Poisson-commutative subalgebras of S(g) associated with involutions
aa r X i v : . [ m a t h . R T ] S e p September 2, 2018
POISSON-COMMUTATIVE SUBALGEBRAS OF S ( g ) ASSOCIATED WITHINVOLUTIONS
DMITRI I. PANYUSHEV AND OKSANA S. YAKIMOVAA
BSTRACT . The symmetric algebra S ( g ) of a reductive Lie algebra g is equipped with thestandard Poisson structure, i.e., the Lie–Poisson bracket. Poisson-commutative subalge-bras of S ( g ) attract a great deal of attention, because of their relationship to integrablesystems and, more recently, to geometric representation theory. The transcendence degreeof a Poisson-commutative subalgebra C ⊂ S ( g ) is bounded by the “magic number” b ( g ) of g . The “argument shift method” of Mishchenko–Fomenko was basically the only knownsource of C with tr . deg C = b ( g ) . We introduce an essentially different construction relatedto symmetric decompositions g = g ⊕ g . Poisson-commutative subalgebras Z , ˜ Z ⊂ S ( g ) g of the maximal possible transcendence degree are presented. If the Z -contraction g ⋉ g ab has a polynomial ring of symmetric invariants, then ˜ Z is a polynomial maximal Poisson-commutative subalgebra of S ( g ) g , and its free generators are explicitly described. C ONTENTS
Introduction 21. Preliminaries on the coadjoint representation 41.1. The Poisson tensor 51.2. Contractions and compatibility 62. Constructing a Poisson-commutative subalgebra Z
63. The algebra Z is polynomial whenever σ is good 104. The extended algebra ˜ Z is polynomial and maximal Poisson-commutative 145. Fancy identities for Poisson tensors 226. Further developments and possible applications 266.1. Quantum perspectives 266.2. Classical applications 27Appendix A. On pencils of skew-symmetric forms 30References 32 Mathematics Subject Classification.
Key words and phrases.
Poisson-commutative subalgebras, coadjoint representation, symmetric pairs.The research of the first author is partially supported by the R.F.B.R. grant } I NTRODUCTION
The ground field k is algebraically closed and of characteristic . A commutative asso-ciative k -algebra A is a Poisson algebra if there is an additional anticommutative bilinearoperation { , } : A × A → A called a
Poisson bracket such that { a, bc } = { a, b } c + b { a, c } , (the Leibniz rule) { a, { b, c }} + { b, { c, a }} + { c, { a, b }} = 0 (the Jacobi identity)for all a, b, c ∈ A . A subalgebra C ⊂ A is Poisson-commutative if {C , C} = 0 . The Poissoncentre ZA of A is defined by the condition ZA = { z ∈ A | { z, a } = 0 ∀ a ∈ A} .Usually, Poisson algebras occur as algebras of functions on varieties (manifolds), andwe are only interested in the case, where such a variety is an affine n -space A n and hence A = k [ A n ] is a polynomial ring in n variables. Two Poisson brackets on A n are said to be compatible , if all their linear combinations are again Poisson brackets.There is a general method for constructing a “large” Poisson-commutative subalgebraof A associated with a pair of compatible brackets, see e.g. [BB02]. Let { , } ′ and { , } ′′ be compatible Poisson brackets on A n . This yields a two parameter family of Poissonbrackets a { , } ′ + b { , } ′′ , a, b ∈ k . As we are only interested in the corresponding Poissoncentres, it is convenient to organise this, up to scaling, in a 1-parameter family { , } t = { , } ′ + t { , } ′′ , t ∈ P = k ∪ {∞} , where t = ∞ corresponds to the bracket { , } ′′ . The centralrank rkc { , } of a Poisson bracket { , } is defined as the codimension of a symplectic leafin general position, see Definition 1. For almost all t ∈ P , rkc { , } t has one and the same(minimal) value, and we set P reg = { t ∈ P | rkc { , } t is minimal } , P sing = P \ P reg . Let Z t denote the centre of ( A , { , } t ) . The key fact is that the algebra Z generated by {Z t | t ∈ P reg } is Poisson-commutative w.r.t. to any bracket in the family. In many cases, thisconstruction provides a Poisson-commutative subalgebra of A of maximal transcendencedegree. We demonstrate this with a well-known important example. Example 0.1.
For any finite-dimensional Lie algebra q , the dual space q ∗ has a Poissonstructure. Here k [ q ∗ ] ∼ = S ( q ) and the Lie–Poisson bracket { , } LP is defined by { ξ, η } LP =[ ξ, η ] for ξ, η ∈ q . The Poisson centre of ( S ( q ) , { , } LP ) coincides with the ring S ( q ) q ofsymmetric q -invariants. The celebrated “argument shift method”, which goes back toMishchenko–Fomenko [MF78], provides large Poisson-commutative subalgebras of S ( q ) starting from the Poisson centre S ( q ) q . Given γ ∈ q ∗ , the γ -shift of argument produces the Mishchenko–Fomenko subalgebra A γ . Namely, for F ∈ S ( q ) = k [ q ∗ ] , let ∂ γ F be the directionalderivative of F with respect to γ , i.e., ∂ γ F ( x ) = dd t F ( x + tγ ) (cid:12)(cid:12)(cid:12) t =0 . OISSON-COMMUTATIVE SUBALGEBRAS 3
Then A γ is generated by all ∂ kγ F with k > for all F ∈ S ( q ) q . The core of this method isthat for any γ ∈ q ∗ there is the Poisson bracket { , } γ on q ∗ such that { ξ, η } γ = γ ([ ξ, η ]) for ξ, η ∈ q , and that this new bracket is compatible with { , } LP . One can prove that rkc { , } t takes one and the same value for all { , } t = { , } LP + t { , } γ with t ∈ k , i.e., k ⊂ P reg , and A γ is generated by all the corresponding centres Z t , t ∈ k . (Actually, P reg = P if and only if γ is regular in q ∗ .) The importance of these subalgebras and their quantum counterpartsis explained e.g. in [FFR, Vi91]. If q is reductive and γ is regular, then A γ is a maximalPoisson-commutative subalgebra of S ( q ) [PY08].Our main object is a certain 1-parameter family of Poisson brackets on the dual of asemisimple Lie algebra g . Let σ be an involution of g and g = g ⊕ g the corresponding Z - grading (or symmetric decomposition ). We also say that ( g , g ) is a symmetric pair . With-out loss of generality, we may assume that the pair ( g , σ ) is indecomposable , i.e., g has noproper σ -stable ideals. Then either g is simple or g = h ⊕ h , where h is simple and σ is apermutation. Our family of Poisson brackets is related to the decomposition: { , } LP = { , } , + { , } , + { , } , , where { , } i,j = [ , ] i,j : g i × g j → g i + j for i, j ∈ Z ≃ { , } , see Section 2 for details. Usingthis, we consider the -parameter family of Poisson brackets on g ∗ :(0 · { , } t = { , } , + { , } , + t { , } , , where t ∈ P and { , } ∞ = { , } , . Each element of this family is a Poisson bracket andhere P reg = k unless g = sl . For sl , one has P reg = P , and this case has to be consideredseparately. Nevertheless, the final result can be stated uniformly, for all simple g , seebelow.Let Z t ( t ∈ P ) denote the centre of ( S ( g ) , { , } t ) and Z the subalgebra of S ( g ) generatedby all Z t with t ∈ P reg . Then { Z , Z } LP = 0 . Moreover, { g , Z } LP = 0 , i.e., Z is a Poissoncommutative subalgebra of S ( g ) g . By [MY, Prop. 1.1], we have tr . deg C
12 (dim g + rk g + rk g ) for any Poisson-commutative subalgebra C ⊂ S ( g ) g . We prove that this upper bound isattained for Z , see Theorem 2.7.The computation of tr . deg Z is completely general and is valid for any σ . However, thisis not the case with more subtle properties. Our goal is to realise whether Z is polynomialand is maximal Poisson commutative in S ( g ) g . For t = 0 in Eq. (0 · g (0) := g ⋉ g ab . The symmetric invariants of g (0) have intensively been studied in [P07’, Y14, Y17]. The output is that there are four “bad”involutions of a simple g in which S ( g (0) ) g (0) is not polynomial. These cases are related to g of type E n . In all other cases, S ( g ) g has a good generating system (= g.g.s. ) for ( g , g ) , say D. PANYUSHEV AND O. YAKIMOVA H , . . . , H l ( l = rk g ), and a set of free generators of S ( g (0) ) g (0) is then obtained from the H i ’svia a simple procedure, see Section 3 for details.In the rest of the introduction, we assume that σ is “good” and g = sl . In particu-lar, there is a g.g.s. for ( g , g ) . This is of vital importance for us, because we then provethat Z is freely generated by the nonzero bi-homogeneous components of all H i ’s and istherefore polynomial, see Theorems 3.3 and 3.6. Let r : S ( g ) g → S ( g ) g be the restrictionhomomorphism related to the embedding g ∗ ֒ → g ∗ = g ∗ ⊕ g ∗ . Furthermore, • Z is a maximal Poisson commutative subalgebra of S ( g ) g if and only if r is onto,see Theorem 4.5. • In general, let ˜ Z be the subalgebra of S ( g ) generated by Z and S ( g ) g . (Hence ˜ Z = Z if and only if r is onto.) We prove that ˜ Z is still polynomial and that it is a maximal Pois-son commutative subalgebra of S ( g ) g , see Theorem 4.12. This statement also embracesthe sl -case, because then Z = ˜ Z is polynomial, etc.In Section 5, we present a Poisson interpretation of the Kostant regularity criterion for g [K63, Theorem 9] and give new related formulas arising from Z -gradings and compat-ible Poisson structures. As a by-product, we describe Z ∞ for all σ .Section 6.1 contains a discussion of possible quantisations of Z and ˜ Z , i.e., their liftingto the enveloping algebra U ( g ) . We conjecture that quantum analogues of these algebrasmay have applications in representation theory, and more explicitly, in the branchingproblem g ↓ g . In Section 6.2, it is explained how to construct a polynomial maximalPoisson-commutative subalgebra of S ( g ) related to a chain of symmetric subalgebras g = g (0) ⊃ g (1) ⊃ g (2) ⊃ . . . ⊃ g ( m ) with [ g ( m ) , g ( m ) ] = 0 .In the Appendix, we gather auxiliary results on the kernels of a 1-parameter family ofskew-symmetric bilinear forms on a vector space.We refer to [DZ05] for generalities on Poisson varieties, Poisson tensors, symplecticleaves, etc. 1. P RELIMINARIES ON THE COADJOINT REPRESENTATION
Let Q be a connected affine algebraic group with Lie algebra q . The symmetric algebra S ( q ) over k is identified with the graded algebra of polynomial functions on q ∗ , and wealso write k [ q ∗ ] for it.Let q ξ denote the stabiliser in q of ξ ∈ q ∗ . The index of q , ind q , is the minimal codi-mension of Q -orbits in q ∗ . Equivalently, ind q = min ξ ∈ q ∗ dim q ξ . By Rosenlicht’s theo-rem [VP89, 2.3], one also has ind q = tr . deg k ( q ∗ ) Q . The “magic number” associated with q is b ( q ) = (dim q + ind q ) / . Since the coadjoint orbits are even-dimensional, the magic OISSON-COMMUTATIVE SUBALGEBRAS 5 number is an integer. If q is reductive, then ind q = rk q and b ( q ) equals the dimension ofa Borel subalgebra. The Lie–Poisson bracket on k [ q ∗ ] is defined on the elements of degree (i.e., on q ) by { x, y } LP := [ x, y ] . The Poisson centre of S ( q ) is S ( q ) q = { H ∈ S ( q ) | { H, x } LP = 0 ∀ x ∈ q } . As Q is connected, we have S ( q ) q = S ( q ) Q = k [ q ∗ ] Q . The set of Q - regular elements of q ∗ is(1 · q ∗ reg = { η ∈ q ∗ | dim q η = ind q } . Set q ∗ sing = q ∗ \ q ∗ reg . We say that q has the codim– n property if codim q ∗ sing > n . By [K63],the semisimple algebras g have the codim– property.1.1. The Poisson tensor.
Let Ω i be the A -module of differential i -forms on A n . Then Ω = L ni =0 Ω i is the A -algebra of regular differential forms on A n . Likewise, W = L ni =0 W i is the graded skew-symmetric algebra of polyvector fields generated by the A -module W of polynomial vector fields on A n . Both algebras are free A -modules. If A n has aPoisson structure { , } , then π is the corresponding Poisson tensor (bivector) . That is, π ∈ Hom A (Ω , A ) is defined by the equality π ( d f ∧ d g ) = { f, g } for f, g ∈ A . Then π ( x ) , x ∈ A n , defines a skew-symmetric bilinear form on T ∗ x ( A n ) ≃ ( A n ) ∗ . Formally, if v = d x f and u = d x g for f, g ∈ A , then π ( x )( v, u ) = π ( d f ∧ d g )( x ) = { f, g } ( x ) . Definition 1.
The central rank of a Poisson bracket { , } on A n , denoted rkc { , } , is theminimal codimension of the symplectic leaves in A n .It is easily seen that if π is the corresponding Poisson tensor, then rkc { , } = min x ∈ A n dim ker π ( x ) = n − max x ∈ A n rk π ( x ) . Example.
For a Lie algebra q and the dual space q ∗ equipped with the Lie–Poissonbracket { , } LP , the symplectic leaves are the coadjoint Q -orbits. Hence rkc { , } LP = ind q .In view of the duality between differential 1-forms and vector fields, we may regard π as an element of W . Let [[ , ]] : W i × W j → W i + j − be the Schouten bracket. The Jacobiidentity for π is equivalent to that [[ π, π ]] = 0 , see e.g. [DZ05, Chapter 1.8]. Lemma 1.1.
Two Poisson brackets { , } ′ and { , } ′′ are compatible if and only if a sole linearcombination, non-proportional to either of the initial brackets, is a Poisson bracket.Proof. In place of Poisson brackets, we may consider the corresponding Poisson tensors.Given two tensors π ′ and π ′′ , consider R = aπ ′ + bπ ′′ with a, b ∈ k × . Then R is a Poissontensor if and only if [[ R, R ]] = 0 . In view of the fact that [[ π ′ , π ′′ ]] = [[ π ′′ , π ′ ]] , this reducesto the condition [[ π ′ , π ′′ ]] = 0 regardless of nonzero a, b . (cid:3) D. PANYUSHEV AND O. YAKIMOVA
Contractions and compatibility.
Let q = h ⊕ V be a vector space decomposition,where h is a subalgebra. For any s ∈ k × , define a linear map ϕ s : q → q by setting ϕ s | h = id , ϕ s | V = s · id . Then ϕ s ϕ s ′ = ϕ ss ′ and ϕ − s = ϕ s − , i.e., this yields a one-parametersubgroup of GL( q ) . The invertible map ϕ s defines a new (isomorphic to the initial) Liealgebra structure [ , ] ( s ) on the same vector space q by the formula(1 · [ x, y ] ( s ) = ϕ − s ([ ϕ s ( x ) , ϕ s ( y )]) . The corresponding Poisson bracket is { , } s . We naturally extend ϕ s to an automorphismof S ( q ) . Then the centre of the Poisson algebra ( S ( q ) , { , } s ) equals ϕ − s ( S ( q ) q ) .The condition [ h , h ] ⊂ h implies that there is the limit of the brackets [ , ] ( s ) as s tendsto zero. The limit bracket is denoted by [ , ] (0) and the corresponding Lie algebra is thesemi-direct product h ⋉ V ab , where [ V ab , V ab ] (0) = 0 . The algebra h ⋉ V ab is called an In¨on ¨u-Wigner or one-parameter contraction of q , see e.g. [PY12, Y14].Having a family of Poisson brackets { , } s on q ∗ associated with the maps ϕ s , it is natu-ral to ask whether these brackets are compatible. Lemma 1.2.
As above, let q = h ⊕ V , where h ⊂ q is a subalgebra. Let s, s ′ ∈ k . (i) If ( q , h ) is a symmetric pair, i.e., [ h , V ] ⊂ V and [ V, V ] ⊂ h , then { , } s = { , } − s and { , } s + { , } s ′ = 2 { , } ˜ s with s = s + ( s ′ ) . (ii) If [ V, V ] ⊂ V , i.e., V is a subalgebra, too, then { , } s + { , } s ′ = 2 { , } ˜ s with s = s + s ′ .Proof. All statements are verified by an easy direct computation. (cid:3)
In this article, we are interested in case (i) of Lemma 1.2 under the assumption that q issemisimple. 2. C ONSTRUCTING A P OISSON - COMMUTATIVE SUBALGEBRA Z Let g be a Z -graded semisimple Lie algebra and σ the corresponding involution of g ,i.e., g = g ⊕ g and σ ( x ) = ( − j x for x ∈ g j . Occasionally, we will need the relatedconnected algebraic groups G and G , i.e., g = Lie ( G ) and g = Lie ( G ) . We may assumethat G ⊂ G . Under the presence of σ , the Lie–Poisson bracket is being decomposed asfollows: { , } LP = { , } , + { , } , + { , } , . More precisely, if x = x + x ∈ g , then { x, y } , = [ x , y ] , { x, y } , = [ x , y ] + [ x , y ] ,and { x, y } , = [ x , y ] . Using this decomposition, we introduce a -parameter family ofPoisson brackets on g ∗ : { , } t = { , } , + { , } , + t { , } , , OISSON-COMMUTATIVE SUBALGEBRAS 7 where t ∈ P = k ∪ {∞} and { , } ∞ = { , } , . It is easily seen that { , } t with t ∈ k × isgiven by the map ϕ s , where s = t (see Section 1.2), and it follows from Lemmas 1.1 and1.2 that all these brackets are compatible. Hence { , } t = { , } + t { , } ∞ , t ∈ P , in accordance with the general method outlined in the introduction, Note that { , } LP = { , } + { , } ∞ . Write g ( t ) for the Lie algebra corresponding to { , } t . Of course, we merelywrite g in place of g (1) . All Lie algebras g ( t ) have the same underlying vector space g . Convention.
We identify g , g , and g with their duals via the Killing form on g . Hence g ∗ ⊕ g ∗ ≃ g ⊕ g . We regard g ∗ as the dual of any algebra g ( t ) and sometimes omit thesubscript ‘ ( t ) ’ in g ∗ ( t ) . However, if ξ ∈ g ∗ , then the stabiliser of ξ in the Lie algebra g ( t ) (i.e.,with respect to the coadjoint representation of g ( t ) ) is denoted by g ξ ( t ) .Let π t be the Poisson tensor for { , } t and π t ( ξ ) the skew-symmetric bilinear form on g ≃ T ∗ ξ ( g ∗ ) corresponding to ξ ∈ g ∗ , cf. Section 1.1. A down-to-earth description is that π t ( ξ )( x , x ) = { x , x } ( t ) ( ξ ) . Set rk π t = max ξ ∈ g ∗ rk π t ( ξ ) . Lemma 2.1.
We have ind g ( t ) = rkc { , } t = rk g , t = ∞ ;dim g + rk g − rk g , t = ∞ . Proof.
We know that rkc { , } LP = rkc { , } = rk g , if g is semisimple.1) If t = 0 , ∞ , then the existence of ϕ s with s = t implies that { , } t is isomorphicto { , } . For t = 0 , one obtains the Poisson bracket of the semi-direct product ( Z -contraction) g (0) = g ⋉ g ab , and it is proved in [P07, Cor. 9.4] that ind ( g ⋉ g ab ) = rk g .2) By definition, rkc { , } ∞ = ind g ( ∞ ) = min ξ ∈ g ∗ dim g ξ ( ∞ ) . Here { , } ∞ represents thedegenerated Lie algebra structure on the vector space g such that [ x + x , y + y ] ∞ =[ x , y ] ∈ g . One easily verifies that if ξ = ξ + ξ ∈ g ∗ , then g ξ ( ∞ ) = g ⊕ g ξ . Therefore, ind g ( ∞ ) = dim g + min ξ ∈ g ∗ dim g ξ = dim g − max ξ ∈ g dim[ g , ξ ] . In the last step, we use the fact that upon the identification of g ∗ and g , the coadjointaction of g ⊂ g ( ∞ ) on g ∗ ⊂ g ∗ ( ∞ ) becomes the usual bracket in g .By a well-known property of Z -gradings, g always contains a regular semisimpleelement of g . If ξ ∈ g is regular semisimple in g and hence in g , then [ g , ξ ] = [ g , ξ ] ⊕ [ g , ξ ] , dim[ g , ξ ] = dim g − rk g , and dim[ g , ξ ] = dim g − rk g . Hence max ξ ∈ g dim[ g , ξ ] = dim g + rk g − rk g , and we are done. (cid:3) D. PANYUSHEV AND O. YAKIMOVA
It follows from Lemma 2.1 that t = ∞ is regular in P if and only if dim g = rk g , i.e., g is Abelian. For the indecomposable pairs, this happens if and only if g = sl . For thisreason, it is necessary to handle the sl -case separately. Example 2.2.
Let g = sl with a standard basis { e, h, f } such that [ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h . Then S ( sl ) sl = k [ h + 4 ef ] . For the unique (up to conjugation) non-trivial σ , one has g = k h and e, f ∈ g . Then Z t ( t = 0 , ∞ ) is generated by h + t − ef . An easycalculation shows that Z = k [ ef ] and Z ∞ = k [ h ] . Here P reg = P , hence Z is generated byall Z t with t ∈ P and Z = k [ h, ef ] . This is a maximal Poisson-commutative subalgebra of S ( g ) and it lies in S ( g ) g .Unless otherwise explicitly stated, we assume below that g = sl . We then obtain a -parameter family of compatible Poisson brackets on g ∗ , with generic central rank beingequal to rk g and P sing = {∞} , where the central rank jumps up to dim g + rk g − rk g .Hence P reg = P \ {∞} = k . For each Lie algebra g ( t ) , there is the related singular set g ∗ ( t ) , sing = g ∗ \ g ∗ ( t ) , reg , cf. Eq. (1 · g ∗ ( t ) , sing = { ξ ∈ g ∗ | rk π t ( ξ ) < rk π t } , which is the union of the symplectic g ( t ) -leaves in g ∗ having a non-maximal dimension.For aesthetic reasons, we write g ∗∞ , sing instead of g ∗ ( ∞ ) , sing .Let Z t denote the centre of the Poisson algebra ( S ( g ) , { , } t ) . Then Z = S ( g ) g . For ξ ∈ g ∗ , let d ξ F denote the differential of F ∈ S ( g ) at ξ . It is standard that for any H ∈ S ( g ) g ,d ξ H ∈ z ( g ξ ) , where z ( g ξ ) is the centre of g ξ .Let { H , . . . , H l } be a set of homogeneous algebraically independent generators of S ( g ) g .By the Kostant regularity criterion for g [K63, Theorem 9],(2 · h d ξ H j | j l i k = g ξ if and only if ξ ∈ g ∗ reg .(Recall that g ξ = z ( g ξ ) if and only if ξ ∈ g ∗ reg [P03, Thm. 3.3].) Set d ξ Z t = h d ξ F | F ∈ Z t i k .Then d ξ Z t ⊂ ker π t ( ξ ) for each t . The regularity criterion obviously holds for any t = 0 , ∞ .That is,(2 ·
2) if t = 0 , ∞ , then ξ g ∗ ( t ) , sing ⇔ d ξ Z t = ker π t ( ξ ) ⇔ dim ker π t ( ξ ) = rk g . A certain analogue of this statement holds for t = 0 , i.e., for g (0) and d x Z , but only forinvolutions σ such that S ( g ) g has a g.g.s. for ( g , g ) , see [Y14].The centres Z t ( t ∈ k ) generate a Poisson-commutative subalgebra with respect to anybracket { , } t , t ∈ P , cf. Corollary A.2. Write Z = alg hZ t i t ∈ k for this subalgebra. Note thatd ξ Z is the linear span of d ξ Z t with t = ∞ . There is a method for estimating the dimensionof such subspaces, see Appendix A. Lemma 2.3.
Suppose that ξ ∈ g ∗ satisfy the properties: OISSON-COMMUTATIVE SUBALGEBRAS 9 (1) dim ker π t ( ξ ) = rk g for all t = ∞ ; (2) the rank of the skew-symmetric form π ( ξ ) | ker π ∞ ( ξ ) equals dim ker π ∞ ( ξ ) − rk g .Then dim d ξ Z = rk g + rk π ∞ ( ξ ) and dim (cid:0) d ξ Z ∩ ker π ∞ ( ξ ) (cid:1) = rk g .Proof. By definition, d ξ Z ⊂ P t = ∞ ker π t ( ξ ) . Then Eq. (2 ·
2) and hypothesis (1) on ξ implythat d ξ Z ⊃ P t =0 , ∞ ker π t ( ξ ) . Observe that we have a -dimensional vector space of skew-symmetric bilinear forms a · π t ( ξ ) on g ≃ T ∗ ξ g ∗ , where a ∈ k , t ∈ P . Moreover, rk π t ( ξ ) =dim g − rk g for each t = ∞ . By Lemma A.1, we have P t =0 , ∞ ker π t ( ξ ) = P t = ∞ ker π t ( ξ ) .Now the desired equalities follow from Theorem A.4. (cid:3) It is not clear yet whether such elements ξ ∈ g ∗ actually exist! However, we will imme-diately see that there are plenty of them. Proposition 2.4.
The hypotheses of Lemma 2.3 hold for generic ξ ∈ g ∗ and therefore tr . deg Z = 12 rk π ∞ + rk g = 12 (dim g − rkc { , } ∞ ) + rk g . Proof.
The first task is to prove that a generic point ξ = ξ + ξ ∈ g ∗ satisfies condition (1) in Lemma 2.3.One can safely assume that ξ is regular for { , } and { , } ∞ . Next, we are lucky that ξ + ξ ∈ g ∗ sing = g ∗ (1) , sing if and only if ξ + s − ξ ∈ g ∗ ( s ) , sing . Therefore,(2 · [ t =0 , ∞ g ∗ ( t ) , sing = { ξ + tξ | ξ + ξ ∈ g ∗ sing , t = 0 , ∞} . Since codim g ∗ ( t ) , sing = 3 for each t ∈ k × , the closure of S t =0 , ∞ g ∗ ( t ) , sing is a proper subset of g ∗ . Hence the condition dim ker π t ( ξ ) = rk g ( t = ∞ ) holds for ξ in a dense open subset.The next step is to check condition (2) , i.e., compute the rank of the restriction of π ( ξ ) to ker π ∞ ( ξ ) . Write ξ = ξ + ξ , where ξ i ∈ g ∗ i . We can safely assume that ξ is regular in g and hence also in g . • For the inner involutions, one has rk g = rk g . Here ker π ∞ ( ξ ) = g and the rank inquestion is dim g − rk g , as required in Lemma 2.3 (2) . • Suppose that σ is outer. Then ker π ∞ ( ξ ) = g ⊕ g ξ with dim g ξ = rk g − rk g . The rankof the form π ( ξ ) on this kernel is equal to dim ker π ∞ ( ξ ) − rk g − dim g ξ = dim ker π ∞ ( ξ ) − rk g . For a generic ξ , where ξ is generic as well, the value in question cannot be smaller than dim ker π ∞ ( ξ ) − rk g . On the other hand, it cannot be larger by Lemma A.3. That is, wehave obtained the required value again!Now, it follows from Lemma 2.3 that tr . deg Z = max ξ ∈ g ∗ dim d ξ Z = 12 (dim g − rkc { , } ∞ ) + rk g . (cid:3) Combining Lemma 2.1 and Proposition 2.4, we obtain(2 · tr . deg Z = 12 (dim g + rk g + rk g ) . Lemma 2.5 ([MY, Prop. 1.1]) . If A ⊂ S ( g ) g and {A , A} LP = 0 , then tr . deg A b ( g ) − b ( g ) + ind g . Note that in our situation, b ( g ) − b ( g ) + ind g = (dim g + rk g + rk g ) . Lemma 2.6.
We have Z ⊂ S ( g ) g .Proof. For all Poisson brackets { , } t with t = ∞ , the commutators [ x , y ] are the same asin g . Hence Z t ⊂ S ( g ) g for each t = ∞ . (cid:3) Aposteriori, this lemma is true for g = sl as well, cf. Example 2.2. Combining previousformulae, together with computations for sl , we obtain the next general assertion. Theorem 2.7.
For any g and any σ , the algebra Z = alg hZ t i t ∈ P reg is a Poisson-commutativesubalgebra of S ( g ) g of the maximal possible transcendence degree, which is given by Eq. (2 · . In Section 3, we provide an explicit set of generators of Z , if S ( g ) g has a good generatingsystem for ( g , g ) . From this, we deduce that Z is a polynomial algebra. Although Z has the maximal transcendence degree among the Poisson-commutative subalgebras of S ( g ) g , it is not always maximal. In Section 4, we construct the extended algebra ˜ Z suchthat Z ⊂ ˜ Z ⊂ S ( g ) g and show that ˜ Z is maximal and still polynomial.3. T HE ALGEBRA Z IS POLYNOMIAL WHENEVER σ IS GOOD
Let { H , . . . , H l } , l = rk g , be a set of homogeneous algebraically independent generatorsof S ( g ) g . Set d i = deg H i . Then P li =1 d i = b ( g ) . Associated with the vector space decom-position g = g ⊕ g , one has the bi-homogeneous decomposition of each H j : H j = d j X i =0 ( H j ) ( i,d j − i ) , where ( H j ) ( i,d j − i ) ∈ S i ( g ) ⊗ S d j − i ( g ) ⊂ S d j ( g ) . Let H • j be the nonzero bi-homogeneouscomponent of H j with maximal g -degree. Then deg g H j = deg g H • j and we set d • j =deg g H • j . Definition 2.
Let us say that H , . . . , H l is a good generating system in S ( g ) g ( g.g.s. for short)for ( g , g ) or for σ , if H • , . . . , H • l are algebraically independent. OISSON-COMMUTATIVE SUBALGEBRAS 11
If the pair ( g , g ) is indecomposable, which we always tacitly assume, then there is no g.g.s. for four involutions related to g of type E n [P07’, Remark 4.3] and a g.g.s. existsin all other cases, see [Y14]. The importance of g.g.s. is clearly seen in the followingfundamental result. Theorem 3.1 ([Y14, Theorem 3.8]) . Let H , . . . , H l be an arbitrary set of homogeneous alge-braically independent generators of S ( g ) g . Then (i) P lj =1 deg g H j > dim g ; (ii) H , . . . , H l is a g.g.s. if and only if P lj =1 deg g H j = dim g ; (iii) if H , . . . , H l is a g.g.s. , then S ( g (0) ) g (0) = k [ H • , . . . , H • l ] is a polynomial algebra. Recall that g (0) = g ⋉ g ab is a Z -contraction of g and ind g (0) = ind g . We continue toassume that g = sl , hence P reg = k and Z = alg hZ t i t ∈ k . Theorem 3.2.
Suppose that { H i } is a g.g.s. for σ . Then the algebra Z is generated by (3 · { ( H j ) ( i,d j − i ) | j = 1 , . . . , l & i = 0 , , . . . , d j } , i.e., by all bi-homogeneous components of H , . . . , H l .Proof. To begin with, Z ( { , } ) = Z ( S ( g )) = k [ H , . . . , H l ] . By the definition of { , } t , wehave Z ( { , } t ) = ϕ − s ( Z ( S ( g ))) for t = 0 , ∞ , where s = t and ϕ s ( H j ) = ( H j ) ( d j , + s ( H j ) ( d j − , + s ( H j ) ( d j − , + . . . Using the Vandermonde determinant, we deduce from this that all ( H j ) ( i,d j − i ) belong to Z and the algebra generated by them contains Z t with t ∈ k \ { } . Moreover, the specific bi-homogeneous components H • , . . . , H • l generate Z , since H , . . . , H l is a g.g.s. Therefore,the polynomials (3 ·
1) generate the whole of Z . (cid:3) However, not every i ∈ { , , . . . , d j } provides a nonzero bi-homogeneous componentof H j . Let us make this precise. Since the case of inner involutions is technically easier,we consider it first. Theorem 3.3.
Suppose that σ ∈ Aut ( g ) is inner, and let H , . . . , H l be a g.g.s. in S ( g ) g with d • j = deg g H j . Then (i) all d • j , j = 1 , . . . , l , are even; (ii) ( H j ) ( i,d j − i ) = 0 if and only if d j − i is even and d j − i d • j ; (iii) the polynomials { ( H j ) ( i,d j − i ) | j = 1 , . . . , l ; & d j − i = 0 , , . . . , d • j } freely generate Z .Proof. (1) Since σ is inner, σ ( H j ) = H j for all j . On the other hand, σ | g = id , σ | g = − id ,and hence σ (( H j ) ( i,d j − i ) ) = ( − d j − i ( H j ) ( i,d j − i ) . This yields (i) and one implication in (ii) . (2) In view of part (1) , the number of non-zero bi-homogeneous components of H j is atmost ( d • j /
2) + 1 . Hence the total number of nonzero bi-homogeneous components of all H j is at most P lj =1 ( d • j /
2) + 1 = (dim g /
2) + rk g .As σ is inner, one also has rk g = rk g . Therefore, tr . deg Z = (dim g / rk g , see Eq. (2 · H j generate Z (Theorem 3.2), we see thatall ( H j ) ( i,d j − i ) with d j − i = 0 , , . . . , d • j are nonzero and algebraically independent. Thus,they freely generate Z . (cid:3) With extra technical details, Theorem 3.3 extends to the outer involutions as well. Let σ be an arbitrary involution of g . It is easily seen that a set of homogeneous generatorsof S ( g ) g can be chosen so that each H j is an eigenvector of σ , i.e., σ ( H j ) = ε j H j = ± H j .Moreover, the set of pairs { ( d j , ε j ) | j = 1 , . . . , l } does not depend on the set of generators,cf. [S74, Lemma 6.1]. However, we need a set of free generators that both is a g.g.s . andconsists of σ -eigenvectors. Lemma 3.4.
If there is a g.g.s. for ( g , g ) , then there is also a g.g.s. that consists of eigenvectorsof σ .Proof. Let H , . . . , H l be a g.g.s. , hence P lj =1 deg g H j = dim g in view of Theorem 3.1.Let A + be the ideal in S ( g ) g generated by all homogeneous invariants of positive degree.Then A := A + / A is a finite-dimensional k -vector space. If H ∈ A + , then ¯ H := H + A ∈ A . As is well-known, F , . . . , F m is a generating system for S ( g ) g if and only if the k -linearspan of ¯ F , . . . , ¯ F m is the whole of A . In our situation, dim k A = l and A = h ¯ H , . . . , ¯ H l i .If H i is not a σ -eigenvector, i.e., σ ( H i ) = ± H i , then we consider the generating set H , . . . , H i − , H i + σ ( H i )2 , H i − σ ( H i )2 , H i +1 , . . . , H l for S ( g ) g that includes l + 1 polynomials. Since ¯ H , . . . , ¯ H i − , ¯ H i +1 , . . . , ¯ H l are linearly inde-pendent in A , we obtain a better generating set by replacing H i with one of the functions H (+) i = H i + σ ( H i )2 or H ( − ) i = H i − σ ( H i )2 . Let us demonstrate that there is actually only one suit-able replacement for H i , and this yields again a g.g.s. Recall that d • j = deg g H • j = deg g H j .(a) Suppose that d • j is even. Then σ ( H • i ) = H • i and H • i cancel out in H ( − ) i . Therefore, deg g H ( − ) i < deg g H i and the sum of g -degrees for H , . . . , H i − , H ( − ) i , H i +1 , . . . , H l is lessthan dim g . By Theorem 3.1, this means that the choice of H ( − ) i in place of H i does notprovide a generating system, and the only right choice is to take H (+) i . Moreover, H • i =( H (+) i ) • and hence H , . . . , H i − , H (+) i , H i +1 , . . . , H l is a g.g.s. (b) If d • j is odd, then we end up with the g.g.s. H , . . . , H i − , H ( − ) i , H i +1 , . . . , H l .The procedure reduces the number of generators that are not σ -eigenvectors, and weeventually obtain a g.g.s. that consists of σ -eigenvectors. (cid:3) OISSON-COMMUTATIVE SUBALGEBRAS 13
Without loss of generality, we can assume that H , . . . , H l is a g.g.s. and σ ( H j ) = ± H j . Lemma 3.5.
For any involution σ ∈ Aut ( g ) , we have (1) σ ( H j ) = H j if and only if d • j is even; (2) rk g = { j | σ ( H j ) = H j } .Proof. (1) The proof is similar to that of Theorem 3.3(i). (2)
This follows from results of T. Springer on regular elements of finite reflectiongroups [S74, Corollary 6.5]. To this end, one has to consider the Weyl group correspond-ing to a Cartan subalgebra t = t ⊕ t ⊂ g ⊕ g such that t is a Cartan in g . (cid:3) Now, we can state and prove the main result of this section.
Theorem 3.6.
Let σ be an involution of g such that S ( g ) g has a g.g.s. Then Z is a polynomialalgebra that is freely generated by the bi-homogeneous components of all { H j } . More precisely, if σ ( H j ) = H j , then d • j is even and the nonzero bi-homogeneous components of H j are ( H j ) ( i,d j − i ) with d j − i = 0 , , . . . , d • j ; if σ ( H j ) = − H j , then d • j is odd and the nonzero bi-homogeneouscomponents of H j are ( H j ) ( i,d j − i ) with d j − i = 1 , , . . . , d • j .Proof. By Lemma 3.5, we may order the basic invariants { H j } such that d • j is even i k := rk g ; odd i > k + 1 . Clearly, if d • j is even, then ε j = 1 and H j has at most ( d • j /
2) + 1 nonzero bi-homogeneouscomponents, while if d • j is odd, then ε j = − and H j has at most ( d • j + 1) / nonzerobi-homogeneous components. Hence the total number of all nonzero bi-homogeneouscomponents is at most k X j =1 (cid:18) d • j (cid:19) + l X j = k +1 d • j + 12 = l X j =1 d • j k + l − k g + rk g + rk g . deg Z . Therefore, all admissible bi-homogeneous components must be nonzero and algebraicallyindependent. (cid:3)
Remark . If there is no g.g.s. for ( g , g ) , then P j deg g H j > dim g for any set of basicinvariants. Hence the number of the bi-homogeneous components of { H j } is bigger than tr . deg Z and these generators of Z are algebraically dependent. Moreover, the algebra Z = Z ( S ( g ⋉ g ab )) , which is contained in Z , is not polynomial [Y17, Section 6], and also H • , . . . , H • l are algebraically dependent, cf. Theorem 3.1. Thus, we cannot say anythinggood about Z in the four “bad” cases. Remark . Recall from the introduction the map r : S ( g ) g → S ( g ) g . If σ is inner, then g contains a Cartan subalgebra of g and r is injective. Hence ( H j ) ( d j , = r ( H j ) = 0 for all j , which also follows from Theorem 3.3. Clearly, r ( S ( g ) g ) ⊂ Z for any σ . Moreprecisely, r ( S ( g ) g ) is freely generated by the r ( H j ) = ( H j ) ( d j , such that σ ( H j ) = H j (i.e., d • j is even). However, for the inner (and some outer) involutions, r ( S ( g ) g ) is a propersubalgebra of S ( g ) g . And this is the reason, why Z appears to be not always a maximalcommutative subalgebra of S ( g ) g .4. T HE EXTENDED ALGEBRA ˜ Z IS POLYNOMIAL AND MAXIMAL P OISSON - COMMUTATIVE
In this section, we assume that g = sl , ( g , g ) is indecomposable, and there is a g.g.s. for ( g , g ) . We write z ( q ) for the centre of a Lie algebra q . An open subset of g ∗ is said to be big , if its complement does not contain divisors.There is an extraordinary powerful tool for proving maximality of certain subalgebras. Theorem 4.1 ([PPY, Theorem 1.1]) . Let F , . . . , F r ∈ S ( g ) be homogeneous algebraically in-dependent polynomials such that their differentials { d F i } are linearly independent on a big opensubset of g ∗ . Then k [ F , . . . , F r ] is an algebraically closed subalgebra of S ( g ) , i.e., if H ∈ S ( g ) isalgebraic over the field k ( F , . . . , F r ) , then H ∈ k [ F , . . . , F r ] . In order to apply this theorem to Z and ˜ Z , we need some properties of divisors in g ∗ . Lemma 4.2.
Let D ⊂ g ∗ be an irreducible divisor. Then there is a non-empty open subset U ⊂ D such that, for each ξ ∈ U , we have (i) ξ g ∗ ( t ) , sing , if t = ∞ ; (ii) if ξ = ξ + ξ with ξ i ∈ g ∗ i , then ξ ∈ ( g ∗ ) reg .Proof. (i) The Lie algebra g (0) = g ⋉ g ab has the codim– property, see [P07’, Theorem 3.3].Hence codim g ∗ (0) , sing > . Recall that dim g ∗ sing = dim g − . Therefore, the union of thesingular subsets g ∗ ( t ) , sing , t ∈ k × , is a subset of codimension , as follows from Eq. (2 · D such that rk π t ( ξ ) = rk π t for each ξ ∈ D and t = ∞ . (ii) Since g is reductive, we also have dim( g ∗ ) sing dim g − . (cid:3) Lemma 4.3.
Suppose that the differentials { d ( H j ) ( i,d j − i ) } are linearly dependent on an irreducibledivisor D ⊂ g ∗ . Then D ⊂ g ∗∞ , sing .Proof. Combining Lemmas 2.3 and 4.2, we see that if the differentials of the ( H j ) ( i,d j − i ) ’sare linearly dependent at a generic point ξ ∈ D , then– either rk π ∞ ( ξ ) < rk π ∞ , OISSON-COMMUTATIVE SUBALGEBRAS 15 – or rk π ∞ ( ξ ) = rk π ∞ , but the restriction of π ( ξ ) to ker π ∞ ( ξ ) does not have the pre-scribed (maximal possible) rank.In the first case, we have ξ ∈ g ∗∞ , sing by the very definition. Let us show that the secondpossibility does not realise. Write ξ = ξ + ξ . By Lemma 4.2 (ii) , we may assume that ξ ∈ g , reg . Since rk π ∞ ( ξ ) = rk π ∞ , we also have ξ ∈ g reg . As in the proof of Proposition 2.4,the rank of π ( ξ ) on ker π ∞ ( ξ ) equals dim ker π ∞ ( ξ ) − rk g . And again the same holds forthe restriction of π ( ξ ) . (cid:3) We also need the following simple but useful observation on g ∗∞ , sing . Lemma 4.4.
The subvariety g ∗∞ , sing is of the form X × g ∗ , where X ⊂ g ∗ is a conical subvariety.Moreover, X ∩ g ∗ reg = ∅ .Proof. Let ξ = ξ + ξ ∈ g ∗ . Since g ξ ( ∞ ) = g ⊕ g ξ , the value rk π ∞ ( ξ ) depends only on ξ = ξ | g . Therefore, g ∗∞ , sing = X × g ∗ , where X = g ∗∞ , sing ∩ g ∗ .It follows from the proof of Lemma 2.1 that min ξ ∈ g dim g ξ = rk g − rk g , and ξ ∈ g ∗∞ , sing if and only if dim g ξ > rk g − rk g . But, if ξ ∈ g ∗ reg , then dim g ξ = rk g and dim g ξ = rk g − rk g . (cid:3) A particularly nice situation occurs if r : S ( g ) g → S ( g ) g is onto. This condition israther restrictive. If σ is inner, then b ( g ) = b ( g ) + (dim g ) / . And since P lj =1 d j = b ( g ) ,the nonzero polynomials { ( H j ) ( d j , } lj =1 cannot form a generating system in S ( g ) g . Hence r cannot be onto for the inner σ . Another observation is that g has to be simple. Thisleads to the following list of suitable symmetric pairs:(4 · ( h ⊕ h , h ) , ( sl n , so n ) , ( sl n , sp n ) , ( so n , so n − ) , ( E , sp ) , ( E , F ) Among them the map r is onto for ( h ⊕ h , h ) , ( sl n +1 , so n +1 ) , ( sl n , sp n ) , ( so n , so n − ) ,and ( E , F ) . But, the pair ( E , F ) is not needed, because it does not have a g.g.s. Theorem 4.5. (1)
If the restriction homomorphism r : S ( g ) g → S ( g ) g is onto, then g ∗∞ , sing does not contain divisors and Z is a maximal Poisson-commutative subalgebra of S ( g ) g . (2) Conversely, if Z is maximal Poisson-commutative, then r is onto.Proof. (1) The list of suitable symmetric pairs is quite short. For each item in the list, g contains a nilpotent element that is regular in g . This implies that every fibre of thequotient morphism g → g //G contains a regular element of g and hence ( g ) ∗ reg ⊂ g ∗ reg .Thus, dim( g ∗ sing ∩ g ) dim g − . Since rk π ∞ ( ξ ) = rk π ∞ for each ξ ∈ g ∗ ∩ g ∗ reg (Lemma 4.4),the subset g ∗∞ , sing does not contain divisors. Therefore, the differentials d ( H j ) ( i,d j − i ) arelinearly independent on a big open subset, in view of Lemma 4.3. Then, by Theorem 4.1, Z is an algebraically closed subalgebra of S ( g ) . Since it is a Poisson-commutative subalgebraof S ( g ) g of the maximal possible transcendence degree, it is also maximal. (2) If r is not onto, then the algebra generated by S ( g ) g and Z is Poisson-commutative,is contained in S ( g ) g , and properly contains Z . (cid:3) Remark . (1) Consider the following four conditions: (a) the restriction homomorphism r : S ( g ) g → S ( g ) g is onto; (b) g contains a regular nilpotent element of g ; (c) g ∗∞ , sing does not contain divisors; (d) Z is a maximal Poisson-commutative subalgebra of S ( g ) g .In the proof of Theorem 4.5(1), we have seen that (a) ⇒ (b) ⇒ (c) ⇒ (d), whereas part(2) of Theorem 4.5 states that (d) ⇒ (a). Thus, all these conditions are equivalent. Onecan also give a direct proof for (b) ⇒ (a) that does not invoke g ( ∞ ) and Z . However, theimplication (a) ⇒ (b) is obtained case-by-case as yet. (2) There is a g.g.s. for ( g , g ) if and only if the restriction homomorphism r : S ( g ) g → k [ g ∗ ] g is onto [P07’, Y14]. Therefore, Z is a polynomial maximal Poisson-commutativesubalgebra of S ( g ) g whenever both r and r are onto.Our ultimate goal is to prove that, in general, ˜ Z = alg h Z , S ( g ) g i is a polynomial max-imal Poisson-commutative subalgebra of S ( g ) g . Unfortunately, the proof requires manytechnical preparations, if g ∗∞ , sing contains divisors (i.e., r is not onto). Lemma 4.7.
Suppose that dim g ∗∞ , sing = n − , and let D ⊂ g ∗∞ , sing be an irreducible componentof dimension n − . Then (i) D = D × g ∗ , where D is a G -stable conical divisor in g ∗ , and D does not containregular elements of g ; (ii) generic elements of D are semisimple, regular in g ≃ g ∗ , and subregular in g ; (iii) rk π ∞ ( ξ ) = rk π ∞ − for generic point ξ ∈ D .Proof. (i) This follows from Lemma 4.4. (ii) If σ is inner, then g contains a Cartan subalgebra t of g and t ∩ D is a W -stabledivisor in t , where W is the Weyl group of ( g , t ) . It is easily seen that any such divisorcontains a subregular element of g .The case of an outer σ is more involved. We use an argument, which is also valid forthe inner case. If t ⊂ g is a Cartan subalgebra of g , then a generic element ν ∈ D ∩ t iseither regular or subregular in g . Consider these two possibilities in turn.(a) Suppose first that ν is regular in g . Then g ν = t and therefore g ν is a sum of a toralsubalgebra and several copies, say k , of sl . Let s i be the i -th copy of sl . Every such s i isdetermined by a root β i of g . That is, s i = g − β i ⊕ ( s i ) σ ⊕ g β i . OISSON-COMMUTATIVE SUBALGEBRAS 17
Moreover, the one-dimensional subspace ( s i ) σ is generated by the coroot β ∨ i . It is alsoclear that ( s i ) σ ⊂ t and β ∨ i is orthogonal to ν . Assume that k > . Then ν is orthogonalto at least two different coroots. Since the number of relevant pairs { β i , β j } is finite, weobtain that D ∩ t lies in a finite union of subspaces of t of codimension > . A contra-diction! Hence k . If k = 0 , then ν is regular in g , which is impossible, see (i) . Thus, k = 1 and ν is subregular in g .(b) Suppose now that D does not contain regular semisimple elements of g . Our goalis to prove that this case does not occur.Here t ∩ D is a union of reflection hyperplanes of W . Let z be one of these hyper-planes and ν ∈ z generic. Then g ν = s ⊕ z , where s ≃ sl and z is the centre of g ν . Here [ z , g ν ] = 0 , since ν ∈ z is generic. Write g ν = h ⊕ z ( g ν ) , where h = [ g ν , g ν ] is semisimple.Then the symmetric pair ( g ν , g ν ) decomposes as ( g ν , g ν ) = ( h , s ) ⊕ ( z ( g ν ) , z ) . The only possibilities for the symmetric pair ( h , s ) are:(4 · ( sl ⊕ sl , sl ) , ( sl , so ≃ sl ) , ( sl , sl ) . For s = [ g ν , g ν ] , the intersection D ∩ ( k ν ⊕ s ) is a conical divisor of k ν ⊕ s that contains ν . If η ∈ s is non-zero semisimple, then ν + η ∈ ( g ) reg is semisimple. Hence ν + η D .Therefore, D ∩ ( k ν ⊕ s ) has to contain a sum ν + e , where e ∈ s is regular nilpotent. Forall pairs in (4 · e is also regular in h . Hence e is a regular element of g ν . Thereby ν + e isa regular element of g . However, this contradicts part (i) .Therefore, case (b) does not materialise and, according to (a), D contains a semisimpleelement ν that is regular in g and subregular in g . Since D ∩ g reg = ∅ , subregularsemisimple elements of g are dense in D . (iii) Since ν is regular in g and subregular in g , we have dim g ν = rk g and dim g ν = rk g + 2 − rk g . The latter precisely means that rk π ∞ ( ν ) = rk π ∞ − for ν in a non-emptyopen subset of D . This completes the proof. (cid:3) Example 4.8.
Let ( g , g ) = ( sl n , so n ) . Then D ⊂ g is the zero set of the Pfaffian. If g consists of skew-symmetric matrices with respect to the antidiagonal, then x = diag( a , . . . , a n − , , , − a n − , . . . , − a ) ∈ D is subregular whenever all a i are nonzero and a i = ± a j for i = j .Recall that { H i } is a g.g.s. for σ such that σ ( H i ) = ε i H i = ± H i for each i . As before, d i = deg H i and l = rk g . Until the end of this section, we assume that d · · · d l . If g issimple, then there is a unique basic invariant of degree d l , i.e., d l − < d l . Lemma 4.9. If g is simple and x ∈ g is subregular, then the differentials { d x H i | i < l } arelinearly independent. Moreover, σ ( H l ) = H l unless ( g , g ) = ( sl k +1 , so k +1 ) , where l = 2 k and d l = 2 k + 1 .Proof. Let e ∈ g be a subregular nilpotent element. Then d e H l = 0 [V68, Corollary 2] and { d e H i | i < l } are linearly independent [Sl80, Chapter 8.2]. If x is subregular and non-nilpotent, then the theory of associated cones developed in [BK79, §
3] shows that Ge ⊂ k × ( Gx ) . This implies that d x H i with i < l are linearly independent, too.The equality σ ( H l ) = H l is obvious for the inner involutions. If σ is outer, then goingthrough the list of outer involutions, one checks that σ ( H l ) = − H l if and only if g = sl k +1 and l = 2 k . Here necessary g = so k +1 . (cid:3) We need below some formulae for the differential and partial derivatives of a homoge-neous polynomial F ∈ S ( g ) = k [ g ∗ ] . If x ∈ g ∗ and d = deg F , then ∂ d − x F is a linear formon g ∗ , i.e., an element of g . In fact, one has(4 · ( d − d x F = ∂ d − x F. By linearity, it suffices to check this for a monomial of degree d . Furthermore, for the op-erator ∂ kx + sx ′ : S m ( g ) → S m − k ( g ) with x, x ′ ∈ g ∗ and s ∈ k , there is the following expansion:(4 · ∂ kx + sx ′ = ∂ kx + (cid:18) k (cid:19) s∂ x ′ ∂ k − x + · · · + (cid:18) ki (cid:19) s i ∂ ix ′ ∂ k − ix + · · · + s k ∂ kx ′ . Lemma 4.10.
Suppose that the restriction homomorphism r is not onto (equivalently, g ∗∞ , sing contains divisors). Then (i) there is x ∈ g ∗ ≃ g such that x is semisimple, regular in g , and subregular in g (i.e., dim g x = rk g + 2 ). Moreover, for a generic x ′ ∈ g ∗ ≃ g , we have y := x + x ′ ∈ g reg ; (ii) lim t →∞ h d y F | F ∈ Z t i k = lim s → ϕ s ( g x + sx ′ )(=: V ) ; (iii) dim( V / V ∩ g ) = rk g − rk g + 1 .Proof. (i) The existence of such an x follows from Lemma 4.7. Then g x = t and if x ′ is ageneric element of g x , then y is regular in g . Hence x + x ′ ∈ g reg for almost all x ′ ∈ g . (ii) By the definition of { , } t , we have Z t = ϕ − s ( S ( g ) g ) if t = 0 , ∞ and s = t . Let ϕ ∗ s : g ∗ → g ∗ be the dual map, i.e., ϕ ∗ s | g ∗ = id , ϕ ∗ s | g ∗ = s − · id . For any F ∈ S ( g ) , we have ϕ s ( d y F ) = d ϕ ∗ s ( y ) ϕ s ( F ) . In particular,d y ϕ − s ( H i ) = ϕ − s ( d ϕ ∗ s ( y ) H i ) , where ϕ ∗ s ( y ) = x + s − x ′ . If s tends to ∞ , then s − tends to . It remains to notice,that for almost all s , the element x + sx ′ is regular and then g x + sx ′ is the linear span of { d x + sx ′ H j } lj =1 , see Eq. (2 · OISSON-COMMUTATIVE SUBALGEBRAS 19 (iii)
The hypothesis that r is not onto excludes the pairs ( h ⊕ h , h ) and ( sl k +1 , so k +1 ) .Hence g is simple and, by Lemma 4.9, d x H , . . . , d x H l − are linearly independent, d x H l is a linear combination of d x H j with j < l , and σ ( H l ) = H l . Since x is semisimple andsubregular, g x = z ( g x ) ⊕ sl and dim z ( g x ) = l − . Hence z ( g x ) = h d x H i | i < l i k .Take j < l and set m j = d j − . Then by Eq. (4 ·
3) and by Eq. (4 ·
4) with k = m j , we have ( m j )! d x + sx ′ H j = ∂ m j x + sx ′ H j = m j X i =0 (cid:18) m j i (cid:19) s i ∂ ix ′ ∂ m j − ix H j , ( m j )! σ ( d x + sx ′ H j ) = ∂ m j x − sx ′ σ ( H j ) = m j X i =0 (cid:18) m j i (cid:19) ( − s ) i ∂ ix ′ ∂ m j − ix σ ( H j ) . It follows that ∂ ix ′ ∂ m j − ix H j ∈ g if and only if either i is even and σ ( H j ) = H j or i is oddand σ ( H j ) = − H j . Therefore, • if σ ( H j ) = H j , then lim s → ϕ s ( d x + sx ′ H j ) = d x H j ∈ g ; while • if σ ( H j ) = − H j , then d x H j ∈ g and ( m j )! ϕ s ( d x + sx ′ H j ) = s ( ∂ m j x H j + m j ∂ x ′ ∂ m j − x H j ) + (terms of degree > w.r.t. s ) = s (cid:0) ( m j )! d x H j + m j ∂ x ′ ∂ m j − x H j (cid:1) + . . . Thus, if σ ( H j ) = − H j , then(4 · lim s → h ϕ s ( d x + sx ′ H j ) i k = h d x H j + 1( m j − · ∂ x ′ ∂ d j − x H j i k . Note that here ∂ x ′ ∂ d j − x H j ∈ g . Write z ( g x ) = z ( g x ) ⊕ z ( g x ) , where z ( g x ) i = z ( g x ) ∩ g i .Then z ( g x ) = h d x H j | σ ( H j ) = − H j i k and z ( g x ) = h d x H j | σ ( H j ) = H j , j = l i k . Hence dim z ( g x ) = rk g − and dim z ( g x ) = rk g − rk g .Let p denote the projection g → g along g . Then p ( V ) = V / ( V ∩ g ) and our goal isto compute dim p ( V ) . By Eq. (4 · z ( g x ) ⊂ p ( V ) .For our further argument, some properties of d x H l ∈ g x are needed. It would be nice tohave d x H l = 0 for x as in (i) . Since this is not always the case, we need a trick.Let ˜ g = g ⊕ c be the central extension of g , where dim c = 1 . We extend the Z -gradingto ˜ g so that c ⊂ ˜ g and ϕ s to ˜ g by letting ϕ s | c = id . Take non-zero z ∈ c and γ ∈ c ∗ . Notethat ˜ g y + γ = ˜ g y for any y ∈ g ∗ . Therefore V ⊕ c = lim s → ϕ s (˜ g x + γ + sx ′ ) . Set ζ = x + γ . Then ζ ∈ ˜ g ∗ is still subregular and z ( ζ ) = 0 . Clearly, there is a linear combination H l = H l + c l − z d l − d l − H l − + . . . + c j z d l − d j H j + . . . + c z d l with c i ∈ k such that ∂ d l − ζ H l = d ζ H l = 0 . Note that z, H , . . . , H l − , H l freely generate Z S (˜ g ) . Let A ζ be the Mishchenko–Fomenko subalgebra of S (˜ g ) associated with ζ . By defini-tion, A ζ is generated by(4 · { z, ∂ k ζ H j ( j < l, k m j ) , ∂ k ζ H l (0 k m l − } . As the total number of these generators is b ( g ) and tr . deg A ζ = b (˜ g ) − b ( g ) [MY,Lemma 2.1], we see that A ζ is freely generated by them. Note that the set in (4 ·
6) containsa basis for the l -dimensional space z ( g x ) ⊕ c = z (˜ g ζ ) . Therefore, F = ∂ m l − ζ H l does notlie in S (cid:0) z ( g x ) ⊕ c (cid:1) . Since ∂ m l ζ H l = 0 , the polynomial F is a ˜ g ζ -invariant in S (˜ g ζ ) [MY,Lemma 1.5]. It is clear that σ ( F ) = F and therefore F ∈ S (˜ g ) ⊕ S ( g ) . Now ˜ g ζ = c ⊕ g x = c ⊕ z ( g x ) ⊕ sl . There is a standard basis { e, h, f } of this sl such that e, f ∈ g (cf.Example 2.2) and F ∈ (4 ef + h ) + S ( z ( g x ) ⊕ c ) .If x ′ ∈ g ∗ is generic enough, then ∂ x ′ F = η + ξ , where η ∈ z ( g x ) and ξ is a non-zeroelement in h e, f i k ⊂ g . Note that in this case ( m l − d ζ + sx ′ H l lies in s∂ x ′ F + s ˜ g . Further, ( m l − ϕ s ( d ζ + sx ′ H l ) = s ( η + ξ ) + m l − s ∂ x ′ ∂ d l − ζ H l + (terms of degree > w.r.t. s ) . Here ∂ x ′ ∂ d l − ζ H l ∈ g . Hence p ( V ) = z ( g x ) + k ( η + ξ ) = z ( g x ) ⊕ k ξ . The desired equality dim p ( V ) = rk g − rk g + 1 follows. (cid:3) Lemma 4.11.
Let y = x + x ′ be as in Lemma 4.10 with x ′ generic. Then the rank of the restrictionof π ( y ) to ker π ∞ ( y ) = g ⊕ g x is equal to dim( g ⊕ g x ) − rk g .Proof. Set U = ker π ∞ ( x ) = g ⊕ g x . Consider the maximal torus t = g x + k l − , where k l − is the centre of g x . The intersection t = g ∩ t = g x is a Cartan subalgebra of g . Further, g = t ⊕ m , where m is the t -stable complement of t in g . The torus t defines a finerdecomposition of U , namely U = m ⊕ t ⊕ ( g α ⊕ g − α ) ⊕ z where g ± α are root spaces and z ≃ k l − rk g .Choose a very particular x ′ , namely as x ′ = ξ α − ξ − α with non-zero root vectors ξ α ∈ g α , ξ − α ∈ g − α under the usual identification g ≃ g ∗ . Then the matrix of ( π ( y ) | U ) with respectto a basis for U adapted to the above finer decomposition has a block form with easy tounderstand blocks (see Fig. 1): • π ( y ) is non-degenerate on m ; • π ( y )( m , t ⊕ g α ⊕ g − α ) = 0 ; • π ( y )( t , g α ⊕ g − α ) = 0 . OISSON-COMMUTATIVE SUBALGEBRAS 21 det = 0 = 0 mt g α ⊕ g − α z ∗ Fig. 1.
The block structure of π ( y ) | U This is enough to see that the rank of π ( y ) on g ⊕ g α ⊕ g − α is at least dim g − rk g + 2 .Hence rk ( π ( y ) | U ) > dim U − rk g . This happens for one, not exactly generic x ′ , however,the generic value cannot be smaller and it also cannot be larger by Lemma A.3. (cid:3) The algebra S ( g ) g is contained in the Poisson centre of S ( g ) g . Let ˜ Z be the (Poisson-commutative) subalgebra of S ( g ) generated by Z and S ( g ) g . If H , . . . , H l is a g.g.s. for ( g , g ) such that σ ( H i ) = ± H i for each i , then ˜ Z is freely generated by ( H j ) ( i,d j − i ) with i = d j and a set of basic invariants ˜ H , . . . , ˜ H rk g ∈ S ( g ) g . In other words, a set of basicinvariants of ˜ Z is obtained from that of Z if one replaces the generators of r ( S ( g ) g ) withthe free generators of S ( g ) g . [Recall that ( H j ) ( d j , = 0 if and only if ε j = 1 and there are rk g such indices j , see Remark 3.8.] Theorem 4.12. (i)
The differentials of the algebraically independent generators of ˜ Z , chosenamong { ( H j ) ( i,d j − i ) } and { ˜ H j } , as above, are linearly independent on a big open subset of g ∗ . (ii) The algebra ˜ Z is a maximal Poisson-commutative subalgebra of S ( g ) g .Proof. (i) Assume that the differentials of the chosen algebraically independent generatorsof ˜ Z are linearly dependent at each point y of an irreducible divisor D ⊂ g ∗ . Since Z ⊂ ˜ Z ,the same holds for d ( H j ) ( i,d j − i ) . Then D ⊂ g ∗∞ , sing by Lemma 4.3 and D = D × g ∗ byLemma 4.7 (i) . Let y = x + x ′ be a generic element of D .Recall that d y Z stands for the linear span of d y F with F ∈ Z . We haved y Z = X t = ∞ d y Z t . According to Lemma 4.2, y ∈ g ∗ ( t ) , reg for each t = ∞ . Hence d y Z t = ker π t ( y ) whenever t = ∞ . By Lemma 4.7 (iii) , rk π ∞ ( y ) = rk π ∞ − . Combining Lemmas 4.7 (ii) , 4.10 (i) , and 4.11, we see that the rank of the restriction of π ( y ) to ker π ∞ ( y ) is equal to dim ker π ∞ ( y ) − rk g .Now Theorem A.4 applies and asserts that(4 · dim( d y Z / d y Z ∩ ker π ∞ ( y )) = 12 rk π ∞ ( y ) = 12 rk π ∞ − . By construction, V ⊂ d y Z ∩ ker π ∞ ( y ) . Recall that ker π ∞ ( y ) = g ⊕ g x . In view of (4 ·
7) andLemma 4.10 (iii) , we have dim( d y Z / d y Z ∩ g ) > rk π ∞ − rk g − rk g + 1) = 12 rk π ∞ + rk g − rk g . The differentials { d x ˜ H j | j = 1 , . . . , rk g } are linearly independent and lie in g . Hence dim d y Z > rk π ∞ + rk g − rk g + rk g = tr . deg Z . Now we see that the differentials of all the generators of ˜ Z are linearly independent at y .A contradiction!Part (ii) follows from (i) and Theorem 4.1. (cid:3) Remark.
In the jargon of completely integrable systems, which is used e.g. in [MF78, B91],Eq. (4 ·
7) means that the restriction of Z to the symplectic leaf of { , } ∞ at y is a “completefamily in involution”.5. F ANCY IDENTITIES FOR P OISSON TENSORS
In this section, the existence of a g.g.s. is of no importance, any indecomposable sym-metric pair ( g , g ) is admitted.Let ω be the standard n -form on g ∗ , where n = dim g , and let π be the Poisson tensor(bivector) of the Lie–Poisson bracket on g ∗ , see Section 1.1. Having a basis { e , . . . , e n } for g , one can write π = X i
By definition, d F ∈ Ω for each F ∈ S ( g ) . Take H , . . . , H l ∈ S ( g ) g . Thend H ∧ . . . ∧ d H l ∈ S ( g ) ⊗ ^ l g . At the same time, V ( n − l ) / π ∈ S ( g ) ⊗ V n − l g ∗ . The volume form ω defines a non-degeneratepairing between V l g and V n − l g . If u ∈ V l g and v ∈ V n − l g , then u ∧ v = c ω with c ∈ k .We write this as u ∧ vω = c and let uω be the element of ( V n − l g ) ∗ such that uω ( v ) = u ∧ vω .For any u ∈ S ( g ) ⊗ V l g , we let u ω be the corresponding element of S ( g ) ⊗ (cid:16)^ n − l g (cid:17) ∗ ∼ = S ( g ) ⊗ ^ n − l g ∗ . There is a Poisson interpretation of the Kostant regularity criterion [K63, Theorem 9], seealso Eq. (2 · Kostant identity (see [Y14]):d H ∧ · · · ∧ d H l ω = ^ ( n − l ) / π. The identity holds if the basic invariants are normalised correctly. It still holds if we apply ϕ − s to both sides.Suppose that σ is outer and σ ( H j ) = − H j . Thend ( H j ) ( d j − , ∈ S d j − ( g ) ⊗ ^ g | {z } I ⊕ g S d j − ( g ) ⊗ ^ g | {z } II . Let d H [1] j stand for the component of the first type. This is a -form on g ∗ . Suppose that σ ( H i ) = H i for i k and σ ( H i ) = − H i for i > k . Then k = rk g here, cf. Lemma 3.5.Let π g denote the Poisson tensor of g . Since g is reductive, V (dim g − rk g ) / π g is non-zero on the big open subset ( g ∗ ) reg . Proposition 5.1. If σ is an inner involution, then (5 ·
1) d ( H ) ( d , ∧ · · · ∧ d ( H l ) ( d l , ω = ^ (dim g ) / π ∞ ⊗ ^ (dim g − l ) / π g . If σ is an outer involution, then (5 ·
2) d ( H ) ( d , ∧ · · · ∧ d ( H k ) ( d k , ⊗ d H [1] k +1 ∧ · · · ∧ d H [1] l ω == V (dim g − l + k ) / π ∞ ⊗ V (dim g − k ) / π g . Proof.
The product d H ∧ · · · ∧ d H l is an l -form on g ∗ with polynomial coefficients. Amongthese coefficients, we are interested in those that have the maximal possible degree in g .It is not difficult to see that the degree in question is equal to b ( g ) − l = ( n − l ) / and thatthe corresponding l -form is either d ( H ) ( d , ∧ · · · ∧ d ( H l ) ( d l , in the inner case ord ( H ) ( d , ∧ · · · ∧ d ( H k ) ( d k , ⊗ d H [1] k +1 ∧ · · · ∧ d H [1] l in the outer case. For the first one, we haved ( H ) ( d , ∧ · · · ∧ d ( H l ) ( d l , ω ∈ S ( n − l ) / ( g ) ⊗ ^ dim g − l g ∗ ⊗ ^ dim g g ∗ . In case of an outer involution σ , the ( n − l ) -vector belongs to S ( n − l ) / ( g ) ⊗ ^ dim g − k g ∗ ⊗ ^ dim g − l + k g ∗ . The right hand side of the Kostant identity is a polyvector with polynomial coefficients ofdegree b ( g ) − l . If ξ ⊗ ( x ∧ y ) is a summand of π and ξ ∈ g , then either x, y ∈ g ∗ or x, y ∈ g ∗ .This justifies the right hand sides of (5 ·
1) and (5 · (cid:3) If σ is inner, then { ( H i ) ( d i , } are algebraically independent. Hence also the right handside of (5 ·
1) is nonzero. In particular, V dim g / π ∞ = 0 in complete accordance withLemma 2.1. If σ is outer, then V (dim g − l + k ) / π ∞ = 0 by Lemma 2.1. It is also clear that V (dim g − k ) / π g = 0 . Therefore the left hand side of (5 ·
2) is nonzero, too.Suppose that σ is inner. Then V (dim g ) / π ∞ = F · x ∧ . . . ∧ x dim g , where F ∈ S dim g ( g ) and { x j } is a basis for g ∗ . The zero set of F is exactly g ∗∞ , sing . Under the identifications g ≃ g ∗ , we have that F ( ξ ) = det(ad ( ξ ) | g ) for ξ ∈ g .Let { ˜ H , . . . , ˜ H l } be a set of suitably normalised basic g -invariants in S ( g ) . Then theysatisfy the Kostant identity with V (dim g − l ) / π g on the right hand side. In other words, if ω is the volume form on g ∗ , thend ˜ H ∧ · · · ∧ d ˜ H l ω = ^ (dim g − l ) / π g . Plugging this identity into (5 · Corollary 5.2.
Keep the assumption that σ is inner and regard ( H j ) ( d j , as an element of S ( g ) .Then d ( H ) ( d , ∧ . . . ∧ d ( H l ) ( d l , = F · d ˜ H ∧ . . . ∧ d ˜ H l , where F is the same as above. Hence the differentials { d ( H i ) ( d i , } are linearly dependent exactlyon the subset g ∗∞ , sing ∪ ( g ∗ ) sing . (cid:3) Proposition 5.3.
Let σ be an outer involution. Then ( H j ) ( d j − , , where k < j l , together witha basis { ξ , . . . , ξ dim g } of g freely generate Z ∞ . Further, there is Q ∈ S ( g ) such that Q · ξ ∧ . . . ∧ ξ dim g ∧ d H [1] k +1 ∧ . . . ∧ d H [1] l ω = ^ (dim g − l + k ) / π ∞ . If Q is regarded as a function on g ∗ , then its zero locus is the maximal divisor of g ∗ contained in g ∗∞ , sing . OISSON-COMMUTATIVE SUBALGEBRAS 25
Proof.
Set P = V dim g i =1 ξ i , P = V lj = k +1 d H [1] j , and P = P ∧ P . By the construction of H [1] j ,we have also P = P ∧ ( V lj = k +1 d ( H j ) ( d j − , ) .Take x ∈ g ∗ . If σ ( H j ) = − H j , then d x H j = d H [1] j ( x ) = d x ( H j ) ( d j − , ∈ g . If y = x + x ′ with x ∈ g ∗ , x ′ ∈ g ∗ , then P ( y ) = P ∧ P ( x ) . We wish to show that P ( y ) = 0 on a big opensubset of g ∗ . This is equivalent to the claim that P ( x ) = 0 on a big open subset of g ∗ .Assume that P is zero on an irreducible divisor X ⊂ g ∗ . By Lemma 4.2 (ii) , x ∈ ( g ∗ ) reg for a generic x ∈ X . If x ∈ g ∗ is regular in g , then the elements d x H i with i l arelinearly independent, see Eq. (2 · P ( x ) = 0 . Thus, dim g x > l + 2 for all x ∈ X and X × g ⊂ g ∗∞ , sing . This settles the claim for the cases, where r is surjective and g ∗∞ , sing doesnot contain divisors.Suppose that dim g ∗∞ , sing = n − . Let x ∈ X be generic. By Lemma 4.7, dim g x = l + 2 .Lemma 4.9 states that the elements d x H j with σ ( H j ) = − H j are linearly independent.Thereby P ( x ) = 0 . The claim is settled.By Theorem 4.1, the subalgebra of S ( g ) generated by ( H j ) ( d j − , with k < j l and ξ i with i dim g is algebraically closed. Since it lies inside Z ∞ and has the sametranscendence degree, dim g + ( l − k ) , it coincides with Z ∞ .Since P is non-zero on a big open subset, we have Q · ξ ∧ . . . ∧ ξ dim g ∧ d H [1] k +1 ∧ . . . ∧ d H [1] l ω = ^ (dim g − l + k ) / π ∞ for some Q ∈ S ( g ) , see e.g. [Y14, Section 2]. Since all the coefficients in the right hand sideare elements of S ( g ) , we have Q ∈ S ( g ) as well. (cid:3) Remark . If σ is inner, then tr . deg Z ∞ = dim g and it is easily seen that Z ∞ = S ( g ) assubalgebra of S ( g ( ∞ ) ) . In particular, Z ∞ is always a polynomial algebra.Combining Proposition 5.3 with Eq. (5 ·
2) and the Kostant identity for g , we obtain thefollowing assertion. Corollary 5.5.
Let ˜ H , . . . , ˜ H k be properly normalised basic g -invariants in S ( g ) . Then d ( H ) ( d , ∧ . . . ∧ d ( H k ) ( d k , = Q · d ˜ H ∧ . . . ∧ d ˜ H k in S ( g ) ⊗ V k g with the same Q as in Proposition 5.3. The differentials d ( H ) ( d , , . . . , d ( H k ) ( d k , are linearly dependent exactly on the union of ( g ∗ ) sing with the zero set of Q . (cid:3) Note that Q is the Pfaffian in the setting of Example 4.8.
6. F
URTHER DEVELOPMENTS AND POSSIBLE APPLICATIONS
We believe that this paper is the beginning of a long exciting journey. Several applicationsof our construction are already available and are presented below. Goals further aheadare stated as conjectures.6.1.
Quantum perspectives.
Let U ( g ) be the enveloping algebra of g . Given a Poisson-commutative subalgebra C ⊂ S ( g ) , it is natural to ask whether there exists a commutativesubalgebra b C ⊂ U ( g ) such that gr( b C ) = C . This question was posed by Vinberg for theMishchenko–Fomenko subalgebras [Vi91], and it is known nowadays as Vinberg’s problem .For the semisimple g , the first conceptual solution was obtained in [R06]. The r ˆole of thesymmetrisation map ̟ : S ( g ) → U ( g ) in that quantisation for the classical g is explainedin [MY]. Conjecture 6.1.
Suppose that there is a g.g.s. for σ . Let b Z be the subalgebra of U ( g ) generatedby ̟ (( H j ) ( i,d j − i ) ) with i l , i d i . Then b Z is commutative and gr( b Z ) = Z . For the symmetric pairs ( gl n + m , gl n ⊕ gl m ) , ( sp n + m ) , sp n ⊕ sp m ) , and ( so n + m , so n ⊕ so m ) ,there might be a connection between b Z and commutative subalgebras of Yangians ortwisted Yangians.The Yangian Y ( gl m ) is a deformation of the enveloping algebra U ( gl m [ z ]) of the currentalgebra gl m [ z ] given by explicit generators and relations. Then U ( gl m ) is a subalgebraof Y ( gl m ) . The facts on Yangians, which are used below, can be found in [M07], see inparticular Chapter 8 therein. The most relevant for us is the centraliser construction ofOlshanski [O91] and Molev–Olshanski [MO00]. For any n , there is an almost surjectivemap Ψ n : Y ( gl m ) → U ( gl n + m ) gl n , where the words “almost surjective” mean that U ( gl n + m ) gl n is generated by the image of Y ( gl m ) and U ( gl n ) gl n . It is known that, for a fixed m , T n > ker Ψ n = 0 . Question 6.2.
Is there a commutative subalgebra
B ⊂ Y ( gl m ) such that gr(Ψ n ( B )) together with Z S ( g ) generate ˜ Z ⊂ S ( gl m ⊕ gl n ) ? Let Y ( sp m ) ⊂ Y ( gl m ) be the twisted Yangian in the sense of G. Olshanski. Here U ( sp m ) ⊂ Y ( sp m ) and there is again an almost surjective map Ψ n : Y ( sp m ) → U ( sp n +2 m ) sp n . Then one can pose an analogous question. A similar situation occurs for Y ( so m ) ⊂ Y ( gl m ) and U ( so n + m ) with n even. OISSON-COMMUTATIVE SUBALGEBRAS 27
Any natural quantisation of Z has to provide a commutative subalgebra b Z ⊂ U ( g ) g . Byadding U ( g ) g one obtains the related quantisation b ˜ Z of ˜ Z . Let V be a finite-dimensionalsimple g -module. Then b ˜ Z acts on the subspace V n ⊂ V of the highest weight vectors of g . Conjecture 6.3.
Let b ˜ Z ⊂ U ( g ) be the subalgebra generated by ̟ (( H j ) ( i,d j − i ) ) with i l , i d i and by U ( g ) g . Then b ˜ Z acts on V n diagonalisably and with a simple spectrum. If Conjecture 6.3 is true, then the action of b ˜ Z produces a solution of the branching prob-lem g ↓ g . There are two renowned examples, where both conjectures are true. Example 6.4 (The Gelfand–Tsetlin construction [GT50, GT50’]) . Let ( g , g ) be one of thesymmetric pairs ( sl n +1 , gl n ) , ( so n +1 , so n ) . Then each H i has at most two nonzero bi-homogeneous components. To be more precise, the Pfaffian in the case of g = so l hasone nonzero component, and all the other generators have exactly two. It follows that ˜ Z is generated by S ( g ) g and S ( g ) g . The quantum analogue b ˜ Z is generated by U ( g ) g and U ( g ) g .For each irreducible finite-dimensional representation V of g , the restriction to g ismultiplicity free. Hence the action of b ˜ Z on V n has a simple spectrum.6.2. Classical applications.
Let us return to the Poisson side of the story.Suppose that there is a g.g.s. for σ . Although ˜ Z (or Z ) is not a maximal Poisson-commutative subalgebra of S ( g ) , it can be included into such a subalgebra in many natu-ral ways. Let C = k [ F , . . . , F b ( g ) ] be a maximal Poisson-commutative subalgebra of S ( g ) .Then necessary S ( g ) g ⊂ C . Suppose further that the F i ’s are homogeneous and their dif-ferentials are linearly independent on a big open subset of g ∗ . For instance, one can take C = A γ with γ ∈ ( g ) ∗ reg , see [PY08]. An easy calculation shows that alg h ˜ Z , Ci = alg h Z , Ci has b ( g ) generators. Indeed, ˜ Z (or Z ) has (dim g + rk g + rk g ) free generators. Then wereplace the generators sitting in S ( g ) (there are rk g of them) with the whole bunch ofgenerators of C . In this way, we obtain
12 (dim g + rk g + rk g ) − rk g + b ( g ) = b ( g ) generators { F i , h j | i b ( g ) , j b ( g ) − b ( g ) } . Furthermore, the differentials { d F i , d h j } are linearly independent at x ∈ g ∗ if and only if dim( d x ˜ Z + d x C ) = b ( g ) . Write x = x + x with x i ∈ g i and suppose that x ∈ ( g ∗ ) reg . Then ( d x ˜ Z ∩ d x C ) ⊂ g , π ( x )( g , d x ˜ Z ) = 0 , and hence d x ˜ Z ∩ d x C = g x . If in addition dim d x ˜ Z = tr . deg Z and dim d x C = b ( g ) , then dim( d x ˜ Z + d x C ) = b ( g ) .In view of Theorem 4.12 (i) , we can conclude that the differentials { d F i , d h j } are linearly independent on a big open subset of g ∗ . Thus, Theorem 4.1 applies and assures that alg h ˜ Z , Ci is a maximal Poisson-commutative subalgebra of g .Arguing inductively, one can produce a maximal Poisson-commutative subalgebra of S ( g ) from a chain of symmetric subalgebras g = g (0) ⊃ g (1) ⊃ g (2) ⊃ . . . ⊃ g ( m ) , where g ( m ) is Abelian and each symmetric pair ( g ( i ) , g ( i +1) ) has a g.g.s. Remark. (i)
For any simple Lie algebra g , there is an involution σ that has a g.g.s. [P07’,Sect. 6]. Therefore our construction of a maximal Poisson-commutative subalgebra of S ( g ) related to a chain of symmetric subalgebras works for any simple g . (ii) In [Vi91, § U ( sl n +1 ) related to the chain sl n +1 ⊃ gl n ⊃ gl n − ⊃ . . . ⊃ gl ⊃ gl , appears as one of these limit subalgebras, see also Example 6.4. The key point of Vin-berg’s construction is that the Poincar´e series of any limit subalgebra is the same as thatof A γ with γ ∈ g ∗ reg . With a few exceptions, our approach produces Poisson-commutativesubalgebras with different Poincar´e series. This can be illustrated by the chain so ⊃ so ⊃ so ⊕ so . Here the degrees of the generators of the related maximal Poisson-commutative subalge-bra are (4 , , , , , opposite to (4 , , , , , in the case of A γ .Another feature is that Z can be used for constructing a Poisson-commutative subalge-bra of S ( g ) . Let ( g , g ) be an arbitrary symmetric pair. If there is a g.g.s. for ( g , g ) , then weare able to consider both algebras, Z and ˜ Z . For η ∈ g ∗ , let Z η , ˜ Z η denote the restrictionsof Z and ˜ Z to g ∗ + η . By choosing η as the origin, we identify g ∗ + η with g ∗ . Then Z η and ˜ Z η are homogeneous subalgebras of S ( g ) . Moreover, they Poisson-commute with g η . Lemma 6.5.
The subalgebras Z η and ˜ Z η are Poisson-commutative.Proof. Take
H, F ∈ Z or H, F ∈ ˜ Z and x ∈ g ∗ . Let h and f be the restrictions of H, F to g ∗ + η . Then d x + η H = d x h + ξ , d x + η F = d x f + ν , where ξ , ν ∈ g . Set ξ = d x h , ν = d x f .Our goal is to show that x ([ ξ , ν ]) = 0 .Since H and F commute w.r.t. any bracket { , } t with t ∈ P , we have in particular x ([ ξ , ν ]) = 0 , as well as ( x + η )([ ξ + ξ , ν + ν ]) = 0 . Both are also g -invariants. Therefore ( x + η )([ ξ , ν + ν ]) = 0 , x + η )([ ν , ξ + ξ ]) = x ([ ν , ξ ]) + η ([ ν , ξ ]) . Now x + η )([ ξ , ν + ν ]) = η ([ ξ , ν ]) and it is clear that x ([ ξ , ν ]) = 0 . (cid:3) OISSON-COMMUTATIVE SUBALGEBRAS 29
Remark . Let ( g , g ) = ( sl n , so n ) . The corresponding involution σ is of maximal rank andany set of generators H , . . . , H l ∈ S ( g ) g is a g.g.s. for σ . The related Poisson-commutativesubalgebra Z appeared, in a way, in work of Manakov [M76]. He stated that the restrictionof Z to g + η with η ∈ g is a Poisson-commutative subalgebra of S ( g ) of the maximalpossible transcendence degree, which is b ( g ) . Below we present a connection betweenhis results and ours. We are grateful to E.B. Vinberg for bringing our attention to the factthat Manakov’s construction involves an involution.Let c ⊂ g be a Cartan subspace. If η ∈ c is generic, then l := g η is reductive andit is also the centraliser of c in g . There are well-known equalities: dim g − dim g =dim l − dim c and rk l = rk g − dim c . Theorem 6.7.
For almost all η ∈ c , we have (i) tr . deg Z η = b ( g ) − b ( l ) + rk l ; (ii) if there is a g.g.s. for σ , then ˜ Z η is a maximal Poisson-commutative subalgebra of S ( g ) l .Besides, if l is Abelian, then ˜ Z η is a maximal Poisson-commutative subalgebra of S ( g ) .Proof. Suppose that η is generic enough. Then • dim d y Z = (dim g + rk g + rk g ) for y in a dense open subset of g + η , and • dim d y ˜ Z = (dim g + rk g + rk g ) for y in a big open subset of g + η .Note that the subspaces d y Z and d y ˜ Z are orthogonal to g w.r.t. the bilinear form π ( y ) = y ([ , ]) . Hence for both of them, the intersection with g has dimension at most dim c . Itis easily seen that actually dim( d y ˜ Z ∩ g ) = dim c . Furthermore,d y Z η ≃ d y Z / ( d y Z ∩ g ) and the same formula holds for ˜ Z . Therefore tr . deg Z η >
12 (dim g + rk g + rk g ) − dim c = 12 (dim g − dim c + rk g − dim c + rk g )= 12 (dim g − dim l + rk l + rk g ) = b ( g ) − b ( l ) + rk l . Since Z η ⊂ S ( g ) l and rk l = ind l , the transcendence degree of Z η cannot be larger than b ( g ) − b ( l ) + rk l by [MY, Prop. 1.1]. Because ˜ Z in an algebraic extension of Z , we alsohave tr . deg ˜ Z η = tr . deg Z η .The difference tr . deg ˜ Z − tr . deg ˜ Z η is equal to dim c . We consider the algebra ˜ Z only ifthere is a g.g.s for σ . In that case the map r is surjective and therefore for certain members H i of the g.g.s. we have H • i ∈ S ( g ) [P07’]. The number of such element is equal to dim c ,and they restrict to constants on g ∗ + η .We see that ˜ Z η is freely generated by ˜ H , . . . , ˜ H rk g ∈ S ( g ) g and the restrictions to η + g of ( H j ) ( i,d j − i ) with < i < d j . Moreover, the differentials of these generators are lin-early independent on a big open subset. According to Theorem 4.1, ˜ Z η is an algebraically closed subalgebra of S ( g ) . By a standard argument, it is a maximal Poisson-commutativesubalgebra of S ( g ) l .Suppose that l is Abelian. Then dim l = rk l and ˜ Z η is a Poisson-commutative subalgebraof S ( g ) of the maximal possible transcendence degree. Here ˜ Z η is maximal in S ( g ) . (cid:3) The statements of Theorem 6.7 are not entirely satisfactory. It would be nice to have anexplicit description of η such that the results hold. In the original setting of Manakov, l istrivial and the equality tr . deg Z η = b ( g ) holds for each regular η ∈ c , see [GDI]. But amore precise assertion requires a further analysis of g ∗ ( t ) , sing and we prefer to postpone it.A PPENDIX
A. O
N PENCILS OF SKEW - SYMMETRIC FORMS
Here we gather some general facts concerning skew-symmetric bilinear forms. Let P bea two-dimensional vector space of (possibly degenerate) skew-symmetric bilinear formson a finite-dimensional vector space V . Set m = max A ∈ P rk A , and let P reg ⊂ P be the set ofall forms of rank m . Then P reg is a conical open subset of P . For each A ∈ P , let ker A ⊂ V be the kernel of A . Our object of interest is the subspace L := P A ∈ P reg ker A . Lemma A.1 ([PY08, Appendix]) . If Ω is a non-empty open subset of P reg , then P A ∈ Ω ker A = L . Corollary A.2.
For all
A, B ∈ P \ { } , we have A (ker B, L ) = 0 and therefore A ( L, L ) = 0 .Proof.
Clearly, the equality A (ker B, L ) = 0 holds if B is a scalar multiple of A . If not, thenwe consider L b := ker( A + bB ) for b ∈ k . Here A (ker B, L b ) = ( A + bB )(ker B, L b ) − bB (ker B, L b ) = 0 . By Lemma A.1, there is an open subset O ⊂ k such that L is spanned by { L b | b ∈ O } .Hence A (ker B, L ) = A (ker B, P b ∈ O L b ) = 0 . (cid:3) Suppose that C ∈ P \ P reg . Then U = ker C may not be a subspace of L . Take A ∈ P \ { } that is not proportional to C and restrict it to U . The resulting skew-symmetric form on U does not change if we replace A with any A + b C , where b ∈ k . Lemma A.3.
Let C , A , and U be as above. Then rk ( A | U ) dim U − (dim V − m ) .Proof. By Corollary A.2, we have A ( U, L ) = 0 . Set r = dim V − m . Because P is irreducible, P reg = P and there is a curve τ : k × → P reg such that lim t → τ ( t ) = C . Hence lim t → (ker τ ( t )) ⊂ ker C, OISSON-COMMUTATIVE SUBALGEBRAS 31 where the limit is taken in the Grassmannian of the r -dimensional subspaces of V . Set U := lim t → (ker τ ( t )) . If t = 0 , then ker τ ( t ) ⊂ L and A (ker τ ( t ) , U ) = 0 . Hence also A ( U , U ) = 0 and U ⊂ ker( A | U ) . It remains to notice that dim U = r . (cid:3) Remark.
Lemma A.3 implies Vinberg’s inequality: if q is Lie algebra, then ind q γ > ind q for any γ ∈ q ∗ , see [P03, Cor. 1.7]. Theorem A.4.
Suppose that P \ P reg = k C with C = 0 and U = ker C . Keep the notationof Lemma A.3 and suppose further that rk ( A | U ) = dim U − dim V + m . Then dim( L ∩ U ) =dim V − m and dim L = (dim V − m ) + (dim V − dim U ) .Proof. Let B ∈ P reg be non-proportional to A . Given A, B ∈ P reg , there is the so-called Jordan–Kronecker canonical form of A and B , see [T91]. Namely, V = V ⊕ · · · ⊕ V d , where A ( V i , V j ) = 0 = B ( V i , V j ) for i = j , and accordingly, A = P A i and B = P B i . Thereare two possibilities for ( A i , B i ) , one obtains either a Kronecker or a
Jordan block here, seefigures below. Assume that dim V i > for each i . A i B i A Jordan block ( λ i ∈ k ) : J ( λ i ) − J ⊤ ( λ i ) ! − II ! , a Kroneckerblock : . . . . . . − . . .. . . − . . . . . . − . . .. . . − , where J ( λ i ) = λ i λ i . . .. . . λ i . In general, there can occur “Jordan blocks with λ i = ∞ ”, but this is not the case here, since B ∈ P is assumed to be regular.Note that if V i gives rise to a Jordan block, then dim V i is even and both A i and B i arenon-degenerate on V i . For a Kronecker block, dim V i = 2 k i + 1 , rk A i = 2 k i = rk B i and thesame holds for every non-zero linear combination of A i and B i . There is a unique λ ∈ k \ { } such that C = A + λB . This λ can be determined as the rootof the equation det( A i + λB i ) = 0 for any Jordan block ( A i , B i ) . This readily follows fromthe uniqueness of the singular line k C ⊂ P . On the other hand, the above matrices showthat the root corresponding to ( A i , B i ) is λ i . Therefore, all λ i ’s are equal and coincide with λ . Let us assume that V i defines a Kronecker block if and only if i d ′ . Then neces-sarily d ′ = dim V − m . Let ker( A i + bB i ) ⊂ V i be the kernel of the bilinear form A i + bB i .Then L = d ′ M i =1 X b : A + bB ∈ P reg ker( A i + bB i ) =: d ′ M i =1 L i . It follows from the above matrix form of a Kronecker block that dim L i = k i + 1 , cf. also[PY08, Appendix].Set C i = A i + λB i for each i ∈ { , , . . . , d } . It is a bilinear form on V i . • If i d ′ , then dim ker C i = 1 . Therefore ker C i ⊂ L i and dim(ker C ∩ L ) = d ′ . • If i > d ′ , then dim ker C i = 2 .Hence dim U = 2( d − d ′ ) + d ′ = 2 d − d ′ . Since U = L di =1 ker C i and the spaces { ker C i } are pairwise orthogonal w.r.t. any form in P , we have A (ker C j , U ) = 0 for j d ′ . Hencethe condition rk ( A | U ) = dim U − dim V + m implies that A i is non-degenerate on ker C i for any i > d ′ . The explicit matrix form of a Jordan block shows that ker C i is spannedby two middle basis vectors of V i . Therefore, A i is non-degenerate on ker C i if and only if dim V i = 2 , and hence C i = 0 .Summing up, we obtain dim L = d ′ X i =1 ( k i + 1) = d ′ + d ′ X i =1 rk C i = d ′ + 12 rk C = (dim V − m ) + 12 (dim V − dim U ) . This completes the proof. (cid:3) R EFERENCES [B91] A. B
OLSINOV . Commutative families of functions related to consistent Poisson brackets,
Acta Appl.Math. , , no. 3 (1991), 253–274.[BB02] A. B OLSINOV and A. B
ORISOV . Compatible Poisson brackets on Lie algebras. (Russian)
Mat. Za-metki (2002), no. 1, 11–34; translation in Math. Notes (2002), no. 1-2, 10–30.[BK79] W. B ORHO and H. K
RAFT . ¨Uber Bahnen und deren Deformationen bei linearen Aktionen reduk-tiver Gruppen,
Comment. Math. Helv. (1979), no. 1, 61–104.[DZ05] J.-P. D UFOUR and N.T. Z
UNG . “Poisson structures and their normal forms”. Progress in Mathe-matics, . Birkh¨auser Verlag, Basel, 2005.[FFR] B. F
EIGIN , E. F
RENKEL , and L. R
YBNIKOV . Opers with irregular singularity and spectra of the shiftof argument subalgebra,
Duke Math. J. , (2010), no. 2, 337–363. OISSON-COMMUTATIVE SUBALGEBRAS 33 [GDI] B. G
AJI ´ C , V. D RAGOVI ´ C , and B. J OVANOVI ´ C . On the completeness of the Manakov integrals, J.Math. Sci. , :6 (2017), 675–685.[GT50] I.M. G ELFAND and M.L. Tsetlin. Finite-dimensional representations of the group of unimod-ular matrices. (Russian)
Doklady Akad. Nauk SSSR (N.S.) (1950), 825–828. English transl. in:I.M. G ELFAND , Collected Papers, vol. II, Springer-Verlag, Berlin, 1988, pp. 653–656.[GT50’] I.M. G
ELFAND and M.L. Tsetlin. Finite-dimensional representations of groups of orthogonal matri-ces. (Russian)
Doklady Akad. Nauk SSSR (N.S.) (1950), 1017–1020. English transl. in: I.M. Gelfand,Collected Papers, vol. II, Springer-Verlag, Berlin, 1988, pp. 657–661.[K63] B. K OSTANT . Lie group representations on polynomial rings,
Amer. J. Math. , (1963), 327–404.[M76] S.V. M ANAKOV . Note on the integration of Euler’s equations of the dynamics of an n -dimensionalrigid body, Funct. Anal. Appl. , (1976), no.4, 93–94 (in Russian).[MF78] A.S. M ISHCHENKO and A.T. F
OMENKO . Euler equation on finite-dimensional Lie groups,
Math.USSR-Izv. (1978), 371–389.[M07] A. M OLEV , “Yangians and Classical Lie Algebras”. Mathematical Surveys and Monographs, ,American Mathematical Society, Providence, RI, 2007.[MO00] A. M
OLEV and G. O
LSHANSKI . Centralizer construction for twisted Yangians,
Selecta Math. , (2000), no. 3, 269–317.[MY] A. M OLEV and O. Y
AKIMOVA . Quantisation and nilpotent limits of Mishchenko–Fomenko subal-gebras, ( arxiv:1711.03917v1 [math.RT] , 32 pp.)[O91] G.I. O
LSHANSKI . Representations of infinite-dimensional classical groups, limits of envelopingalgebras, and Yangians, In:
Topics in Representation Theory , Advances in Soviet Math. (A.A. Kirilloved.), vol. , AMS, Providence RI, 1991, 1–66.[P03] D. P ANYUSHEV . The index of a Lie algebra, the centraliser of a nilpotent element, and the nor-maliser of the centraliser,
Math. Proc. Camb. Phil. Soc. , , Part 1 (2003), 41–59.[P07] D. P ANYUSHEV . Semi-direct products of Lie algebras and their invariants,
Publ. RIMS , , no. 4(2007), 1199–1257.[P07’] D. P ANYUSHEV . On the coadjoint representation of Z -contractions of reductive Lie algebras, Adv.Math. , (2007), 380–404.[PPY] D. P ANYUSHEV , A. P
REMET and O. Y
AKIMOVA . On symmetric invariants of centralisers in reduc-tive Lie algebras,
J. Algebra , (2007), 343–391.[PY08] D. P ANYUSHEV and O. Y
AKIMOVA . The argument shift method and maximal commutative subal-gebras of Poisson algebras,
Math. Res. Letters , , no. 2 (2008), 239–249.[PY12] D. P ANYUSHEV and O. Y
AKIMOVA . On a remarkable contraction of semisimple Lie algebras,
An-nales Inst. Fourier (Grenoble), , no. 6 (2012), 2053–2068.[R06] L. G. R YBNIKOV , The shift of invariants method and the Gaudin model,
Funct. Anal. Appl. (2006),188–199.[Sl80] P. S LODOWY . ”Simple singularities and simple algebraic groups”, Lect. Notes Math. , Berlin:Springer, 1980.[S74] T.A. S
PRINGER . Regular elements of finite reflection groups,
Invent. Math. (1974), 159–198.[T91] R.C. T HOMPSON . Pencils of complex and real symmetric and skew matrices,
Linear Algebra and itsAppl. , (1991), 323–371.[V68] V.S. V ARADARAJAN . On the ring of invariant polynomials on a semisimple Lie algebra,
Amer. J.Math. , (1968), 308–317. [Vi91] E.B. V INBERG . Some commutative subalgebras of a universal enveloping algebra,
Math. USSR-Izv. (1991), 1–22.[VP89] З.B. Vinberg, V.L. Popov . “Teori(cid:31) Invariantov” , V: Sovrem. probl. matematiki. Fun-damentalьnye napravl., t. 55, str.
Moskva: VINITI
OPOV and E.B. V
INBERG . “Invariant theory”, In:
Algebraic Geometry IV (Encyclopae-dia Math. Sci., vol. 55, pp.123–284) Berlin Heidelberg New York: Springer 1994.[Y14] O. Y
AKIMOVA . One-parameter contractions of Lie-Poisson brackets,
J. Eur. Math. Soc. , (2014),387–407.[Y17] O. Y AKIMOVA . Symmetric invariants of Z -contractions and other semi-direct products, Int. Math.Res. Notices , (2017) 2017 (6): 1674–1716.(D. Panyushev) I
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