation relations L i f α = ε Y i ( α ) f α L i and the definition of K α imply that V f α is an invariant subspace for the action of K ± α . Therefore in (9.36) [ k ] ε α ε k − α K α − ε − kα K − α ε α − ε − α w k ∈ V f α .We claim that one can choose a quantum coadjoint transformation e g ∈ G such that φ ◦ e π ( e gη ) ∈ µ − M + ( u ) and vectors ε k − α K α − ε − kα K − α ε α − ε − α w k are all non–zero in (9.36).Let H ′ be the connected subgroup of H which corresponds to the Lie subalgebra h ′ ⊂ h and H ⊂ H be the connected subgroup corresponding to the Lie subalgebra h ′⊥ ⊂ h so that H = HH (direct product of subgroups). By Lemma 7.3 for η ∈ Spec( Z ), φ ◦ e π ( η ) ∈ µ − M + ( u ) and for any h ′ ∈ H ′ one can find a quantum coadjoint transformation e g ∈ G such that φ ◦ e π ( e gη ) ∈ µ − M + ( u ) andthe H = Spec( Z )–component of e gη in Spec( Z +0 ) × Spec( Z ) × Spec( Z − ) is equal to h ′ η , where η ∈ H .Since any α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } has a non–zero projection onto h ′ , and the number ofsuch roots α is finite, one can find h ′ ∈ H ′ and a quantum coadjoint transformation e g ∈ G suchthat the H = Spec( Z )–component of e gη in Spec( Z +0 ) × Spec( Z ) × Spec( Z − ) is equal to h ′ η and( e gη ( K mα )) m = ( η h ′ ( K mα )) m = ε − k ) α for all α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } , k = 1 , . . . , m −
1, and forall roots ( e gη ( K mα )) m of degree m of e gη ( K mα ). Thus we have ( e gη ( K mα )) m = ε − k ) α for all α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } , k = 1 , . . . , m −
1, and for all roots ( e gη ( K mα )) m of degree m of e gη ( K mα ).Since K mα = e gη ( K mα ) in U e gη ( g ) the numbers ( e gη ( K mα )) m exhaust all possible eigenvalues of K α in V , and hence the operators K α − ε − k ) α acting in V are invertible for all α ∈ ∆ m + , α
6∈ { γ , . . . , γ l ′ } , k = 1 , . . . , m −
1. Therefore the operators ε k − α K α − ε − kα K − α ε α − ε − α = ε k − α K − α ( K α − ε − k ) α ) ε α − ε − α are invertible as well. Thus, if e g is chosen as above, the vectors ε k − α K α − ε − kα K − α ε α − ε − α w k are all non–zero in (9.36), and from(9.36) we obtain the following relation Q X k =1 a k e k − α w k = 0 , where w k ∈ V f α are non–zero vectors.Applying successively f α to the above relation Q − a Q w Q − k = 0for a non–zero vector w Q − k ∈ V f α . This is a contradiction. Thus the vectors e kα w i , k = 0 , . . . , m − i = 1 , . . . , P are linearly independent. The last assertion implies that dim V ≥ m dim V f α . Sincethe Jordan blocks of f α in V have size at most m we also have the opposite inequality, dim V ≤ m dim V f α . Thus dim V = m dim V f α , and hence all Jordan blocks of f α in V have size m . Thiscompletes the proof. (cid:3) From now on we assume that e g ∈ G is fixed as in the previous lemma. We already proved thatthe U e gη ( m − )–module homomorphism σ : V → Hom C ( U e gη ( m − ) , V ′I ) ≃ U e gη ( m − ) ⊗ V ′I is an imbedding. Thus V is a submodule of the free U e gη ( m − )–module U e gη ( m − ) ⊗ V ′I .Let β , . . . , β D be the normal ordering of ∆ + opposite to (6.18). Then by Lemma 8.8 the elements f r D β D . . . f r β f n γ . . . f n l ′ γ l ′ , where r i , n j ∈ N , 0 ≤ r i , n j ≤ m − i = 1 , . . . D , j = 1 , . . . , l ′ , and r i canbe strictly positive only if β i ∈ ∆ m + , β i
6∈ { γ , . . . , γ l ′ } , form a linear basis of U η ( m − ). Hence theelements(9.37) f r L β iL . . . f r β i f n γ . . . f n l ′ γ l ′ ⊗ v k , { β i , . . . , β i L } = ∆ m + \ { γ , . . . , γ l ′ } , β i < . . . < β i L , r i , n j ∈ N , 0 ≤ r i , n j ≤ m − i = 1 , . . . L , j = 1 , . . . , l ′ form a linear basis of U e gη ( m − ) ⊗ V ′I .Our aim now is to show that the image of σ in U e gη ( m − ) ⊗ V ′I contains a subspace spanned bygenerating vectors. This will justify that σ is surjective. First we describe the image of the subspace V I in U e gη ( m − ) ⊗ V ′I under the homomorphism σ . Lemma 9.14.
The image of the subspace V I ⊂ V in U e gη ( m − ) ⊗ V ′I under the composition of thehomomorphism σ and of the isomorphism Hom C ( U e gη ( m − ) , V ′I ) ≃ U e gη ( m − ) ⊗ V ′I is the linear subspace X with the linear basis (9.38) f m − β iL . . . f m − β i f n γ . . . f n l ′ γ l ′ ⊗ v k , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, { β i , . . . , β i L } = ∆ m + \ { γ , . . . , γ l ′ } , β i < . . . < β i L .Proof. Recall that there is an isomorphism of U e gη ( m − ) regarded as the left regular representationand of U e gη ( m − ) ∗ regarded as the dual to the right regular representation. This isomorphism isinduced by the associative bilinear form on U e gη ( m − ) mentioned in the proof of Proposition 8.5.Using this isomorphism, commutation relations (5.9) and the fact that J is an ideal in U e gη ( m − ),the image of X under the isomorphism U e gη ( m − ) ⊗ V ′I ≃ U e gη ( m − ) ∗ ⊗ V ′I ≃ Hom C ( U e gη ( m − ) , V ′I )is identified with the linear subspace with the basis(9.39) f n γ . . . f n l ′ γ l ′ v k , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 35 where v k are regarded as images of the corresponding elements of V under σ . By the first part ofthe proof of Theorem 9.12 elements (9.39), where v k are regarded as elements of V , form a linearbasis of V I . Hence elements (9.39) form a linear basis of the image of V I in Hom C ( U e gη ( m − ) , V ′I )under σ , and elements (9.38) form a linear basis of the image of V I in U e gη ( m − ) ⊗ V ′I under thehomomorphism σ . (cid:3) From now on we shall identify Hom C ( U e gη ( m − ) , V ′I ) and U e gη ( m − ) ⊗ V ′I as left U e gη ( m − )–modules.Now we show that the image of σ contains some special elements which in fact generate U e gη ( m − ) ⊗ V ′I . Lemma 9.15.
The image of σ in U e gη ( m − ) ⊗ V ′I contains elements of the form (9.40) f n γ . . . f n l ′ γ l ′ ⊗ v k + x, ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, where x ∈ J ⊗ V ′I . Proof.
We shall use the notation of Lemma 9.14. Recall that using the injective homomorphism σ and Lemma 9.13 the module V can be regarded as a free U e gη ( f β iL )–submodule of U e gη ( m − ) ⊗ V ′I .Elements (9.38) belong to that submodule and each of elements (9.38) is annihilated by f β iL . Sinceall Jordan blocks of f β iL in V have size m the image of V in U e gη ( m − ) ⊗ V ′I must also contain elementswhich are mapped to elements (9.38) under the action of f m − β iL . Such elements have the form f m − β iL − . . . f m − β i f n γ . . . f n l ′ γ l ′ ⊗ v k + x L , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, x L ∈ f β iL U e gη ( m − ) ⊗ V ′I . Now we proceed by induction. Assume that for some 0 < p < L the image of σ in U e gη ( m − ) ⊗ V ′I contains elements of the form(9.41) f m − β ip . . . f m − β i f n γ . . . f n l ′ γ l ′ ⊗ v k + x p +1 , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, where x p +1 ∈ J p ⊗ V ′I , and J p is the ideal in U e gη ( m − ) generated by the elements f β ip +1 , . . . , f β iL .Since J p is an ideal, V p = J p ⊗ V ′I ⊂ U e gη ( m − ) ⊗ V ′I is invariant under the action of f β ip .Moreover, by commutation relations (5.9) and by the Poincar´e–Birkhoff–Witt theorem for U e gη ( m − ),both U e gη ( m − ) ⊗ V ′I and the subspace V p are free modules over U e gη ( f β ip ), and there is a Jordan basis f nβ ip w t , n = 1 , . . . , m − t = 1 , . . . , S for the action of f β ip on U e gη ( m − ) ⊗ V ′I such that f nβ ip w t , n = 1 , . . . , m − t = 1 , . . . , R ≤ S is a Jordan basis of V p . Actually as w t one can take the followingelements f r L β iL . . . f r p +1 β ip +1 f r p − β ip − . . . f r β i f n γ . . . f n l ′ γ l ′ ⊗ v k , ≤ r i , n i ≤ m − , k = 1 , . . . , M, and for the elements w t with 1 ≤ t ≤ R at least one r i is non–zero for p + 1 ≤ i ≤ L .Now recall that V can be regarded as a free U e gη ( f β ip )–submodule of the free U e gη ( f β ip )–module U e gη ( m − ) ⊗ V ′I . Hence there is an f β ip –Jordan basis of U e gη ( m − ) ⊗ V ′I such that the image of V under σ in U e gη ( m − ) ⊗ V ′I consists of Jordan blocks of that basis. An arbitrary f β ip –Jordan basis of U e gη ( m − ) ⊗ V ′I has the form w s = m − X r =1 S X t =1 a str f rβ ip w t ,w s = m − X r =1 S X t =1 a str f r +1 β ip w t , (9.42) · · · · · · · · · · · · · · · · · · · · · · · · w m − s = S X t =1 a st f m − β ip w t , where s = 1 , . . . , S , a str ∈ C , and det a st = 0.Assume that the coefficients a str are chosen in such a way that w qs for q = 1 , . . . , m − s = 1 , . . . , K ≤ S form a linear basis of the image of V in U e gη ( m − ) ⊗ V ′I under σ .Since V p ⊂ U e gη ( m − ) ⊗ V ′I is a U e gη ( f β ip )–submodule and by the construction of the elements w t the quotient U e gη ( m − ) ⊗ V ′I /V p is a U e gη ( f β ip )–module spanned by the classes of the elements w qs for q = 1 , . . . , m − s = 1 , . . . , S .Let A be the rank of the matrix a st , s = 1 , . . . , S , t = R + 1 , . . . , S . One can find indexes s i , i = 1 , . . . , A such that the classes of the elements w qs i for q = 1 , . . . , m − i = 1 , . . . , A form an f β ip –Jordan basis of U e gη ( m − ) ⊗ V ′I /V p . Thus by the construction of the basis the quotient U e gη ( m − ) ⊗ V ′I /V p is a free U e gη ( f β ip )–module, and the image of V in it has the f β ip –Jordan basisgiven by the classes of w qs i , q = 1 , . . . , m − s i ∈ { , . . . , K } .Therefore the image of V in U e gη ( m − ) ⊗ V ′I /V p is a free U e gη ( f β ip )–module. This image containsthe classes of elements (9.41) which are non–zero by construction and which are annihilated by theaction of f β ip . Hence that image must also contain the classes of elements f m − β ip − . . . f m − β i f n γ . . . f n l ′ γ l ′ ⊗ v k + x ′ p , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, x ′ p ∈ f β ip U e gη ( m − ) ⊗ V ′I which are mapped to the classes of elements (9.41) under the action of f m − β ip , and the image of V in U e gη ( m − ) ⊗ V ′I must contain elements f m − β ip − . . . f m − β i f n γ . . . f n l ′ γ l ′ ⊗ v k + x p , ≤ n j ≤ m − , j = 1 , . . . , l ′ , k = 1 , . . . , M, where x p ∈ J p − ⊗ V ′I . This completes the proof. (cid:3)
Now using the relations f mγ i = a i = 0, the fact that J is an ideal in U e gη ( m − ) and applyingappropriate products of powers of elements f γ i to elements (9.40) we deduce that the image of σ in U e gη ( m − ) ⊗ V ′I contains elements of the form(9.43) y k = 1 ⊗ v k + x, k = 1 , . . . , M, where x ∈ J ⊗ V ′I . Lemma 9.16.
Elements (9.43) are linearly independent over U e gη ( m − ) and generate U e gη ( m − ) ⊗ V ′I over U e gη ( m − ) .Proof. Assume that elements (9.43) are linearly dependent over U e gη ( m − ). Let(9.44) M X k =1 z k y k = 0 , z k ∈ U e gη ( m − )be a relation between them in U e gη ( m − ) ⊗ V ′I .Consider the corresponding relation in ( U e gη ( m − ) ⊗ V ′I ) / ( J ⊗ V ′I ), M X k =1 z k ⊗ v k = 0 , where z k are the classes of the elements z k in U e gη ( m − ) / J . The last relation obviously implies z k = 0,and hence z k ∈ J . Therefore z k y k ∈ J ⊗ V ′I .Now from (9.44) we derive the following relation in ( J ⊗ V ′I ) / (cid:0) J ⊗ V ′I (cid:1) ,(9.45) M X k =1 z k ⊗ v k = 0 , EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 37 where z k are the classes of the elements z k in J / J . Clearly, (9.45) yields z k = 0, and hence z k ∈ J .Finally simple induction and the fact that J is nilpotent imply that z k = 0. Thus elements(9.43) are linearly independent over U e gη ( m − ). The number of elements (9.43) is equal to the rankof U e gη ( m − ) ⊗ V ′I over U e gη ( m − ). Hence elements (9.43) generate U e gη ( m − ) ⊗ V ′I over U e gη ( m − ). Thiscompletes the proof of the lemma (cid:3) By the previous lemma σ is surjective. This completes the proof of the theorem. (cid:3) From the previous theorem and the fact that dim U e gη ( m − ) = m dim m − we immediately obtainthe following corollary. Corollary 9.17.
Assume that the conditions of Theorem 9.12 are satisfied. Then the dimension ofany finite–dimensional U η ( g ) –module V is divisible by m dim m − , and dim V = m dim m − dim V χ . Proposition 7.4 implies that 2dim m − +dim Σ s = dim G . Therefore dim m − = (dim G − dim Σ s ).By Proposition 7.1 Σ s is transversal to the set of conjugacy classes in G . Therefore by Proposition 4.3and by the definition of maps φ and e π for η ∈ Spec( Z ), φ ◦ e π ( η ) ∈ µ − M + ( u ) we have dim G − dim Σ s ≤ dim O η , where O η is the G –orbit of η . In [12] De Concini, Kac and Procesi formulated the followingconjecture. Conjecture 9.18. (De Concini, Kac and Procesi (1992))
The dimension of any finite–dimensional irreducible U η ( g ) –module V is divisible by m dim O η . By Proposition 4.4 it suffices to verify this conjecture in case of elements η ∈ Spec( Z ) such that πη ∈ G is exceptional. Recall that by the discussion above for η ∈ Spec( Z ), φ ◦ e π ( η ) ∈ µ − M + ( u )the dimension of any finite–dimensional U η ( g )–module V is divisible by m (dim G − dim Σ s ) . Remindalso that the map φ is induced by the conjugation action. Combining these facts and Corollary 9.17with the description of the quantum coadjoint action orbits in Proposition 4.3 in terms of the finitecovering π we deduce that for η ∈ Spec( Z ) such that πη ∈ G is conjugate to an element from λ ( µ − M + ( u )) the dimension of any U η ( g )–module V is divisible by m dim m − .Let G = [ C∈ C ( W ) G C . be the Lusztig partition of G (see [28]; we use the notation of [39], Section 4). Here C ( W ) ⊂ W is acertain subset of the set of conjugacy classes W in W . By Theorem 5.2 in [39] for every C ∈ C ( W )there is a system of positive roots ∆ + associated to s ∈ C in ∆ such that all conjugacy classes in thestratum G C intersect the corresponding transversal slice Σ s , s ∈ C at some points of sH N s , and forany g ∈ G C dim Z G ( g ) = dim Σ s , and hence dim m − = (dim G − dim Σ s ) = dim O g , where O g is the conjugacy class of any g ∈ G C .On the other hand every element of sH N s is conjugate to an element of λ ( µ − M + ( u )). Indeed, let sh n s , h ∈ H , n s ∈ N s be such element. Recalling that by Proposition 7.2 s = m − um − for some m, m ∈ N we have sh n s = m − um − h n s . Conjugating the element in the r.h.s. by m we obtain that sh n s is conjugate to um − h n s m = h unh , n ∈ N, where h ∈ H is any element such that h h = h , and we used the fact that H normalizes N andcommutes with the element u . The r.h.s. of the last equality belongs to λ ( µ − M + ( u )) by definition.Therefore from the above discussion, Corollary 9.17 and Propositions 6.1 and 6.2 in [39] we deducethe following theorem. Theorem 9.19.
Let η ∈ Spec( Z ) be an element such that πη ∈ G C , C ∈ C ( W ) . Let q be thenumber introduced in Proposition 6.1 in [39] , and d the number defined in Proposition 6.2 in [39] for the conjugacy class C . Assume that the order m of the root of unity ε is not divisible by q if q isdefined, and suppose that there is a positive integer n such that ε nd − = 1 . Then the dimension ofany finite–dimensional U η ( g ) –module V is divisible by m dim O η . A categorial equivalence
In this section we establish an equivalence between categories of finite–dimensional representationsof quantum groups and of q-W algebras at roots of unity. This is a version of Skryabin equivalencefor quantum groups at roots of unity (see [34]).In this section we assume that the conditions of Theorem 9.12 are satisfied. We shall also use thenotation introduced in that theorem. For given η ∈ Spec( Z ), φ ◦ e π ( η ) ∈ µ − M + ( u ) we assume that aquantum coadjoint transformation e g ∈ G is fixed as in Theorem 9.12 and denote ξ = e gη ∈ Spec( Z ).Let χ be a character of U ξ ( m − ), C χ the corresponding representation of U ξ ( m − ). Denote by Q χ the induced left U ξ ( g )–module, Q χ = U ξ ( g ) ⊗ U ξ ( m − ) C χ . Let W sε,ξ ( G ) = End U ξ ( g ) ( Q χ ) opp be thealgebra of U ξ ( g )–endomorphisms of Q χ with the opposite multiplication. The algebra W sε,ξ ( G ) iscalled a q-W algebra associated to s ∈ W . Denote by U ξ ( g ) − mod the category of finite–dimensionalleft U ξ ( g )–modules and by W sε,ξ ( G ) − mod the category of finite–dimensional left W sε,ξ ( G )–modules.Observe that if V ∈ U ξ ( g ) − mod then the algebra W sε,ξ ( G ) naturally acts on the finite–dimensionalspace V χ = Hom U ξ ( m − ) ( C χ , V ) = Hom U ξ ( g ) ( Q χ , V ) by compositions of homomorphisms. Theorem 10.20.
The functor E Q χ ⊗ W sε,ξ ( G ) E establishes an equivalence of the category offinite–dimensional left W sε,ξ ( G ) –modules and the category U ξ ( g ) − mod . The inverse equivalence isgiven by the functor V V χ . In particular, the latter functor is exact, and every finite–dimensional U ξ ( g ) –module is generated by Whittaker vectors.Proof. Let E be a finite–dimensional W sε,ξ ( G )–module. First we observe that by the definition of thealgebra W sε,ξ ( G ) we have W sε,ξ ( G ) = End U ξ ( g ) ( Q χ ) opp = Hom U ξ ( m − ) ( C χ , Q χ ) = ( Q χ ) χ as a linearspace, and hence ( Q χ ⊗ W sε,ξ ( G ) E ) χ = E . Therefore to prove the theorem it suffices to check thatfor any V ∈ U ξ ( g ) − mod the canonical map f : Q χ ⊗ W sε,ξ ( G ) V χ → V is an isomorphism.Indeed, f is injective because otherwise by Proposition 8.10 its kernel would contain a non–zero Whittaker vector with respect to χ . But all Whittaker vectors of Q χ ⊗ W sε,ξ ( G ) V χ belong tothe subspace 1 ⊗ V χ , and the restriction of f to 1 ⊗ V χ induces an isomorphism of the spaces ofWhittaker vectors of Q χ ⊗ W sε,ξ ( G ) V χ and of V .In order to prove that f is surjective we consider the exact sequence0 → Q χ ⊗ W sε,ξ ( G ) V χ → V → W → , where W is the cokernel of f , and the corresponding long exact sequence of cohomology,0 → Ext U ξ ( m − ) ( C χ , Q χ ⊗ W sε,ξ ( G ) V χ ) → Ext U ξ ( m − ) ( C χ , V ) → Ext U ξ ( m − ) ( C χ , W ) →→ Ext U ξ ( m − ) ( C χ , Q χ ⊗ W sε,ξ ( G ) V χ ) → . . . . Now recall that f induces an isomorphism of the spaces of Whittaker vectors of Q χ ⊗ W sε,ξ ( G ) V χ and of V . By Theorem 9.12 the finite–dimensional U ξ ( g )–module Q χ ⊗ W sε,ξ ( G ) V χ is free over U ξ ( m − ). EPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF UNITY AND Q-W ALGEBRAS 39
Since U ξ ( m − ) is Frobenius Q χ ⊗ W sε,ξ ( G ) V χ is also injective over U ξ ( m − ), and henceExt U ξ ( m − ) ( C χ , Q χ ⊗ W sε,ξ ( G ) V χ ) = 0. Therefore the initial part of the long exact cohomology sequencetakes the form 0 → V χ → V χ → W χ → , where the second map in the last sequence is an isomorphism. Using the last exact sequence wededuce that W χ = 0. But if W were non–trivial it would contain a non–zero Whittaker vector byProposition 8.10. Thus W = 0, and f is surjective. This completes the proof of the theorem. (cid:3) Next we study some further properties of q-W algebras at roots of unity and of the module Q χ .First we prove the following lemma. Lemma 10.21.
The left U ξ ( g ) –module Q χ is projective in the category U ξ ( g ) − mod .Proof. We have to show that the functor Hom U ξ ( g ) ( Q χ , · ) is exact. Let V • be an exact complex offinite–dimensional U ξ ( g )–modules. Since by Theorem 9.12 objects of U ξ ( g ) − mod are U ξ ( m − )–free,and U ξ ( m − ) is Frobenius we have V • = U ξ ( m − ) ⊗ V • ≃ U ξ ( m − ) ∗ ⊗ V • , where V • is an exact complex of vector spaces and the action of U ξ ( m − ) on U ξ ( m − ) ∗ is induced bymultiplication from the right on U ξ ( m − ).Now by Frobenius reciprocity we have obvious isomorphisms of complexes,Hom U ξ ( g ) ( Q χ , V • ) ≃ Hom U ξ ( g ) ( Q χ , U ξ ( m − ) ∗ ⊗ V • ) = Hom U ξ ( m − ) ( C χ , U ξ ( m − ) ∗ ⊗ V • ) ≃≃ Hom C ( U ξ ( m − ) ⊗ U ξ ( m − ) C χ , V • ) = V • , where the last complex is exact. Therefore the functor Hom U ξ ( m − ) ( Q χ , · ) is exact. (cid:3) The following proposition is an analogue of Theorem 2.3 in [34] for quantum groups at roots ofunity.
Proposition 10.22.
Let η ∈ Spec( Z ) , φ ◦ e π ( η ) ∈ µ − M + ( u ) and assume that a quantum coadjointtransformation e g ∈ G is fixed as in Theorem 9.12. Denote ξ = e gη ∈ Spec( Z ) and b = m dim m − . Let χ be a character of U ξ ( m − ) , C χ the corresponding representation of U ξ ( m − ) . Then Q bχ ≃ U ξ ( g ) asleft U ξ ( g ) –modules, U ξ ( g ) ≃ Mat b ( W sε,ξ ( G )) as algebras and Q χ ≃ ( W sε,ξ ( G ) opp ) b as right W sε,ξ ( G ) –modules.Proof. Let E i , i = 1 , . . . , C be the simple finite–dimensional modules over the finite–dimensionalalgebra U ξ ( g ). Denote by P i the projective cover of E i . Since by Theorem 9.12 the dimension of E i is divisible by b we have dim E i = br i , r i ∈ N , where r i is the rank of E i over U ξ ( m − ) equal to thedimension of the space of Whittaker vectors in E i . By Proposition 2.1 in [34] U ξ ( g ) = Mat b (End U ξ ( g ) ( P ) opp ) , where P = L Ci =1 P r i i . Therefore to prove the second statement of the proposition it suffices to showthat P ≃ Q χ . Since by the previous lemma Q χ is projective we only need to verify that r i = dim Hom U ξ ( g ) ( P, E i ) = dim Hom U ξ ( g ) ( Q χ , E i ) . Indeed, by Frobenius reciprocity we havedim Hom U ξ ( g ) ( Q χ , E i ) = dim Hom U ξ ( m − ) ( C χ , E i ) = r i . This proves the second statement of the proposition. From Proposition 2.1 in [34] we also deducethat P b ≃ U ξ ( g ) as left U ξ ( g )–modules. Together with the isomorphism P ≃ Q χ this gives the firststatement of the proposition.Using results of Section 6.4 in [30] and the fact that Q χ is projective one can find an idempotent e ∈ U ξ ( g ) such that Q χ ≃ U ξ ( g ) e as modules and W sε,ξ ( G ) ≃ eU ξ ( g ) e as algebras.By the first two statements of this proposition one can also find idempotents e = e , e , . . . , e b ∈ U ξ ( g ) such that e + . . . + e b = 1, e i e j = 0 if i = j and e i U ξ ( g ) = eU ξ ( g ) as right U ξ ( g )–modules.Therefore e i U ξ ( g ) e = eU ξ ( g ) e as right eU ξ ( g ) e –modules, and Q χ ≃ U ξ ( g ) e = b M i =1 e i U ξ ( g ) e ≃ ( eU ξ ( g ) e ) b ≃ ( W sε,ξ ( G ) opp ) b as right W sε,ξ ( G )–modules. This completes the proof of the proposition (cid:3) Corollary 10.23.
The algebra W sε,ξ ( G ) is finite–dimensional, and dim W sε,ξ ( G ) = m dim Σ s .Proof. By Proposition 7.4 2dim m − + dim Σ s = dim G . Therefore by the definition of Q χ wehave dim Q χ = m dim G − dim m − = m dim m − +dim Σ s . Finally from the last statement of the previoustheorem one obtains that dim W sε,ξ ( G ) = dim Q χ /m dim m − = m dim Σ s . (cid:3) By the results of the discussion before Theorem 9.19 in the end of Section 9 for any η ∈ Spec Z such that π ( η ) ∈ G C , π ( η ) is conjugate to an element from λ ( µ − M + ( u )), where u corresponds to s ∈ C .Combining this observation and Propositions 6.1 and 6.2 in [39] we deduce from Proposition 10.22the following theorem on the structure of the algebra U η ( g ). Theorem 10.24.
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