Slices for biparabolics of index one
aa r X i v : . [ m a t h . R T ] N ov SLICES FOR BIPARABOLICS OF INDEX 1 Anthony Joseph
Donald Frey Professional ChairDepartment of MathematicsWeizmann Institute of Science2 Herzl StreetRehovot, 76100, [email protected]
Florence Fauquant-Millet
Universit´e de Lyon, F-42023 Saint-Etienne, FranceLaboratoire de Math´ematiques de l’Universit´e de Saint-EtienneFacult´e des Sciences et Techniques23, rue du Docteur Paul MichelonF-42023 Saint-Etienne C´edex 02 Francefl[email protected] Words: Invariants, Slices, Nil-cones.AMS Classification: 17B35
Abstract
Let a be an algebraic Lie subalgebra of a simple Lie algebra g with index a ≤ rank g .Let Y ( a ) denote the algebra of a invariant polynomial functions on a ∗ . An algebraic slicefor a is an affine subspace η + V with η ∈ a ∗ and V ⊂ a ∗ a subspace of dimension index a such that restriction of function induces an isomorphism of Y ( a ) onto the algebra R [ η + V ]of regular functions on η + V .Slices have been obtained in a number of cases through the construction of an adaptedpair ( h, η ) in which h ∈ a is ad-semisimple, η is a regular element of a which is an eigenvectorfor h of eigenvalue minus one and V is an h stable complement to (ad a ) η in a ∗ . The classicalcase is for g semisimple [15], [16]. Yet rather recently many other cases have been provided.For example if g is of type A and a is a “truncated biparabolic” [11] or a centralizer [12].In some of these cases (particular when the biparabolic is a Borel subalgebra) it was found[12], [13], that η could be taken to be the restriction of a regular nilpotent element in g .Moreover this calculation suggested [12] how to construct slices outside type A when noadapted pair exists.This article makes a first step in taking these ideas further. Specifically let a be atruncated biparabolic of index one. (This only arises if g is of type A and a is the derivedalgebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes arecoprime.) In this case it is shown that the second member of an adapted pair ( h, η ) for a is the restriction of a particularly carefully chosen regular nilpotent element of g .A by-product of the present analysis is the construction of an invariant associated to apair of coprime integers. Date : April 21, 2019. Work supported in part by Israel Science Foundation Grant, no. 710724. Introduction
Unless mentioned to the contrary the base field K is assumed algebraically closed ofcharacteristic zero.1.1. Invariants.
Let a be a finite dimensional Lie algebra, S ( a ) its symmetric algebraand K ( a ) the field of fractions of S ( a ). If A is algebra in which a acts by derivations, set A a = { a ∈ A | xa = 0 , ∀ x ∈ a } . It is a subalgebra of A .Given ξ ∈ a ∗ , set a ξ = { a ∈ a | aξ = 0 } , that is the stabilizer of ξ under co-adjoint action.It is a Lie subalgebra of a .Define index a := min ξ ∈ a ∗ dim a ξ . Set a ∗ reg = { ξ ∈ a ∗ | dim a ξ = index a } , called the setof regular elements of a ∗ .A problem of Dixmier [3, Problem 4] suggests that C ( a ) := K ( a ) a , is always a puretranscendental extension of K .One may further ask under what conditions is Y ( a ) := S ( a ) a a polynomial algebra.1.2. Slices.
In [13, Sect. 7] we focused some attention on refinements of these questions.Here it is convenient to assume that S ( a ) admits no proper semi-invariants. In this case C ( a ) is just the field of fractions of Y ( a ). Moreover under this hypothesis, Ooms andVan den Bergh [17, Prop. 4.1] have shown that the growth rate (that is Gelfand-Kirillovdimension) of Y ( a ) takes its maximum possible value, namely index a .Under the above hypothesis define a rational slice to be an affine translate η + V ⊂ a ∗ of a vector subspace of V of a ∗ such that the restriction of functions gives on injection θ of Y ( a ) into the algebra of regular functions R [ η + V ] on η + V and induces an isomorphismof fields of fractions. Observe that R [ η + V ] identifies with S ( V ∗ ) and then comparison oftranscendence degrees implies that dim V = index a . We call η the base point of the slice η + V .We suggested that a rational slice always exists [13, 7.11].Define an algebraic slice to be a rational slice for which θ is an isomorphism. Obviouslythis implies that Y ( a ) is a polynomial algebra; but we found an example ([13, 11.4, Example2]) for which the converse is false.In view of this counter-example it would seem appropriate to suggest that if Y ( a ) ispolynomial then there exists an affine subspace η + V ⊂ a ∗ such that restriction of functionsgives an embedding Y ( a ) ֒ → R [ η + V ] ∼ → S ( V ∗ ) whose image takes the form S ( V ∗ ) G forsome finite (pseudo-reflection) group G acting linearly on V ∗ .Finally we remark that the notions of a rational or algebraic slice were given ([13, Sect.7] natural geometric interpretations in the case when A is a connected algebraic group withLie algebra a , that is when a is algebraic. In particular A ( η + V ) must be dense [13, 7.9](but not necessarily open [13, 11.4, Example 3]) in a ∗ . Thus η + V must meet most regularorbits (defined as those of codimension equal to index a ). However even in the case of analgebraic slice not every regular orbit need pass [8, 8.12(ii)] through η + V , nor need everyorbit meeting η + V be regular [13, 11.4, Example 3]. In particular the base point η neednot be regular. LICES 3
Adapted Pairs.
An adapted pair ( h, η ) for a finite dimensional Lie algebra consistsof a regular element η ∈ a ∗ and an element h ∈ a such hη = − η with respect to co-adjointaction. Such pairs are rather hard to find, it being particularly difficult to check regularity.Assume that a is an algebraic Lie algebra. Then in the above we may use Jordandecomposition to show that the (adjoint) action of h on a can be taken to be reductivewithout loss of generality. As a η is h stable we may define { m i } index a i =1 , to be the set ofeigenvalues (counted with multiplicities) of − h acting on a η . These can be rather arbitraryand may depend on the choice of the adapted pair [14, 8.3]. However suppose that S ( a )admits no proper semi-invariants and that Y ( a ) is polynomial. Then by [14, Cor. 2.3] thedegrees of the homogeneous generators of Y ( a ) are the m i + 1 : i = 1 , , . . . , index a , andmoreover η + V is an algebraic slice for any h stable complement V to a η in a ∗ . (This isactually proved under a slightly weaker hypothesis which allows a to be the centralizer g x in a semisimple Lie algebra g .) Thus the { m i } generalize the so-called “exponents” definedclassically for a semisimple.In the above situation every element of η + V is regular (see for example [13, 7.8]) by astandard deformation argument.1.4. The Nilpotent Cone.
One would like to have a systematic way to construct alge-braic slices. In this we make the rather bold suggestion below. It should be regarded moreas a signpost rather than a serious conjecture.Suppose that a is an algebraic subalgebra of a semisimple Lie algebra g . Let G be theadjoint group of g and A the unique closed subgroup whose Lie algebra is a .Assume that index a ≤ rank g . This is the case if a is a biparabolic subalgebra [5], [7]or a centralizer (via the now known truth of the Elashvili conjecture [2], [6], [21]).Further assume that g admits a Chevalley antiautomorphism κ such that a and κ ( a )are non-degenerately paired through the Killing form K on g . This is clearly the case for(truncated) biparabolics. It is well-known for a centralizer - see [14, 4.5,4.6] for detailsand references. It is less clear that this condition is really necessary. We use it mainly forconvenience.Under the above hypothesis we may and will identify a ∗ with the subalgebra κ ( a ) of g .Let k be the kernel of the restriction map g ∗ → a ∗ . Identifying g ∗ with g through theKilling form we may view k as a subspace of g .Obviously k is A stable. In view of the above identifications we may write A ( ξ + k ) = Aξ + k , for all ξ ∈ a ∗ . In particular if ξ ∈ a ∗ reg , then codim A ( ξ + k ) ≤ rank g .Let N ( g ) denote the cone of ad-nilpotent elements of g . As is well-known, codim N ( g ) = rank g . Moreover N ( g ) is irreducible and admits only finitely many G orbits. Inparticular N ( g ) reg consists of a dense open orbit.Now let ( h, η ) be an adapted pair for a . The relation hη = − η , forces η ∈ N ( g ).However it is almost never the case that η ∈ N ( g ) reg . (For example, in [11, Sect. 10] adetailed study of the nilpotent orbit to which η belongs was made in the case of an adaptedpair ( h, η ) of a truncated biparabolic in type A .)In view of the above codimensionality estimates we propose the Suggestion . Suppose ( h, η ) is an adapted pair for a . Then η + k ∩ N ( g ) reg is non-empty. ANTHONY JOSEPH AND FLORENCE MILLET
Remarks . This just means that there is some pre-image of η ∈ g ∗ lying in N ( g ) reg . Itcould be proved by showing that the codimension of N ( g ) ∩ A ( η + k ) in g ∗ equals rank g ,though this is likely to be rather difficult if even true. It suggests that one should construct η as the restriction of some element of N ( g ) reg . The fact that η itself does not belongto N ( g ) reg is just a consequence of having made a particular choice of its pre-image in g .However it is this choice which allows one to guess η , itself a rather hard task as explainedin [11, 1.3]. A main point that lies behind our suggestion is that there may be a more“canonical” choice which leads to a slice making sense for a biparabolic or centralizer of anarbitrary semisimple Lie algebra. For the moment it is not too clear if this can be divined.Hopefully the present article will provide a clue. Here we should also stress that adaptedpairs are far from unique even up to the obvious conjugation [11, 1.4]. Our suggestion isalso partly motivated by Question (5) of [11, Sect. 11].1.5. Let us recall some cases in which the above suggestion has a positive answer.Choose a Cartan subalgebra h of g and let ∆ ⊂ h ∗ denote the set of non-zero roots of g with respect to h . For each α ∈ ∆ let x α be a non-zero vector in g of weight α .let π ⊂ ∆ be a choice of simple roots and set ∆ + = ∆ ∩ N π , n = P α ∈ ∆ + K x α and b = h + n , which is a Borel subalgebra. Let N, H, B be the corresponding closed subgroupsof G .First assume that a is a centralizer, that is of the form g x . (Here we can assume x nilpotent without loss of generality and we shall always do this.)Suppose ξ is a regular element of ( g x ) ∗ , for example coming from the second factor in anadapted pair. Then under the identifications made in 1.4, it follows from [12, Lemma 2.2](which was inspired by the proof of the Vinberg inequality) and the truth of the Elashviliconjecture that x + tξ is a regular element of g ∗ for all t belonging to a cofinite subsetΩ ⊂ K . However in general x + tξ will not be nilpotent. Rather for ξ in general position x + tξ : t ∈ Ω will be semisimple [12, 5.5].Conversely if g is of type A , then we may take x in Jordan form (defined by an orderedpartition x of n ) and then “complete” it to a standard regular nilpotent element. Moreprecisely up to conjugation we can write P α ∈ π ′ x α , with π ′ ⊂ π corresponding to x . Set y ′ = P α ∈ π \ π ′ x α . Then x + y ′ = P α ∈ π x α , which is the standard presentation of a regularnilpotent element.Now consider y ′ as an element of ( g x ) ∗ through the Killing form K . (With respect towhat we said in 1.4 we can arrange for x, y := κ ( x ) to generate a Jacobson-Morosov s-triple containing x . In this g y = κ ( g x ), contains y ′ and is non-degenerately paired to g x through K , so identifies with ( g x ) ∗ .) A basic result proved in [12, Thm. 4.7] is that thereexists h ′ ∈ h ∩ g x making ( h ′ , y ′ ) an adapted pair for g x . Now clearly K ( x, g x ) = 0 andso x ∈ k . We conclude that for this particular adapted pair the suggestion of 1.4 has apositive answer.Notice that this construction makes sense for nilpotent orbits generated by a subset ofthe simple root vectors (called Bala-Carter orbits or orbits of Cartan type) for any simpleLie algebra. However y ′ obtained in this fashion is seldom regular in ( g x ) ∗ and in factregularity requires a very careful choice of π ′ . An interesting case is when card π ′ = 1, LICES 5 say π ′ = { α } . Notice that if α is a long root, then g x α is conjugate to the centralizerof the highest root vector which is also a standard truncated parabolic subalgebra. Nowforgetting type A n , which is just the case when the Coxeter number is odd, a simple rootsystem contains a distinguished long root defined in terms of its Dynkin diagram. This isthe central root in type A n +1 , the root with three neighbours in types D, E and the uniquelong root with a short root neighbour in types
B, C, F, G . (For a further interpretationrelating this construction to the highest root, see [13, 2.14].)If one chooses the (long) simple root α as above, then in all cases (except E ) the element y ′ as defined above can be completed to an adapted pair [12, Sect. 6]; but this generallyfails if one takes α to be an arbitrary long simple root. This again verifies our suggestionfor that particular pair, showing in addition that the question is rather delicate. In type E , the element y ′ is not regular [12, 6.14] in ( g x ) ∗ and it is not known if the latter algebraadmits an adapted pair. After Yakimova [22] the invariant algebra Y ( g x ) is not polynomial.Our suggestion was also found to hold for some adapted pairs for the (truncated) Borelsubalgebra in type A . In this case the specification of a Borel subalgebra implies a choice ofa set π of simple roots and it was found that the regular nilpotent element in the conclusionof the suggestion was obtained from P α ∈ π x − α through conjugation by a rather carefullychosen element of the Weyl group W := N G ( H ) /H . It turned out that this element of W made sense for all simple Lie algebras and through its use we were able to construct[13, Thm. 9.4] an algebraic slice for a truncated Borel in all types except C, B n , F eventhough an adapted pair does not exist (outside type A ). Here the base point η was notregular but still satisfied the conclusion of Suggestion 1.4. Obviously we should like to takethese last observations further.We remark that the index of a (truncated) parabolic (resp. biparabolic) was calculatedin [5] (resp. [7]) and that in most cases (all cases for types A, C ) the invariant algebrawas shown to be polynomial ([4], [7]). In type A an adapted pair was constructed for alltruncated biparabolics [11]. For a centralizer g x of a simple Lie algebra g the invariantalgebra was shown [18] to be polynomial in many cases (all cases in types A, C ), whilst intype A , or for a long root vector outside type E , the above construction of an adaptedpair (which has the additional property of being “compatible”) allows one to prove veryeasily [12, Thm. 3.5] the polynomiality of Y ( g x ).1.6. The purpose of the present article is to verify our suggestion for (truncated) bi-parabolics of index one. As noted in [9, 2.2,2.3], these are described as the derived algebrasof maximal parabolic subalgebras in type A for which the Levi factor consists of two blocksof coprime sizes p, q . In this we shall take p < q with the smaller block in the top left handcorner. The parabolic is assumed to have Levi factor having these two blocks and withnilradical m being the lower left hand corner block and thus is spanned by root vectorsin which the “non-compact” simple root, namely α p in the Bourbaki notation [1, PlancheI], occurs with coefficient − m . The truncated parabolic p is just thederived algebra of the above. (Though it might be more appropriate to denote it by p ′ ,this would just be cumbersome and in any case we do not need to refer to the parabolicitself.) We denote by P the closed subgroup of G with Lie algebra p . An adapted pair for ANTHONY JOSEPH AND FLORENCE MILLET p was constructed in [9]. A rather unusual (but easily proven - see 1.8) aspect of the indexone case is that such a pair is unique up to conjugation by an element of P . Moreover via[8, Cor. 8.7], every regular orbit meets the resulting slice at exactly one point and eventransversally (see [13, Prop. 7.8(ii)] for example). This is the exact analogue of the resultobtained in the semisimple case by Kostant [15], [16].1.7. The proof that our suggestion holds in the above case is obtained from the combi-natorial analysis given in the following two sections. This turned out to be surprisinglydifficult though ultimately we believe the solution is rather elegant. However unlike theBorel case and the case p = 1, the element in its conclusion is obtained from a standardnilpotent element not just by conjugation through W but rather by an element of the form n w b , with n w a representative of w ∈ W lying in N G ( H ) and b ∈ B . Moreover we give arecipe for computing w , but at present its meaning is unclear.1.8. Define p as in 1.6. Let us recall the construction of an adapted pair for p given in [9].Let h ′ denote the set of diagonal matrices lying in p . One has dim h ′ = n − π := { α i } n − i =1 be the set of simple roots for sl ( n ) labelled as in Bourbaki [1, PlancheI] with respect to the Borel subalgebra b being the set of upper triangular matrices of tracezero.Identify p ∗ with p − := κ ( p ). Recall that we are assuming b ⊂ p − . The nilradical m of p isa complement to p − in g and identifies via the Killing form with the kernel of the restrictionmap g ∗ → p ∗ , that is m = k , in the notation of 1.4. Under the present conventions m isspanned by those vectors corresponding to roots in which α p appears with a coefficient of −
1. (This convention should be recalled in 2.6 b).)Recall (cf [9, 2.5]) the notion of the Kostant cascade B (of positive strongly orthogonalroots) defined for any semisimple Lie algebra. For sl ( n ) this is just { α i + α i +1 + , . . . , + α n − i } [( n − / i =1 .In particular B ∩ π = φ , if and only if n is even and then this intersection is { α n/ } .By definition, the Levi factor of p is isomorphic to sl ( p ) × sl ( q ). Let − B ′ denote itsKostant cascade.Since q, p are coprime, exactly one of the integers p, q, n is even and so ( B ∪ B ′ ) ∩ ( π ∪− π )consists of exactly one element, say α . Set B := B ∪ B ′ \ { α } . One may remark that upto signs B ∪ B ′ is a choice of simple roots [9, 2.6]. In particular B ∪ B ′ is a basis for h ∗ .A main result of [9] is that η := X β ∈ B x β , ( ∗ )is regular (see [9, 3.7]) in p ∗ and that there exists a unique h ∈ h ′ such that hη = − η , withrespect to co-adjoint action. Thus ( h, η ) is an adapted pair for p .It is checked in [9, 3.7] that K x α is a complement to p η in p ∗ . It is further checked[9, 3.3] that hx α = mx α , where m + 1 is the degree, namely p + q + pq − , of the uniquehomogeneous generator f of Y ( p ). Moreover f is irreducible. Indeed otherwise it couldnot be the generator of Y ( p ). We remark that for p = 1 there is a rather precise andelegant description of f discovered independently by Dixmier and Joseph (see [4, 15] and LICES 7 references therein). However even for the case p = 2 , q = 3 a simple description of f is notknown.Let N ( p ), or simply N , denote the zero set of f in p ∗ . By the above remarks N isan irreducible closed subvariety of codimension one in p ∗ . In more general terms if a is analgebraic Lie algebra, then N ( a ) is defined to be the nilfibre of the categorical quotientmap a ∗ → a ∗ //A . It is seldom irreducible even for a truncated biparabolic [11, 1.4].The relation hη = − η , forces P η ⊂ N . Since P η has codimension index p = 1 in p ∗ ,it follows from the above that P η is open dense in N and consequently P η = N reg . Inparticular if ( h ′ , η ′ ) is a second adapted pair for p , then η ′ ∈ P η . Moreover since h isuniquely determined by η , we conclude that there exists p ∈ P such that ph = h ′ , pη = η ′ .Through the identifications made in 1.4 we may consider η as an element of g ∗ . Thenthe relation hη = − η forces η ∈ N ( g ), but one cannot conclude that P η ⊂ N ( g ). Again η is not regular in N ( g ). Here one should stress that by identifying p ∗ with κ ( p ) we arechoosing a particular pre-image of η ∈ g ∗ .The main result of the present paper is that there exists ξ ∈ m such that η + ξ ∈ N ( g ) reg .This can be expressed as saying that there is some pre-image of η ∈ g ∗ lying in N ( g ) reg .We do not believe that this to be a priori obvious. However one may remark that sinceindex p = 1, it follows that P η + m has codimension 1 in g ∗ . On the other hand since g is semisimple the codimension of g ∗ \ g ∗ reg in g ∗ is 3 (as is well-known - see [10, 2.6.14], forexample). It follows that ( P η + m ) ∩ g ∗ reg = φ , however this is not quite what we require.1.9. Recall the notation of 1.5. Set x = P α ∈ π x α . Set n ′ = [ n , n ]. We need the followingwell-known technical result. We give a proof for completeness. Lemma.
N x = x + n ′ .Proof. The inclusion
N x ⊂ x + n ′ , is trivial.The converse will be proved by an easy induction. For all β ∈ ∆ + , we may write β = P α ∈ π k α α and we set o ( β ) = P α ∈ π k α . Let N β be the closed subgroup of N with Liealgebra K x β . For all k ∈ N + , set N k := Y β ∈ ∆ + | o ( β ) ≥ k N β . ( ∗ )Clearly N k is a closed subgroup of N with Lie algebra n k := X β ∈ ∆ + | o ( β ) ≥ k K x β . Set n k := X β ∈ ∆ + | o ( β )= k K x β . Then n k = X ℓ ≥ k n ℓ . ANTHONY JOSEPH AND FLORENCE MILLET
Suppose we have shown that x + n k +1 ⊂ N k x , which is of course trivial for k sufficientlylarge. If k = 1, we are done. Otherwise use of the well-known relation n k = [ n , n k − ],together with the induction hypothesis gives the assertion for k replaced by k − (cid:3) Remark 1 . It is clear that
N x is dense in x + n ′ . On the other hand N is unipotentgroup acting linearly on its Lie algebra n . Thus N x is closed in n by a result of Rosenlicht[20, Thm. 2] and from this the lemma follows. Actually Rosenlicht attributes (withoutreference) the required version of the result to Kostant the latter having given a “compli-cated Lie algebra argument”, of which we believe the above is an extract (see [15, Thm.3.6]). Remark 2 . The result is even easier for sl ( n ) since closure is not needed. Take x ′ ∈ x + n ′ and let V be the standard sl ( n ) module of dimension n . Choose a basis in V so that x hasJordan block form. From this one immediately verifies that V, x ′ V, x ′ V, . . . , is a completeflag and so there exists a basis for V such that x ′ has Jordan block form. Thus x ′ (as wellas x ) is regular. Consequently dim N x ′ = dim N − dim C N ( x ′ ) ≥ dim N − dim C G ( x ′ ) = | ∆ | − rank g = dim n ′ . Thus N x ′ must be dense in the irreducible variety x + n ′ and henceopen. As a special case, N x is open dense in x + n ′ . Consequently these orbits must meetand so x ′ ∈ N x .1.10. Let P denote the set of pairs of coprime positive integers and S the set of allfinite ordered sequences of ones and minus ones. Our construction gives a map (possiblysurjective) of P into S , through the signature of a meander (see 2.1 and 3.3). We believethis to be quite new though whether it has any arithmetical significance is another matter.It would be interesting to determine the image and fibres of this map.1.11. V. Popov has informed us of work of particularly the Russian school on algebraic andrational slices. Although this has practically no intersection with our present paper (beingconcerned mainly with case where g is a reductive group acting on a finite dimensionalmodule V ) it is nevertheless appropriate to give a sketch of their results of which [19]provides in particular a useful summary.Adopting the terminology of [19] we call a linear action of a Lie algebra a on a finitedimensional vector space co-regular if the algebra of invariant regular functions on V ispolynomial.What we call an algebraic slice in [13, 7.6], the Russian school had called a Weierstrasssection. (This terminology comes for the case g = sl (3) acting on a simple ten dimen-sional module for which such a section was exhibited by Weierstrass.) The existence of aWeierstrass section (trivially) forces the action to be co-regular.A fairly comprehensive study of Weierstrass sections was given in [19, Sect 2] for a co-regular action of a semisimple Lie algebra g acting on a finite dimensional vector space V particularly if either g or V is simple. A notable general result [19, Thm. 2.2.15] is thata Weierstrass section exists if the zero fibre N V ( g ) of the categorical quotient map admits LICES 9 a regular element (in a sense analogous 1.1). Moreover the converse holds if the set ofnon-regular elements in V is of codimension ≥ g semisimpleit is not improbable that they extend to the general case as we already partly verified in[14]. Moreover it is interesting to note that in all our examples (with g solvable) wherewe found [13] a Weierstrass section for which the base point was not regular, the set ofnon-regular elements (in g ∗ ) was indeed of codimension 1.Apart from these general considerations, when in comes to actually finding a Weierstrasssection the results reported in [19] and own own work [11, 12, 13] are of a quite differentnature not least because they are mainly obtained on a case by case basis and whilst [19,Sect 2] concentrates on the semisimple case, our own work concerns the non-reductive case.Indeed it is not easy to find coregular actions, rather difficult to find regular elements inthe zero fibre and even harder to exhibit Weierstrass sections if no such elements exist.Just to exemplify the last of these, Popov [19, 2.2.16] notes that for the action sl ( n ) on n copies of its defining n dimensional module, the invariant algebra is generated by the(obvious) determinant and as a consequence the nilfibre has no regular elements, whilsta Weierstrass section obtains by sending all off-diagonal entries to zero and all diagonalentries besides the first to zero. On the other hand our examples [13] come from truncatedBorels of simple Lie algebras outside types A, C, B n , F and this for the adjoint action. Inthese cases they are many generators and a Weierstrass section is not so easy to describe.Classifying Weierstrass sections for co-regular actions of non-reductive groups is a wideopen problem. Acknowledgement . The authors would like to thank Anna Melnikov for Latex instructionand Vladimir Popov for his comments on some points in the manuscript.A preliminary version of this result was presented by Florence Fauquant-Millet at theWorkshop ”Problems and Progress in Lie Algebraic Theory ” held on 7-8 July 2010 in theWeizmann Institute. 2.
The Combinatorial Construction g denotes the simple Lie algebra sl ( n ), with n >
2. Let h denote thediagonal matrices in g . It is a Cartan subalgebra. Let ( , ) denote the Cartan scalarproduct on h .Set I = { , , . . . , n − } , ˆ I = I ∪ { n } . Following Bourbaki [1, Planche I], we choose anorthonormal basis ε i : i ∈ ˆ I in R n and set α i = ε i − ε i +1 : ∀ i ∈ I . Then π = { α i } i ∈ I is asimple root system for g .Let p, q be positive integers with sum n . We assume that p ≤ q . Following a suggestion(see [9, 2.6, Remark] of G. Binyamini we use the Dergachev-Kirillov meanders on the set { ε , ε , . . . , ε n } to describe the support of the second element η of the adapted pair ( h, η )defined in 1.8. This is instead of using the action of the group < i, j > defined in [4] (see [9,2.2]) on the set π := { α , α , . . . , α n − } of simple roots. Here a meander is interpreted as an orbit of the group Γ generated by involutions σ, τ defined as follows. For all i = 1 , , . . . , n ,set σ ( i ) = n + 1 − i and τ ( i ) = (cid:26) p + 1 − i : 1 ≤ i ≤ p,n + p + 1 − i : p + 1 ≤ i ≤ n. One checks that τ σ ( k ) = p + k , where it is understood that any integer is reduced modulo n so that it lies in [1 , n ]. It follows that { , , . . . , n } is a single Γ orbit O if and only if p, q are coprime.2.2. Assume from now on that p, q are coprime.By a slight abuse of language we say that an end point of O is an element of O fixed byeither σ or τ . One easily checks that O has exactly two end points a, b .If p is odd, we can set a = ( p + 1) /
2. If in addition n (resp. q ) is odd we can set b = ( n + 1) / b = p + ( q + 1) / a < b . If p is even we can set b = ( n + 1) / a = p + ( q + 1) /
2. In this case b < a . We call a (resp. b ) the starting(resp. finishing) point of O .We define a bijection ϕ : ˆ I ∼ → ˆ I as follows. First note that the starting point a isalways a τ fixed point. Then set ϕ (1) = a, ϕ (2) = σ ( a ) , ϕ (3) = τ σ ( a ) , . . . . (This may bea little confusing since the domain which identifies with { , , . . . , n } and the target whichidentifies with O are both denoted by ˆ I .)Set β i = ε ϕ ( i ) − ε ϕ ( i +1) : i ∈ I . By our conventions β is a positive (resp. negative) rootif p odd (resp. even). Again ( β i , β i +1 ) <
0, for all i ∈ I \ { n − } , whilst the remainingscalar products between distinct elements, vanish. Hence Π := { β i } i ∈ I is a simple rootsystem and in particular W conjugate to π .Recall 1.8. One easily checks that up to signs there is a unique subset of Π which is theKostant cascade B for sl ( n ) and again up to signs Π r B is the opposed Kostant cascade B ′ for the Levi factor sl ( p ) × sl ( q ) of p . In particular relative to π the elements of B (resp. B ′ ) are positive (resp. negative roots). It is the analysis of these signs which is the maincombinatorial content behind the construction of a further simple root system Π ∗ . Thisis the main step in achieving our goal of finding a regular nilpotent element y of g , whoserestriction to p is η .2.3. Towards the above goal we define a turning point of O to be an element ϕ ( t ) : t ∈ ˆ I such that t − σ ( t ) is of opposite sign to t − τ ( t ). Here we include the end points of O in itsset of turning points, that is to say when one of the above integers is zero. The remainingturning points are called internal turning points.The observation in 2.2 can be expressed as saying that for all i ∈ I one has either β i ∈ B ∪ B ′ or β i ∈ − ( B ∪ B ′ ). Notice further that up to signs if β i − ∈ B , then itssuccessor β i ∈ B ′ and vice-versa. Let us now make precise how these signs vary. Indeedtaking account of the fact that the elements of B (resp. B ′ ) are positive (resp. negative)roots, the following fact is easily verified. Lemma.
Suppose β t − ∈ B ∪ B ′ (resp. β t − ∈ − ( B ∪ B ′ ) ), then β t ∈ − ( B ∪ B ′ ) (resp. β t ∈ B ∪ B ′ ) if and only if ϕ ( t ) is an internal turning point of O . LICES 11
Remark . By our conventions if p is odd, then β ∈ B and if p is even, then β ∈ − B .2.4. It is easy to compute the set of turning points of O . They form two disjoint “con-nected” sets, namely A := [[ p/
2] + 1 , p ] and B := [[ n/
2] + 1 , p + [( q + 1) / p is oddboth have cardinality ( p + 1) / a ∈ A and b ∈ B . If p is even, then A hascardinality p/
2, whilst B has cardinality 1 + p/ a and b .The Γ orbit O viewed as starting at a and finishing at b acquires a linear order withsmallest element a and largest element b being the natural order on ˆ I translated under ϕ .It induces a linear order on the set T of turning points. Lemma.
With respect to the above linear order on T the nearest neighbour(s) of an elementof A lie(s) in B and vice-versa.Proof. Since | A | ≤ | B | with equality unless both a, b lie in B in which case | A | + 1 = | B | ,it is enough to show that the set of successors of an element of B first meets A . Take b ′ ∈ B and assume that σ ( b ′ ) (resp. τ ( b ′ )) is a successor of b ′ . Since B lies in an intervalof width ≤ p/
2, whilst τ σ (resp. στ ) is translation by p (resp. − p ), it follows that the setof successors of b ′ with respect to powers of τ σ (resp. στ ) meets the interval [1 , p ] beforeit meets B again.Finally observe that the image under τ of any of the above p -translates of b ′ (that is tosay after starting at b ′ and until [1 , p ] is reached) do not lie in B . On the other hand when[1 , p ] is reached then the set of successors first meets A since A ∪ τ ( A ) = [1 , p ]. Hence theassertion of the lemma. (cid:3) p + 1 turning points of which p − p is odd (resp. even) we label them as ϕ ( t i ) : i = 1 , , . . . , p + 1 (resp. i = 0 , , . . . , p ),where the t i are strictly increasing. Set J := 1 , , . . . , [( p + 1) / A = { ϕ ( t j − ) : j ∈ J } .For all k = 0 , , . . . , p , set ǫ i = ( − k − , for all i = t k , t k + 1 , . . . , t k +1 −
1. That is ǫ i = 1(resp. ǫ i = −
1) in the interval in which an element of A (resp. B ) is followed by an elementof B (resp. A ). Corollary. B ∪ B ′ = { ǫ i β i : i ∈ I } . In particular the ǫ i β i : i ∈ I which lie in B (resp. B ′ ), are positive (resp. negative) roots. i ∈ I such that β i ∈ ± π . We call this theexceptional index e and β e the exceptional value.From the corollary we see that B ∪ B ′ cannot be a simple root system because successivescalar products acquire the wrong sign as the (internal) turning points are crossed. Ouraim is to change the ǫ i β i to new elements β ∗ i , so thata) Π ∗ := { β ∗ i } i ∈ I , is a simple root system,b) Suppose β ∗ i = ǫ i β i . Then expressed as a sum of elements of π , the ”non-compact”root α p appears in β ∗ i with a negative coefficient (hence with coefficient − β ∗ e = ǫ e β e . d) ǫ i β i ∈ N Π ∗ , for all i ∈ I \ { e } . Equivalently the ǫ i β i : i ∈ I \ { e } are positive rootswith respect to Π ∗ .The meaning of these conditions is as follows. Set y ′ = X i ∈ I x β ∗ i . Condition a) means that y ′ is a regular nilpotent element of sl ( n ) and hence can beconjugated by an element w of the Weyl group W = S n to a standard nilpotent element y := P α ∈ π x α . Condition d) means that the x ǫ i β i , for i non-exceptional, either alreadyoccur in y ′ or can be added to y ′ as commutators of the x β ∗ i : i ∈ I . In particularby Lemma 1.9, the new element y ′′ obtained by adding these commutators, namely the { x ǫ i β i : ǫ i β i = β ∗ i } i ∈ I \{ e } , is again regular nilpotent. Finally b) and c) imply that y ′′ restricted to p coincides with η as defined in 1.8. Let B denote the Borel subgroup of G defined with respect to π .Suppose conditions a)-d) are satisfied. We conclude by the above, Lemma 1.9 andCorollary 2.5 that there exists w ∈ W and b ∈ B such that the restriction of y := n w by is η . Moreover we shall give a (fairly) explicit expression for Π ∗ and this determines w .Since P contains the opposed Borel subgroup B − rather than B one should consider y defined above as the negative element of a principal s-triple. In the present work this hasno particular significance.The construction of Π ∗ and the proof that it satisfies a)-d) above is given in the nextsection. The proof is illustrated by Figures 1-7. It should also be possible for the readerto reconstruct the analysis from just these figures. Remarks . One could imagine that a simpler way to satisfy these conditions mightbe possible by the following approach. Recall that the ǫ i β i : i ∈ I form a basis for h ∗ .Thus we can choose c i ∈ N + : i ∈ I such that there is a unique element h ∈ h satisfying h ( ǫ i β i ) = c i , ∀ i ∈ I , and that this element is regular. Then ∆ ∗ + := { α ∈ ∆ | h ( α ) > } isa choice of positive roots for ∆ and so defines a set Π ∗ of simple roots in which d) willbe satisfied by construction and even in the overly strong form that ǫ i β i ∈ N Π ∗ , for all i ∈ I . It is not so obvious if and how we can choose the c i : i ∈ I , to ensure that b)is satisfied. Again c) will not be satisfied in general; but in our approach we modify oursolution weakening this overly strong form of d) to recover condition c). A postiori onemay recover a good choice of the c i : i ∈ I by setting h ( α ) = 1 , ∀ α ∈ Π ∗ .2.7. Relative to π , the roots in B ′ have a zero coefficient of α p . Thus by Corollary 2.5, allelements of { ǫ i β i } i ∈ I have a non-negative coefficient of α p , which is hence in { , } . Thiscoefficient is non-zero only if β i ∈ ± B . By our conventions (see 2.2) β i ∈ ± B , if and onlyif i is odd. Thus β i has a non-zero coefficient of α p only if i is odd. In particular neighbours β i , β i +1 cannot both have a non-zero coefficient of α p .Fix t ∈ I . Call t a nil point if the coefficient of α p in β t is non-zero and a boundarypoint if ϕ ( t ) or ϕ ( t + 1) is a turning point. Call β t a boundary (resp. nil) value if t is aboundary (resp. nil) point. (By Corollary 2.5 and our convention in 1.6 it follows that β t is a nil value if and only if x ǫ t β t belongs to the nilradical of p − .) LICES 13
The unique boundary value to an end point is called an end value.
Lemma. (i) Suppose ϕ ( t ) is an internal turning point, then t and t − cannot be both nil points.(ii) Suppose t ∈ I is a nil boundary point with ϕ ( t ) ∈ B (resp. ϕ ( t + 1) ∈ B ). Then ϕ ( t + 1) ∈ A (resp. ϕ ( t ) ∈ A ).(iii) Suppose ϕ ( t ) ∈ A . Then t − or t must be a nil point, in particular must be a nilpoint if p is odd.Proof. (i) follows from the remarks in the first paragraph above.(ii) Since t is a nil point, we must have σ ( ϕ ( t )) = ϕ ( t + 1). Now suppose i := ϕ ( t ) ∈ B .Then i ≥ [ n/
2] + 1. Thus t is a nil point if and only if σ ( i ) ≤ p . Again i ≤ p + [( q + 1) / σ ( i ) = n + 1 − i ≥ [ q/
2] + 1 ≥ [ p/
2] + 1. Consequently ϕ ( t + 1) = σ ( i ) ∈ A , as required.The proof of the second case is exactly the same.(iii) Since i := ϕ ( t ) ∈ A , we have i ≤ p and so σ ( i ) ≥ p + 1. Thus either β t or β t − musthave a non-zero coefficient of α p . Hence (iii). (cid:3) Remarks . Since β i = ε ϕ ( i ) − ε ϕ ( i +1) we may regard β i as lying between the elements ϕ ( i ) , ϕ ( i + 1) of ϕ ( ˆ I ). We say that β i − , β i are the neighbours of ϕ ( i ) : i ∈ ˆ I in ± ( B ∪ B ′ )and that ϕ ( i ) , ϕ ( i + 1) : i ∈ I are the neighbours of β i ∈ ± ( B ∪ B ′ ). Then (iii) of thelemma can be expressed as saying that every element of A has exactly one nil boundaryvalue neighbour, whereas (ii) of the lemma can be expressed as saying that if a nil boundaryvalue has an element of B as a neighbour, then it is sandwiched between an element of A and an element of B . By (ii) and (iii) of the lemma every nil boundary value has a uniqueelement of A as a neighbour. Finally an end value is non-nil only if it is the (unique)neighbour of an element of B . For example if p = 2 , q = 5, both end-points are non-nil.However an end value can be nil even if it is a neighbour of an element of B . For exampleif p = 2 , q = 3, the starting value is nil.2.8. Intervals.
Let ϕ ( s ) , ϕ ( t ) ∈ T be turning points with s < t . The subset I s,t := { s, s + 1 , . . . , t − } is called an interval. If ϕ ( t ) is the immediate successor to ϕ ( s ) in T , itis called a simple interval. Otherwise it is called a compound interval.The sum ι s,t := X i ∈ I s,t β i , is called a simple (resp. compound) interval value if I s,t is simple (resp. compound). Theset { β i : i ∈ I s,t } is called the support of ι s,t or of I s,t . Lemma.
Let I r,s be a simple interval. There is exactly one i ∈ I r,s such that β i has anon-zero coefficient of α p .Proof. Observe that ι r,s = ε ϕ ( r ) − ε ϕ ( s ) . ( ∗ )Since ϕ ( r ) ∈ A and ϕ ( s ) ∈ B or vice-versa, it follows from 2.4 that the coefficient of α p in the above sum equals one or minus one. Moreover there can be no cancellations of coefficients of α p in the sum because only alternate terms can have a non-zero coefficientand these are either all positive roots or all negative roots since the indices lie betweensuccessive turning points. Hence the assertion. (cid:3) The Sign of Simple Interval Values.
Take ϕ ( s ) ∈ T and let ϕ ( t ) be its immediatesuccessive in T . The sign of the simple interval value ι s,t is said to be positive (resp.negative) if ϕ ( s ) ∈ A (resp. ϕ ( s ) ∈ B ).A positive (resp. negative) interval value is a positive (resp. negative) root relative to π with the coefficient of α p being 1 (resp. − ǫ i , Corollary 2.5 andLemma 2.8.2.10. The Exceptional Value.
Recall the exceptional value β e : e ∈ I defined in 2.6. Itdefines a unique simple root α ∈ π and one has β e = α , up to a sign . Lemma.
The exceptional value β e is never a nil value, equivalently α = α p . It is aboundary value to some unique turning point, which is either(i) internal,or(ii) an end point lying in B .Proof. Since p, q are coprime, exactly one of the integers p, p + q, n = p + q , call it m ,is even. Then β e = ± α m/ . For this to be a nil value we would need α m/ = α p , that is m/ p , which is impossible since p < n/
2. This proves the first assertion.One checks from the description of the turning points in 2.4 that either m/ m/ β e would be nil by2.8, contradicting the first part. On the other hand by Lemma 2.7(iii) an end-point lyingin A is nil. Hence the second assertion. (cid:3) Remark . On may check from 2.2, that (ii) holds if and only if p = 1.2.11. Isolated values.
We call t ∈ I an isolated point if both ϕ ( t ) and ϕ ( t +1) are turningpoints. By Lemma 2.8 an isolated point is necessarily nil. If t is an isolated point we call β t an isolated value. 3. The Description of Π ∗ ∗ by changing some of the ǫ i β i . Here it is convenientto write β ∗ i = ǫ i β ′ i and to say that a value is changed if β ′ i = β i . Let us describe thosevalues that are changed. Here we impose three general rules. The first two are1) Change only boundary values and change only those which are non-nil.2) Change exactly one of the boundary values at each internal turning point. Remark . In the initial stage (up to 3.8) end values will not be changed. However if anend value is exceptional (and hence non-nil and so the unique neighbour of an element of B ), then it will be changed in the final stage (3.9). LICES 15 i, j ∈ ˆ I , with i < j , set β i,j = ε ϕ ( i ) − ε ϕ ( j ) , which we recall is a root (and positivewith respect to Π). In this notation β i = β i,i +1 . Again if ϕ ( i ) , ϕ ( j ) ∈ T , we have β i,j = ι i,j .Now let β i be a boundary value which is to be changed (according to rules 1) and 2)- in particular β i is a non-nil boundary value). Then there is an internal turning point,say ϕ ( t s ) which either equals ϕ ( i + 1) or ϕ ( i ). (Both possibilities cannot simultaneouslyarise since otherwise by Lemma 2.8, β i would be a nil boundary value.) In the first (resp.second) case we shall say that β i is above (resp. below) ϕ ( t s ).In the first case we replace β i by β ′ i defined by adding to β i “a compound interval valuewhich is an odd sum of simple interval values below ϕ ( t s )”, that is to say we set β ′ i = β i + β t s ,t r = β t s − ,t r : r − s ∈ N + 1 . ( ∗ )In the second case we replace β i by β ′ i defined by adding to β i “a compound intervalvalue which is an odd sum of simple interval values above ϕ ( t s )”, that is to say we set β ′ i = β i + β t r ,t s = β t r ,t s +1 : s − r ∈ N + 1 . ( ∗∗ )We may summarize the above by saying that in both cases the added interval value ison the opposite side of the turning point to the element in question and is a sum of an oddnumber of simple interval values.Finally (recall) that we set β ∗ i = ǫ i β ′ i . Lemma.
Suppose β ′ i = β i . Then ǫ i β ′ i is a root. Moreover expressed as a sum of elementsof π , the non-compact root α p has coefficient − in ǫ i β ′ i .Proof. Since β i is a non-nil boundary value the coefficient of α p in it is zero.By definition ǫ i changes sign as each turning point is crossed. Thus if ǫ i = 1 (resp. ǫ i = − β i is negative (resp. positive).Then by the second paragraph of 2.9 the coefficients of α p of the successive simple intervalvalues added to ǫ i β i are {− , , − , . . . } , whereas by construction the number of such simpleintervals is odd. (cid:3) Signature.
To complete our description of Π ∗ we must now specify which boundaryvalues are to be changed (which will specify t s and how t r in equations ( ∗ ) and ( ∗∗ ) isdetermined).The above data will be completely determined by the signature of the orbit O definedas follows.Recall that by the choices made in 2.5 and by Lemma 2.7, at each turning point ϕ ( t j − ) : j ∈ J , which we recall lies in A , one has that either β t j − − or β t j − is nil (but not both).In the first case we set sg( j ) = − j ) = 1 (to specify that the nil boundaryvalue is below the turning point).If sg(1) = 1 (which is always the case if p is odd) then there is a unique increasingsequence j , j , . . . , j r ∈ J with j = 1, such thatsg( i ) = ( − u − , ∀ i = j u , j u + 1 , . . . , j u +1 − , ∀ u = 1 , , . . . , r − , sg( j r ) = ( − r − . If sg(1) = − p is even) then there is a unique increasingsequence j , j , . . . , j r ∈ J with j = 1, such thatsg( i ) = ( − u , ∀ i = j u , j u + 1 , . . . , j u +1 − , ∀ u = 1 , , . . . , r − , sg( j r ) = ( − r . We say that the signature at the turning point ϕ ( t j − ) ∈ A is positive (resp. negative)if sg( j ) = 1 (resp. sg( j ) = − O is defined to be the set { sg ( i ) } [ p/ i =1 . In the notation of 1.10,it lies in S and defines a map of the set of coprime pairs P into S .3.4. We assume until the end of 3.9, that the signature at the first turning point in A ispositive. This is always the case if p is odd by virtue of Lemma 2.7(iii). Then the easiestcase to describe is when there are no signature changes. This is illustrated in Figure 1,where the given pattern is repeated as many times as there are turning points in A . c ϕ ( t j − ) Aϕ ( t j ) Aϕ ( t j +1 ) B ❢❢ c rrrrrrrrr Figure 1.This shows the basic repeating pattern for positive signature. Turning points are labelledby their type, that is A or B , and nil values are encircled. The dots on the vertical centralline label a subset of ˆ I = { , , . . . , n } . In the language of 3.4, the turning point ϕ ( t j − ) ∈ A has positive signature. In the terminology of 3.6, the thickened lines describe the linksbetween the elements of β ∗ i : i ∈ I and define a sub-chain linking β ∗ t j − − to β ∗ t j . Thenon-nil boundary values that are changed carry the symbol c which is given the subscript or depending on whether rule 1) or 2) of 3.1 is applied as described in 3.5. The map χ defined in 3.5 takes ϕ ( t j − ) to ϕ ( t j ) . A mirror reflection perpendicular to the verticalaxis gives the basic repeating pattern for negative signature. LICES 17
The next easiest case is when there is one signature change, namely from positive tonegative. After that there is the case when there are two signature changes, namely frompositive to negative to positive. From then on it is simply a repetition of the procedurefor two signature changes. The first two cases can be considered as a degeneration of thethird by simply eliminating terms. Thus it will suffice to describe the case of two signaturechanges, to make precise what is meant by degeneration and to describe the modificationneeded if the signature of O is initially negative which can happen if p is even.Fix a positive odd integer u ≤ r and set j = j u , k = j u +1 , ℓ = j u +2 . The first and secondcases above correspond to k not being defined and k being defined but ℓ not being defined.Assume k is defined. Then by definition ϕ ( t k − − ) ∈ A and admits a nil boundaryvalue just below, namely β t : t = t k − − . If ℓ is not defined let ϕ ( s ) be the last turningpoint or (end point) ϕ ( n ), otherwise set s = t ℓ − . In both cases ϕ ( s ) ∈ B .If t >
1, set β ′ t − = β t − + β t,s = β t − ,s . ( ∗ )Observe that β t,s is defined even when t = 1 and is a compound interval value ι t,s whichis a sum of 2 m + 1 : m = ℓ − k adjacent simple interval values ι , ι , . . . , ι m +1 , starting at ι = ι t k − ,t k − .Suppose ℓ is defined (and hence so is β s ). If ι is not reduced to an isolated value, set β ′ s = β s + β t,s = β t,s +1 . ( ∗∗ )Otherwise set β ′ s = β s + m +1 X i =3 ι i = β t k − ,s +1 . ( ∗ ∗ ∗ )3.5. In the remaining cases added interval values will be simple and we only have to specifythe non-nil boundary values which are changed. We need only describe these between theturning points t j − and t ℓ − in the notation of 3.4 since the pattern just repeats itself.For this there is a simple algorithm.Notice first that if a simple interval value ι is to be added to a non-nil boundary value β i , then this simple interval value is uniquely determined by β i itself via the rule in 3.2.We define a map χ from the set of internal turning points of A to the set of turningpoints of B .At every internal turning point ϕ ( t ) ∈ A , (so then t = t v − for some v ∈ J ) thereis exactly one non-nil boundary value β u and by rule 1) of 3.1, it must be changed, thatis β ′ u = β u . By the rule described in 3.2 there is a unique turning point ϕ ( t ′ ) ∈ B sothat β ′ u = β u + ι t,t ′ , where we have defined ι t,t ′ := ι t ′ ,t if t ′ < t . We set t ′ = χ ( t ) and χ ( ϕ ( t )) = ϕ ( χ ( t )) = ϕ ( t ). Notice that t ′ > t (resp. t < t ′ ) if the signature of ϕ ( t ) ∈ A is positive (resp. negative) and we say that ϕ ( t ′ ) ∈ B is a subsequent (resp. previous)turning point to ϕ ( t ) ∈ A .Observe further that there is a unique boundary value β w to ϕ ( t ′ ) ∈ B such that thescalar product ( β ′ u , β w ) is strictly positive. With one possible exception (within a double signature change) described below, we set β ′ w = β w , that is this particular boundary valueis left unchanged. Then if b ′ := ϕ ( t ′ ) is an internal turning point, its second boundaryvalue β w ′ (and for which ( β ′ u , β w ′ ) is strictly negative) should be changed by rule 2) of3.1 unless it is a nil boundary value . Let us show that the latter can occur at most once(within a double signature change). It results from ι (as defined in 3.4) being reduced toan isolated value. In this case we shall compute w ′ explicitly.Suppose that β w ′ is a nil boundary value. Then by Lemma 2.7(ii) it is an isolated value.Suppose its second neighbour a ′ ∈ A lies above b ′ , so then w ′ = t ′ − , w = t ′ . Thismeans that the signature at a ′ = ϕ ( w ′ ) is positive. Then t ′ = χ ( t ) implies that ϕ ( t ) ∈ A is a subsequent turning point to b ′ with negative signature. By definition of k this forces t = t k − . Consequently t ′ = t k − , and then β w ′ = ι , by definition of the latter. Inparticular ι is reduced to an isolated value. Then we set β ′ w ′ = β w ′ and β ′ w = β w + ι .This is the only case (within a double signature change) that we leave unchanged theunique neighbour β w ′ of ϕ ( t ′ ) for which ( β ′ u , β w ′ ) is strictly negative.A similar argument to the above shows that a ′ cannot lie below b ′ . Indeed this wouldimply that the signature of a ′ is negative, whilst t ′ = χ ( t ) implies that ϕ ( t ) ∈ A is aprevious turning point to b ′ with positive signature. However the construction of 3.4 hasthe property that if ϕ ( t ) ∈ A has positive signature then the immediate subsequent turningpoint in A to ϕ ( χ ( t )), which is a ′ in the present application, has positive signature, so unlikethe previous case we obtain to a contradiction.Figure 2 compares the cases when ι is not and is reduced to an isolated value. From itone may see why 3.4( ∗∗ ) has been replaced by 3.4( ∗ ∗ ∗ ). c , c c c , c , c c c , AA AAϕ ( t k − ) ϕ ( t k − ) ϕ ( t ℓ − ) ϕ ( t k − ) ϕ ( t k − ) ϕ ( t ℓ − ) B BBB ❤❤❤ ❤❤❤rrrrrrrrrrrr rrrrrrrrrrrr
LICES 19
Figure 2.In the notation of 3.4 this compares the cases when ι is not isolated (on the left) and ι isisolated (on the right). The same conventions as in Figure 1 apply, where in addition theadditional subscript to c , c refers to the application of the rule described in 3.4. Theparticular case here corresponds to taking ℓ = k + 1 . For the general case one must extendthe two outermost lines in each diagram downwards and insert a further ℓ − ( k + 1) copiesof the basic repeating pattern for negative signature. This is illustrated by Figure 3 in whichthe case ℓ = k + 2 is considered. We remark that one may have an isolated point in a region of negative signature. Thisis illustrated in Figure 4.One checks from 3.4, 3.5 that the map χ defined above is an injection from the set ofinternal turning points lying in A to the set of turning points in B . In all cases the latterset has cardinality one greater than the former. Thus the cokernel of χ is a singleton whichwe call the undecided element d ∈ B . It is clear that the above algorithm just leaves atmost one boundary value of d undecided. The exact location of d depends on the signatureof O as we now explain.Suppose p is odd and recall ϕ ( t ) has positive signature. If ϕ ( t ) is not defined, then d = ϕ ( t ). It is a finishing point in B and we leave its unique neighbour unchanged. If ϕ ( t ) is defined and has positive signature, then d = ϕ ( t ) and we change β t by adding asimple interval value, namely ι t ,t .Finally suppose that ϕ ( t ) has a negative signature. This corresponds to having k = 2in 3.4. Then in the notation of 3.4 one has d = ϕ ( s ). If d is a finishing point we leave itsunique neighbour unchanged. Otherwise we change β s by the rules described in 3.4( ∗ ∗ ∗ )or 3.4( ∗∗ ), depending on whether ι is reduced to an isolated value or not. One may remark that when ϕ ( t ) is defined the solutions given in the above two para-graphs would result if we were to treat ϕ ( t ) as if it were an internal turning point. Suppose p is even. If ϕ ( t ) has positive signature, Then d = ϕ (1) and we leave its uniqueneighbour unchanged. The case when ϕ ( t ) has negative signature will be postponed to3.10.Note that if ϕ ( t ′ ) is an end-point, namely t ′ = 1 (resp. t ′ = n ), its unique boundaryvalue, that is β (resp. β n − ) is left unchanged by the above procedure. However this willbe modified in 3.9.This (nearly !) completes our description of Π ∗ . What can happen however is thatcondition c) of 2.6 can sometimes fail and this will need a further modification to bedescribed in 3.9.3.6. Define β ′ i : i ∈ I through the rules described in 3.1 - 3.5, set β ∗ i = ǫ i β ′ i and Π ∗ = { β ∗ i } i ∈ I .Here we show that Π ∗ satisfies condition a) of 2.6 by exhibiting an ordering so thatnearest neighbours have a strictly negative scalar product (which we call a link) and that noother non-zero scalar products exists between distinct elements. We call such a successionof links, a sub-chain. In this we shall assume that k and ℓ of 3.4 are defined, otherwise one just obtains adegeneration of that case.Retain the notation of 3.4. Our construction gives a link between β ∗ t j − − (if it isdefined) and β ∗ t j − which is connected via a sub-chain to the elements in the support of I t j − ,t j taken in the reverse order. Furthermore the last element in this chain, namely β ∗ t j − is linked to β ∗ t j which is connected via a sub-chain of elements in the support of I t j ,t j +1 taken in their natural order to β ∗ t j +1 − , by repeating the pattern in Figure 1 theappropriate number of times. This process is repeated till one reaches β ∗ t k − − which liesjust above the last turning point in A with positive signature. Thus β ∗ t j − − is connectedvia a sub-chain to β ∗ t k − − .A similar (reversed) phenomenon occurs in a region of negative signature. In particular β ∗ t k − is connected via a sub-chain to β ∗ t ℓ − − , which is in turn linked via 3.4( ∗ ) to β ∗ t k − − .This can be illustrated by simply making a mirror reflection of Figure 1 perpendicular toits main axis (in simple language turning it upside down).If ι is not reduced to an isolated value, then β ∗ t k − is connected via a sub-chain to β ∗ t k − which is in turn linked to β ∗ t ℓ − via 3.4( ∗∗ ), the latter being connected by a sub-chain to β ∗ t ℓ − − .Thus using a line to designate a link or a sub-chain we may summarize the above as β ∗ t j − − − β ∗ t k − − − β ∗ t ℓ − − − β ∗ t k − − β ∗ t k − − β ∗ t ℓ − − β ∗ t ℓ − − . ( ∗ )Except for the two extreme terms the links or sub-chains between these elements areillustrated in the left hand side of Figure 3. LICES 21 β ∗ t k − − β ∗ t k − − β ∗ t k − β ∗ t k − β ∗ t k − β ∗ t ℓ − − β ∗ t ℓ − β ∗ t ℓ − − β ∗ t ℓ − β ∗ t k − β ∗ t k − β ∗ t k − − AA AA AAϕ ( t k − ) ϕ ( t k − ) ϕ ( t ℓ − ) ϕ ( t k − ) ϕ ( t k − ) ϕ ( t ℓ − ) B BBBB B ❤❤❤❤❤ ❤❤❤❤❤rrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrr
Figure 3.The left (resp.) right figure illustrates the links or sub-chains in ( ∗ ) , ( ∗∗ ) ) (resp. ( ∗ ) , ( ∗ ∗∗ ) ) of 3.4. The conventions of Figures 1,2 apply. Compared to Figure 2 one has ℓ = k + 2 . If ι is reduced to an isolated value, one checks (taking account of the alternating signsof the ǫ i ) that β ∗ t ℓ − is linked to β ∗ t k − − via 3.4( ∗ ∗ ∗ ). Moreover the latter is connectedby a sub-chain to β ∗ t k − which is linked to β ∗ t k − in turn linked to β ∗ t k − . (See Figure 3).In this we remark that β ′ t k − = β t k − + ι and β ′ t k − = β t k − + ι .As before we may summarize the above as β ∗ t j − − − β ∗ t k − − − β ∗ t ℓ − − − β ∗ t k − − β ∗ t k − − β ∗ t k − − β ∗ t k − − − β ∗ t ℓ − − β ∗ t ℓ − − . ( ∗∗ ) Except for the two extreme terms the links or sub-chains between these elements areillustrated in the right hand side of Figure 3.In both cases β ∗ t j − − is connected via a sub-chain to β ∗ t ℓ − − and the process is thenrepeated. Notice that in both cases the sub-chain passes through all the β ∗ of the left handor right hand side of the figure.We may summarize the above by the Lemma.
Condition a) of 2.6 is satisfied by Π ∗ . Remark . This result can be read off more easily though less rigorously from Figure 4which is a paradigm for the general case (except when ι is an isolated value; but then onecombines it with Figure 3).3.7. We define a partial order ≤ on I as follows. The smallest elements are those i ∈ I for which β i is unchanged. If ι as defined in 3.4 is reduced to an isolated value say β j − ,then we set β j > β j − recalling that β j − is left unchanged and β ′ j = β j + β j − .If m = 1 in 3.4( ∗ ∗ ∗ ) , then β s occurring there is changed by a simple interval valuenamely ι . However for the present purposes it is convenient to view this as a compoundinterval value. Then the largest elements of i ∈ I are just those for which β i is changed byadding a compound interval value, namely when i = t − , s in the notation of 3.4, withinthat double signature change. Continuing with this convention we may easily observe thatif β ′ j = β j + ι is a simple interval value, then the β i in the support of ι are unchanged (seeFigure 2, for example). Thus if we let j ∈ I for which β j is changed by a simple intervalvalue, to be the second largest elements of I, it follows that ≤ is well-defined and lifts to atotal order (which we also denote by ≤ ) giving the Lemma.
With respect to ≤ the transformation taking the ǫ i β i to β ∗ i is triangular with oneson the diagonal. Proposition. Π ∗ satisfies conditions a), b) and d) of 2.6.Proof. Condition a) is just Lemma 3.6. Condition b) is verified by Lemma 2.8 and the factthat the added interval values are always a sum of an odd number of sequentially adjacentsimple interval values.For condition d) we remark that the ǫ i β i : i ∈ I are roots and therefore are either positiveor negative roots with respect to Π ∗ . Yet they must be positive roots by Lemma 3.7. (cid:3) LICES 23 c c c , c c c c c , c , c c ϕ ( t j − ) ϕ ( t k − ) ϕ ( t k − ) ϕ ( t ℓ − ) ϕ ( t ℓ − ) AAAAAAAA BBBBBB ❢❢❢❢❢❢❢❢❢❢❢ rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr
Figure 4.A diagrammatic presentation of the proof of Lemma 3.6, though of course one couldincrease the size of the regions of positive and negative signature. The conventions arethose of Figures 1,2. The diagram on the left describes what happens if the last elementof B is an end point. Again if ϕ ( t ℓ − ) is replaced by ϕ ( n ) = b , that is to say ℓ is notdefined, all lines which lie partly or completely below the end point b are removed. If thefirst element of A is an end point then one omits the dashed lines starting just above thefirst element of B . In this case the lines start at the undecided point, namely ϕ ( t ) . β e is an unchangedboundary value (that is β ′ e = β e ) of a unique turning point ϕ ( t ) ∈ B which in particulardoes not have an isolated value as a neighbour. This may be either an internal turningpoint or an end point. The latter case is essentially a degeneration of the former which weconsider first.The procedure to obtain Π ∗ must be modified in the above situation. To be transparentwe first recall some features of the description of Π ∗ and then define the modified simpleroot system which we shall denote by Π ∗∗ . Here we recall that Π = { β i } i ∈ I is a simple rootsystem of type A n − . In this β i − , β i +1 will be said to be the neighbours of β i .In what follows ι , ι are interval values but not necessarily those defined in 3.4.Let β f denote the second boundary value of the internal turning point ϕ ( t ) definedabove. By our assumption and rule 2) of 3.1, this boundary value must be changed, thatis to say we have β ′ f = β f + ι , where ι is an interval value. Moreover this interval value starts at β e and so ι − β e isa root. Since β f is a non-nil boundary value, Lemma 2.8 forces it to admit a neighbour β f ′ in the simple interval containing β f . Moreover again by Lemma 2.8 either β f ′ is not aboundary value or it is nil and so by rules 1), 2) of 3.1, it is unchanged. That is β ′ f ′ = β f ′ . From the definition of ι given in 3.4, 3.5, it follows that β f ′ + β f + ι is root.The fact that β ′ e = β e , means that either there is a unique i ( e ) ∈ I , or simply i , such that β ′ i = β i + ι , for some interval value ι , with β ′ i − β e being a root, or β ∗ e is at the end point of the Dynkindiagram for Π ∗ and in this case we say that i ( e ) is not defined.Since β e is a non-nil boundary value, Lemma 2.8 forces it to admit a neighbour β e ′ inthe simple interval containing β e . The same argument for β f ′ given above shows that β ′ e ′ = β e ′ . From the definition of ι given in 3.4, 3.5, it follows that ι − β e and ( ι − β e ) − β e ′ areroots. LICES 25
Since β i is a non-nil boundary value to an element of A , it admits by Lemma 2.8, a uniqueneighbour β i ′ in the same interval. Moreover β i ′ cannot be a nil boundary value (becausethen it would be a boundary value to an element of B and this would contradict Lemma2.7(ii)) and it cannot be a non-nil boundary value either (because this would contradictLemma 2.8). Hence it is not a boundary value and so is unchanged by rule 1) of 3.1, thatis β ′ i ′ = β i ′ . In particular the fact that β i and β i ′ are neighbours implies that β ′ i and β ′ i ′ are neighboursand we designate this as β ′ i − β ′ i ′ , with the sign being that of the scalar product. (For thestarred quantities β ∗ i = ǫ i β i , one recalls that all scalar products have non-positive signs.)As in 3.6, one checks that β e ′ = β ′ e ′ is linked via ι through a chain defined by non-vanishing scalar products of neighbours to β ′ f = β f + ι . We write this as β ′ e ′ − . . . + β ′ f ,with the signs having the same meaning as before. Thus (previous to our proposed modi-fication) we obtain the chain β ′ i ′ − β ′ i + β ′ e − β ′ e ′ − . . . + β ′ f − β ′ f ′ . ( ∗ )Now we make the following modification (using a double prime to make the distinctionclear and writing β ∗∗ i := ǫ i β ′′ i , with Π ∗∗ = { β ∗∗ i } i ∈ I ). Here exactly three double primedelements are distinct from the single primed elements and only these are described below.Set β ′′ i = β i + ι − β e , β ′′ e = − ι , β ′′ f = β f . Using the above observations and in particular the linking role of ι , we obtain the chain β ′′ i ′ − β ′′ i + β ′′ e ′ − . . . − β ′′ e + β ′′ f − β ′′ f ′ . ( ∗∗ )If i ( e ) is not defined then the first two terms in both ( ∗ ) and ( ∗∗ ) are absent.The transition between ( ∗ ) and ( ∗∗ ) above is illustrated in the passage of the left to theright hand side of Figure 5 (resp. Figure 6) when ι = ι is a simple (resp. compound)interval value. Observe how the long link between β ′ e ′ and β ′ f , propagated in the firstinstance through ι , becomes transformed to a long link between β ′′ e ′ and β ′′ e similarlypropagated in the first instance through ι . β i ′ β i ( e ) β e ′ β e β f β f ′ c c c ′ c ′ A AB B ❤ ❤rrrrrrrr rrrrrrrr
Figure 5.Passing from left to right illustrates the transition from ( ∗ ) to ( ∗∗ ) in 3.9, when ι = ι and is a simple interval value. The conventions of Figure 1 apply. The new changed valuesaccording to the discussion following ( ∗ ) of 3.9 are indicated by a prime. LICES 27 β i ′ β i ( e ) β e ′ β e β f β f ′ c , c , c ′ , c ′ , c c c c A AA AB BB B ❤ ❤❤ ❤❤ ❤rrrrrrrrrrrr rrrrrrrrrrrr
Figure .Passing from left to right illustrates the transition from ( ∗ ) to ( ∗∗ ) in 3.9, when ι = ι and is a compound interval value. The conventions of Figure 1 apply. The new changedvalues according to the discussion following ( ∗ ) of 3.9 are indicated by a prime. Finally let us suppose that ϕ ( t ) is an end point lying (necessarily) in B . Let ϕ ( s ) be theclosest turning point to ϕ ( t ). By Lemma 2.4 we have ϕ ( s ) ∈ A . Denote the interval value ι t,s simply by ι .Suppose that the unique nil boundary value to the turning point ϕ ( s ) lies in the supportof I t,s (rather than in the support of the adjacent interval). If ϕ ( s ) admits a secondboundary value say β i , then this is changed to β ′ i := β i + ι , noting here that i = i ( e ).Moreover we obtain the chain described in ( ∗ ) except that the two terms on the right handside are not defined. Then parallel to the above we write β ′′ i = β i + ι − β e , β ′′ e = − ι. This gives the chain described in ( ∗∗ ) except that the two terms on the right hand side arenot defined.If either the unique nil boundary value to ϕ ( s ) does not lie in the support of I t,s (that isto say that it lies in the support of the adjacent interval or ϕ ( s ) is also an end point) then i ( e ) is not defined and we obtain ( ∗ ) with just the two central terms. Then we set β ′′ e = − ι and we obtain ( ∗∗ ) with just the two central terms. Theorem.
In the above construction of Π ∗∗ , conditions a)-d), of 2.6 hold.Proof. Retain the notation of 3.9.Comparison of ( ∗ ) and ( ∗∗ ) shows that condition a) for Π ∗ (verified in Proposition 3.8)implies condition a) for Π ∗∗ .To show that condition d) holds observe that we still have ǫ i β i = β ∗ i + P j ∈ I N β ∗ j , for all i ∈ I \ { e } and so these elements are positive roots with respect to Π ∗ .By contrast β ′′ e = − ι = − β e − . . . , and so ǫ e β e does not lie in N Π ∗∗ . However conditiond) does not require this.Thus condition d) holds for Π ∗∗ .Observe that ι expressed as an element of π admits a positive (resp. negative) coefficientof α p if β e lies above (resp. below) ϕ ( t ) ∈ B and that ǫ e = 1 (resp. − β ∗∗ e and hence for Π ∗∗ .Finally condition c) holds by construction. (cid:3) p is even, is similar. We sketch the necessary changes below.Recall the notation of 3.3 and in particular that j = 1. Before we had assumed thefirst of the two possibilities in 3.3, namely that sg( j ) = 1, to hold. Now we assume thesecond possibility, namely that sg( j ) = −
1, to hold. Fix a positive odd integer u ≤ r andset j = j u , k = j u +1 , ℓ = j u +2 .When only j is defined or when all three are defined the solution we adopt is just the“mirror image” of that described in 3.4. Indeed reading indices in the opposite direction(more precisely applying the involution i n − i to I it follows that for example a negativeto positive to negative signature change becomes a positive to negative to positive signaturechange. (Here we do not mean to imply a configuration produced by a coprime pair willtransform to a configuration produced by another coprime pair. Indeed our formalismallows for some configurations not necessarily coming from coprime pairs. See 3.11.)It remains to consider the case of a negative to positive signature change, that is whenjust j, k above are defined. Surprisingly this is not quite a degeneration of the case whenall three are defined.We remark that to describe a negative to positive signature change repeated more thanonce (say twice to be specific) then we match a negative to positive to negative signaturechange with a negative to positive signature change (as illustrated by the right hand sideof Figure 7 below).Set t = t k − and let s be the last turning point which occurs in A . One easily checksthat the undecided element d is just ϕ ( t ). We leave β t unchanged (which is just what wewould do if ℓ were defined) and set β ′ t − = β t − + ι t,s . The result is illustrated in Figure 7.In view of this last change we have labelled the changed element β ′ t − by c , in Figure 7. LICES 29 c c , c , c c c c ϕ ( t j − ) ϕ ( t ) ϕ ( t k − ) ϕ ( s ) A AAAAB BBBB ❢ ❢❢❢❢❢❢❢rrrr rrrrrrrrrrrrrrrrrrrr
Figure 7.This illustrates the negative to positive signature change as discussed and in the notationof 3.10, more precisely in the special case when j = k − . The conventions are those ofFigures 1,2. The left hand side describes the situation when t j − is a starting point.The right hand side describes the situation when ϕ ( t k − ) has positive signature but ispreceded by a component of negative signature and this being repeated any number of times.Here the top half of the diagram must and does match the negative to positive to negative configuration which can be obtained by inverting Figure 4. Curiously the bottom half of thediagram does not match the positive to negative to positive configuration from Figure 4,nor indeed need it do so. An end point of Π ∗ in the sense of its Dynkin diagram is β ∗ t . Forthe diagram on the left a second end point is β ∗ t j − − . The modification of this construction when the exceptional value is not changed is exactlyas in 3.9. The extension of Theorem 3.9 to this case is proved similarly.3.11. Although the solution we gave to Conditions a)-d) of 2.6 is unambiguous, one caneasily check that other solutions can exist for certain coprime pairs p, q .The number of coprime pairs grows at most linearly with n whilst the number of possiblesignatures grows exponentially. Thus to obtain all possible signatures one would need tohave increasingly large gaps between turning points. In particular we do not claim thatthe particular arrangements described in Figure 4 actually arise from a coprime pair.3.12. Combining Theorem 3.9 with the remarks in 1.8, 2.6 and 3.10, we obtain the Corollary.