Center of U(n), Cascade of Orthogonal Roots and a Construction of Lipsman–Wolf Bertram Kostant Dedicated to Joe, a special friend and valued colleague 2 Abstract. Let G be a complex simply-connected semisimple Lie group and let g = 1 0 LieG. Let g = n− +h+n be a triangular decomposition of g. One readily has that 2 n CentU(n) is isomorphic to the ring S(n) of symmetric invariants. Using the cascade n n B of strongly orthogonal roots, some time ago we proved (see [K]) that S(n) is a a J polynomial ring C[ξ ,...,ξ ] where m is the cardinality of B. The authors in [LW] 1 m 1 introduce a very nice representation-theoretic method for the construction of certain 2 n elements in S(n) . A key lemma in [LW] is incorrect but the idea is in fact valid. In ] n T our paper here we modify the construction so as to yield these elements in S(n) and R use the [LW] result to prove a theorem of Tony Joseph. . h t Key words: cascade of orthogonal roots, Borel subgroups, nilpotent coadjoint action. a m MSC (2010) codes: representation theory, invariant theory. [ 1. Introduction 1 v 4 1.1. Let g be a complex semisimple Lie algebra and let 9 4 4 g = n +h+n − . 1 0 2 be a fixed triangular decomposition of g. Let ∆ ⊂ h∗ be the set of h roots in g. The 1 Killing form (x,y) on g, denoted by K, induces a nonsingular bilinear form (µ,ν) on : v h∗. For each ϕ ∈ ∆ let e ∈ g be a corresponding root vector. The root vectors can i ϕ X and will be chosen so that (e ,e ) = 1 for all roots ϕ. ϕ −ϕ r a If s ⊂ g is any subspace stable under adh let ∆(s) = {ϕ ∈ ∆ | e ∈ s}. ϕ The set ∆ of positive roots is then chosen so that ∆ = ∆(n), and one puts ∆ = + + − −∆ . If s is a Lie subalgebra, then S(s) and U(s) are respectively the symmetric and + enveloping algebras of s. Our concern here is with the case where s = n. Let b = h+n so that b is a Borel subalgebra of g. Let G be a Lie group such that LieG = g and let H,N,B be Lie subgroups corresponding, respectively, to h,n,b. Then S(n) is a B-module since B = HN normalizes N. Let m be the maximal number of strongly orthogonal roots. Then we proved the following some time ago, generalizing a result of Dixmier (case where g is of type A ), ℓ 1 Theorem A. There exists ξ ∈ S(n)N,i = 1,...,m, so that i S(n)N = C[ξ ,...,ξ ] 1 m is a polynomial ring in m-generators. Furthermore, S(n)N ∼= CentU(n) so that one has a similar statement for CentU(n). We will present an algebraic-geometric proof of a much stronger statement than Theorem A and relate it to a representation-theoretic construction, due to Lipsman– Wolf, of certain elements in S(n)N. See [K], [LW]. A key tool is the cascade B = {β ,...,β } of orthogonal roots which will now be defined. 1 m 1.2. Let Π ⊂ ∆ be the set of simple positive roots. For any ϕ ∈ ∆ and + + α ∈ Π there exists a nonnegative integer n (ϕ) such that α ϕ = X nα(ϕ)α. α∈Π Let Π(ϕ) = {α ∈ Π | n (ϕ) > 0}. α Then Π(ϕ) is a connected subset of Π and hence defines a simple Lie subalgebra g(ϕ) of g. We will say that ϕ is locally high if ϕ is the highest root of g(ϕ). Obviously the highest roots of all the simple components of g are locally high. Remark 1. If g is of type A , but only in this case, are all ϕ ∈ ∆ locally high. ℓ + Let ϕ ∈ ∆ be locally high and let + Π(ϕ)o = {α ∈ Π(ϕ) | (α,ϕ) = 0}; let g(ϕ)o be the semisimple Lie algebra having Π(ϕ)o as its set of simple roots. We ′ ′ will then say that a root ϕ ∈ ∆ is an offspring of ϕ if ϕ is the highest root of a + simple component of g(ϕ)o. Remark 2. One notes that an offspring of a locally high root ϕ is again locally high and that it is strongly orthogonal to ϕ. A sequence of positive roots ′ ′ C = {β ,...,β } 1 k 2 will be called a cascade chain if β′ is a highest root of a simple component of g, and 1 ′ ′ if 1 < j ≤ k, then β is an offspring of β . Now let B be the set of all positive roots j j−1 β which are members of some cascade chain. Let W be the Weyl of (h,g). Theorem 1. The cardinality of B is m and B = {β ,...,β } 1 m is a maximal set of strongly orthogonal roots. Furthermore, if s is the W-reflection βi of h corresponding to β , then the long element w of W may be given by i o w = s ···s . (1.1) o β1 βm B is the cascade of orthogonal roots. 1.3. One has the vector space direct sum g = n ⊕b. (1.2.) − Let P : g → n be the projection defined by (1.2). Since b is the K-orthogonal subspace to n in g we may identify n with the dual space n∗ to n, so that for v ∈ n and x ∈ n, − − one has hv,xi = (v,x). The coadjoint action of N on n may then be given so that if − u ∈ N, then on n − Coadu = P Adu. (1.3.) In fact, using (1.2) the coadjoint action of N on n extends to an action of B on n , − − so that if b ∈ B and v ∈ n , one has b·v = P Adb(v). In addition we can regard S(n) − as the ring of polynomial functions on n . Since B normalizes N the natural action − of N on S(n) extends to an action of B on S(n) where if f ∈ S(n), b ∈ B, and v ∈ n , − one has (b·f)(v) = f(b−1 ·v). (1.4) Recalling m = cardB, let r be the commutative m-dimensional subalgebra of n spanned by e for β ∈ B and let R ⊂ N be the commutative unipotent subgroup β corresponding to r. In the dual space let r ⊂ n be the span of e for β ∈ B. For − − −β any z ∈ r , β ∈ B, let a (z) ∈ C be defined so that − β z = X aβ(z)e−β, (1.5) β∈B and let × r = {τ ∈ r | a (τ) 6= 0, ∀β ∈ B}. − − β 3 As an algebraic subvariety of n clearly − r× ∼= (C×)m. (1.6) − Also for any z ∈ n let O be the N-coadjoint orbit containing z. Let N ⊂ N − z z be the coadjoint isotropy subgroup at z and let n = LieN . Since the action is z z algebraic, N is connected and hence as N-spaces z ∼ O = N/N . (1.7) z z × Theorem 2. Let τ ∈ r . Then (independent of τ) N = R so that (1.7) becomes − τ ∼ O = N/R. (1.8) τ In particular dimO = dimn−m (1.9) τ and O is a maximal dimensional coadjoint orbit of N. τ Now consider the action of B on n . In particular consider the action of H on − n . Obviously − r× ∼= (C×)m, (1.10) − × andfurthermore r isanorbitofH. In additionH permutes themaximalN-coadjoint − × orbits O , τ ∈ r . More precisely, τ − × Theorem 3. For any a ∈ H and τ ∈ r , one has − a·O = O . (1.11) τ a·τ 1.4. If V is an affine variety, A(V) will denote its corresponding affine ring of functions. Note that S(n) = A(n ). Let Q(n ) be the quotient field of S(n). − − Theorem 4. There exists a unique Zariski open nonemtpy orbit X of B on n . − In particular X = n . (1.12) − Furthermore X is an affine variety so that S(n ) ⊂ A(X) ⊂ Q(n ). (1.13) − − × Moreover n ⊂ X, and in fact one has a disjoint union − X = ⊔ O (1.14) τ∈r× τ − 4 so that all N-coadjoint orbits in X are maximal and isomorphic to N/R. Let Λ ⊂ h∗ be the H-weight lattice and let Λ ⊂ Λ be the root lattice. Let ad Λ ⊂ Λ be the sublattice generated by the cascade B. Since the elements of B are B ad mutually orthogonal note that Λ = ⊕ Z β (1.15) B β∈B is a free Z-module of rank m. If M is an H-module, let Λ(M) ⊂ Λ be the set of H-weights occurring in M. Note that if M is a B-module, then MN is still an H-module. Recalling the definition × of r and (1.6), note that − × Λ(A(r )) = Λ − B (1.16) and each weight occurs with multiplicity 1. We can now give more information about X and its affine ring A(X). Define a × B action on r by extending the H-action so that N operates trivially. Next define a − B-actionon N/R, extending the N-action by letting H operate by conjugation, noting that H normalizes both N and R. With these structures and the original action on X, we have the following. Theorem 5. One has a B-isomorphism × X → N/R×r − of affine varieties so that as B-modules A(X) ∼= A(N/R)⊗A(r×). (1.17) − Furthermore, taking N-invariants, one has an H-module isomorphism A(X)N ∼= A(r×) (1.18) − so that, by (1.16), Λ(A(X)N) = Λ (1.19) B and each H-weight occurs with multiplicity 1. Recalling (1.13) one has the N-invariant inclusions S(n)N ⊂ A(X)N ⊂ Q(n )N (1.20) − of H-modules so that Λ(S(n)N) ⊂ Λ(A(X)N) ⊂ Λ(Q(n )N). (1.21) − 5 But since S(n) is a unique factorization domain, any u ∈ Q(n ) may be uniquely − written, up to scalar multiplication as u = f/g (1.22) where f and g are prime to one another. Furthermore, it is then immediate (since N is unipotent) that if u is N-invariant, one has f,g ∈ S(n)N. If, in addition, u is an H-weight vector, the same is true of f and g so that, using Theorem 5, one readily concludes the following. Theorem 6. Every H-weight in Λ(S(n)N) occurs with multiplicity 1 in S(n)N. In fact Λ(Q(n ) = Λ and every weight γ in Λ(Q(n ) occurs with multiplicity 1 in − B − Q(n )N and is of the form − γ = ν −µ (1.23) where µ,ν ∈ Λ(S(n)N). For any γ ∈ Λ let ξ ∈ QN be the unique (up to scalar multiplication) H- B γ n − weight vector with weight γ. Thus if γ ∈ Λ , we may uniquely write (up to scalar B multiplication ξ = ξ /ξ (1.24) γ ν µ where µ,ν ∈ Λ(S(n)N) and ξ and ξ are prime to one another. Let ν µ Λ = {λ ∈ Λ | λ be a dominant weight}. dom Remark 3. By the multiplicity 1-condition note that if ν ∈ Λ(S(n)N), then ξ is necessarily a homogeneous polynomial. Define degν so that ξ ∈ Sdegν(n). ν ν Furthermore, clearly ξ is then a highest weight vector of an irreducible g-module in ν Sdegν(g) and in particular ν ∈ Λ . That is, dom Λ(S(n)N) ⊂ Λ ∩Λ . (1.25) dom B 1.5. If ν ∈ Λ(S(n)N), it follows easily from the multiplicity-1 condition and the uniqueness of prime factorization that all the prime factors of ξ are again weight ν vectors in S(n)N. Let P = {ν ∈ Λ(S(n)N) | ξ be a prime polynomial in S(n)N}. (1.26) ν We can then readily prove Theorem 7. One has cardP = m where, we recall m = cardB, so that we can write P = {µ ,...,µ }. (1.27) 1 m 6 Furthermore the weights µ in P are linearly independent and the set P of prime i polynomials, ξ , i = 1,...,m, are algebraically independent. In addition, one has a µi bijection Λ(S(n)N) → (N)m, ν 7→ (d (ν),...,d (ν)) (1.28) 1 m such that, writing d = d (ν), up to scalar multiplication, i i ξ = ξd1 ···ξdm (1.29) ν µ1 µm and (1.29) is the prime factorization of ξ for any ν ∈ Λ(S(n)N. Finally, ν S(n)N = C[ξ ,...,ξ ] (1.30) µ1 µm so that S(n)N is a polynomial ring in m-generators. Remark 4. One may readily extend part of Theorem 7 to weight vectors in Q(n)N. In fact one easily establishes that there is a bijection Λ(Q(n )N) → (Z)m, γ 7→ (e (γ),...,e (γ)) − 1 m so that writing e (γ) = e one has i i ξ = ξe1 ···ξem. (1.31) γ µ1 µm Separating the e into positive and negative sets yields ξ and ξ of (1.24). i ν µ 1.6. Let ν ∈ Λ(S(n)N). Then by Theorem 6 and (1.25) one has ν ∈ Λ ∩Λ B dom so that there exists nonnegative integers b , β ∈ B such that β ν = Xbββ. (1.31a) β∈B Remark 5. The nonnegativity follows from dominance since one must have (ν,β) ≥ 0 for β ∈ B. We wish to prove Theorem 8. One has X bβ = degν, (1.32) β∈B 7 × and as a function ξ | r does not vanish identically and up to a scalar ν − ξν | r×− = Y ebββ. (1.33) β∈B Proof. Let Sdegν(n)(ν) be the ν weight space in Sdegν(n). It does not reduce to zero since ξ ∈ Sdegν(n)(ν). Let Γ be the set of all maps γ : ∆ → N such that ν + X γ(ϕ) = degν ϕ∈∆+ (1.34) X γ(ϕ)ϕ = ν. ϕ∈∆+ Then if eγ = Y eγ(ϕ), ϕ ϕ∈∆+ the set {eγ | γ ∈ Γ} is clearly a basis of Sdegν(n)(ν) and consequently unique scalars s exist so that γ ξν = Xsγeγ. (1.35) γ∈Γ But by Theorem 5 there exists x ∈ X such that ξ (x) 6= 0. However since X is ν × × B-homogeneous, the H-orbit r is contained in X and there exists t ∈ r such that − − x = u · t for some u ∈ N. But since ξ is N-invariant one has ξ (t) 6= 0. But from ν ν (1.34) this implies that Xsγeγ(t) 6= 0. (1.36) γ∈Γ But eγ(t) = 0 for any γ ∈ Γ such that γ(ϕ) 6= 0 for ϕ ∈/ B. Thus there exists γ′ ∈ Γ such that ′ γ (ϕ) = 0 for all ϕ ∈/ B and eγ′(t) 6= 0. (1.37) ′ But by the independence of B one has that γ is unique and hence one must have ′ γ (β) = b . A similar argument yields (1.33). QED β 2. A representation-theoretic construction, due to Lipsman–Wolf, of certain elements in S(n)N 2.1. Let λ ∈ Λ and let V be a finite-dimensional irreducible g-module with dom λ highest weightλ. Then, correspondingly, V is a U(g)-module with respect to a surjec- λ tion π : U(g) → EndV . Let 0 6= v ∈ V be a highest weight vector. Also let V∗ be λ λ λ λ λ 8 the contragredient dual g-module. The pairing of V and V∗ is denoted by hv,zi with λ λ ∗ v ∈ V and z ∈ V . (We will use this pairing notation throughout in other contexts.) λ λ But as one knows V∗ is g-irreducible with highest weight λ∗ ∈ Λ given by λ dom ∗ λ = −w λ. (2.1) o But then by (1.1) and the mutual orthogonality of roots in the cascade ∗ ∨ −λ = λ− Xλ(β )β. β∈B That is ∗ ∨ λ+λ = Xλ(β )β (2.2) β∈B and hence ∗ λ+λ ∈ Λ ∩Λ . (2.3) B dom On the other hand, regarding U(g)∗ as a g-module (dualizing the adjoint action on U(g)) it is clear that if f ∈ U(g)∗ defined by putting, for u ∈ U(g), f(u) = huvλ,zλ∗i, (2.4) then f is n-invariant and (2.5) f is an h weight vector of weight λ+λ∗. Now it is true (as will be seen below) that λ+λ∗ ∈ Λ(S(n)N). It is the idea of Lipsman–Wolf to construct ξλ+λ∗ using f. The method in [L−W] is to symmetrize f and restrict to S(n). However Lemma 3.7 in [L-W] is incorrect (one readily finds counterexamples). But the idea is correct. One must modify f suitably and this we will do in the next section. 2.2. Assume s is a finite-dimensional Lie algebra. Let U (s), j = 1,..., be the j standard filtration of the enveloping algebra U(s). Let 0 6= f ∈ U(s)∗. We will say that k ≥ −1 is the codegree of f if k is maximal such that f vanishes on U (s). But k−1 then if k is the codegree of f and if x ∈ s, i = 1,...,k, and σ is any permutation of i {1,...,k}, then (x ···x −x ···x ) ∈ U (s) so that 1 k σ(1) σ(k) k−1 f(x ···x ) = f(x ···x ). (2.6) 1 k σ(1) σ(k) But this readily implies that there exists a unique element f ∈ Sk(s) such that for (k) any u ∈ U (s) one has k f (u) = f(u) (2.7) (k) e 9 where u ∈ Sk(s) is the image of u under the Birkhoff–Witt surjection U (s) → Sk(s). k Noew let s = g and let f be given by (2.4). Let k be the codegree of f. Identify g with g∗ using the Killing form. Then f ∈ (Sk(g))N and is an H-weight vector of (k) ∗ weight λ+λ . On the other hand, by (1.2), U (g) = U (n )⊕U (g)b. (2.8) k k − k−1 However b · v ⊂ Cv so that f vanishes on U (g)b. But this readily implies λ λ k−1 f ∈ S(n)N. We have proved (k) Theorem 9. Let f be given by (2.4) and let k be the codegree of f. Then λ+λ∗ ∈ Λ(S(n)N). Furthermore k = deg(λ+λ∗) and up to scalar multiplication f(k) = ξλ+λ∗. (2.9) The inclusion (1.25) is actually an equality Λ(S(n)N) = Λ ∩Λ . (2.10) dom B This equality is due to Tony Joseph and I was not aware of it until read it in [J]. However, the equality (2.10) follows immediately from the modified Lipsman–Wolf construction Theorem 9. Indeed let ν ∈ Λ ∩Λ . To show ν ∈ Λ(S(n)N, it suffices dom B to show that e (ν) ≥ 0 (2.11) i ∗ in (1.31) for any i = 1,...,m. But putting λ = ν, one has λ + λ = 2ν and by Theorem 9 one has all e (2ν) ≥ 0. But clearly e (2ν) = 2e (ν). This proves (2.11). i i i The results in this paper will appear in [K1] in Progress in Mathematics, in honor of Joe. References [J] Anthony Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. of Alg., 48, No.2, (1977), 241–289. [K] The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group, paper in honor of I.M. Gelfand, to appear in the Moscow Math Journal, edited by Victor Ginzburg, Spring 2012. [K1] Bertram Kostant, CentU(n), cascade of orthogonal roots, and a construction of Lipsman-Wolf,arXiv/0178653,[math.RT],Jan12,2011; toappearinLie Groups, Structures, and Representation Theory, in honor of Joseph A. Wolf, Progress in Mathematics, 2012. [LW] RonaldLipsmanandJosephA.Wolf,Canonicalsemi-invariantsandthePlancherel formula for parabolic groups, Trans. Amer. Math. Soc. 269 (1982), 111–131. 10