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Generators and Relations of the Affine Coordinate Rings of Connected Semisimple Algebraic Groups

Vladimir L. Popov

Vienna, Preprint ESI 972 (2000)

Supported by Federal Ministry of Science and Transport, Austria Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL: http://www.esi.ac.at

December 15, 2000

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS OF CONNECTED SEMISIMPLE ALGEBRAIC GROUPS

Vladimir L. Popov December 13, 2000 Abstract. Let G be a connected semisimple algebraic group over an algebraically closed field of characteristic zero. A canonical presentation by generators and relations of the algebra of regular functions on G is found. The conjectures of D.Flath and J.Towber, [FT], are proved.

(1) Let G be a connected semisimple algebraic group over an algebraically closed field k of characteristic zero. We obtain here a canonical presentation of the algebra k[G] of regular functions on G (i.e., the affine coordinate ring of G) by means of generators and relations. In the simplest case when G = SL2 one has the isomorphism µ : k[A, B, C, D]/(AD − BC − 1) −→ k[G], (1.1) where µ(A)(

·

¸ a b ) = a, ..., c d

and our presentation is obtained from (1.1) by rewriting it in the form k[G] ' (S ⊗k S − )/I,

(1.2)

where S = k[µ(A), µ(C)] ' k[A, C], S − = k[µ(B), µ(D)] ' k[B, D] and I = (µ(A)⊗ µ(D)−µ(B)⊗µ(C)−1). The meaning of (1.2) is that S is the algebra of invariants of 1991 Mathematics Subject Classification. 20G05. Key words and phrases. Semisimple algebraic group, maximal torus, Borel subgroup, weight, Weyl module, orbit, invertible sheaf. Research partly supported by Grant # MQZ000 from the International Science Foundation, by the Max-Planck-Institut f¨ ur Mathematik, and The Erwin Schr¨ odinger International Institute for Mathematical Physics (Vienna, Austria).

1

Typeset by AMS-TEX

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VLADIMIR L. POPOV

the maximal unipotent subgroup of SL2 consisting of all upper triangual matrices and acting on G by right translation, and S − is the algebra of invariants of the opposite maximal unipotent subgroup. We show that there is an analogue of (1.2) for any connected semisimple algebraic group G. The algebras S and S − and the ideal I case can be described quite explicitly. This gives the canonical presentation of the algebra k[G] by generators and relations. The canonical generators of the ideal I are of degree 2 and not homogeneous. If the monoid of dominant weights of G is freely generated, the algebras S and S − are quadratic. Thus, for instance, if G is simply connected, we obtain an explicit canonical presentation of the algebra k[G] by generators and relations, in which all relations have degree 2. This gives a canonical presentation of the group variety G as an intersection of quadrics in an affine space (as a parallel, recall that an abelian variety can be canonically presented as an intersection of quadrics in a projective space given by the Riemann equations, [K], [LB]). In a nonsimply connected case one has to combine this peresentation for the simply connected covering of G with the corresponding action of the fundamental group of G which is easy to control. For G = SLn , GLn , SOn and Spn the analogue of (1.2) was obtained by D.Flath and J.Towber in [FT] by means of the direct bulky computations of the corresponding Laplace decompositions, minors and the algebraic identities between them. They formulated two general conjectures which we prove here, cf. Section (4). The problem of describing the ideal I in general case was formulated in [F]. The algebras of regular functions on the linear algebraic groups are the basic objects of the theory of quantum groups. I believe that the canonical character of our presentation of k[G] is a true indication that there exists a nice general procedure of quantization of k[G] based on this presentation. I also think that the results similar to the ones proven here can be obtained for the infinite dimensional groups as well, cf. [KP]. Finally, I would like to mention that our results have the aplications in the theory of coherent tensor operators, cf. [F]. In this preprint, the proofs (and even some formulations, see Remark in Section (13)) are heavily dependent on the assumption that the ground field has characteristic zero. It is remarkable however that all main results remain true without this assumption. In particular, this gives an approach to quantization of k[G] in positive characteristic, the case in which only a little is known at the moment. Of course the characteristic free proofs (and some formulations) become more complicated. This material will appear in a forthcoming final version of this paper. (2) We fix the following notation: ◦ ◦ ◦ ◦ ◦ ◦

B a Borel subgroup of G; T a maximal torus of B; U the unipotent radical of B; X(H) the group of characters of an algebraic group H written additively; tλ the value of λ ∈ X at t ∈ T ; P++ ⊂ X the monoid of the highest weights of the simple G-modules;

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

3

R(λ) a simple G-module with the highest weight λ ∈ P++ ; λ∗ the highest weight of R(λ)∗ ; U − the maximal unipotent subgroup of G opposite to U ; w0 an element of the normalizer of T such that w0 U w0−1 = U − ; g the Lie algebra of G; U(g) the universal enveloping algebra of g; gα the root line in g corresponding to a root α; t the Lie algebra of T ; Vλ the weight space of a T -module V corresponding to a weight λ; M H the fixed point set of an action of a group H on a set M ; H ·x the H-orbit of a point x; H ·X := {H ·x | x ∈ X}; Hx the H-stabilizer of a point x; k[X] the algebra of regular functions of an algebraic variety X; k(X) the field of of rational functions of an algebraic variety X.

(3) We consider k[G] as the G-module with respect to the action defined by the left translations. We set S = {f ∈ k[G] | f (gu) = f (g), g ∈ G, u ∈ U }

(3.1)

and, for any λ ∈ P++ , Sλ = {f ∈ S | f (gb) = bλ f (g), g ∈ G, b ∈ B}.

(3.2)

It is well known that Sλ is a simple submodule of k[G] with the highest weight λ∗ with respect to B and that the submodules Sλ give a P++ -grading of S: S= Algebra

⊕ Sλ , Sλ ·Sγ = Sλ+γ .

λ∈P++

(3.3)

S − = {f ∈ k[G] | f (gu) = f (g), g ∈ G, u ∈ U − }

(3.4)

Sλ− = {f ∈ S − | f (gt) = tw0·λ f (g), g ∈ G, t ∈ T }.

(3.5)

is obtained from S by the right translation by w0 . It is a submodule of k[G]. If Sλ− is the right translation of Sλ by w0 , then Sλ− is a simple submodule of k[G] with the highest weight λ∗ with respect to B and

The submodules Sλ− give a P++ -grading of S − : S− =

− . ⊕ Sλ− , Sλ−·Sγ− = Sλ+γ

λ∈P++

(3.6)

(4) Consider the homomorphism of algebras µ : S ⊗k S − , f ⊗ h 7→ f h.

The conjectures of D.Flath and J.Towber, [FT], are formulated as follows:

(4.1)

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VLADIMIR L. POPOV

(Sur) µ is surjective; (Ker) ker µ is generated by (ker µ)G . We shall prove (Sur) in Section (6) and (Ker) in Section (7). In Section (12) we describe the algebra structure of (ker µ)G and find the invariant generators of ker µ. Combining this with the presentations of S and S − by generators and relations described in Section (13), we obtain a presentation of k[G] by generators and relations for any connected semisimple algebraic group G such that the monoid P++ is freely generated. In particular this gives a presentation of k[G] for any simply connected semisimple algebraic group G. If G is not simply connected semisimple group and ˜ −→ G is its universal covering, then k[G] is obtained from k[G] ˜ by taking π : G ˜ ˜ The the invariants of the action of the finite central subgroup ker π of G on k[G]. ˜ structure of this action is quite explicit with respect to our presentation of k[G] and, if necessary, can be used in the concrete cases for obtaining a presentation of k[G] as well. (5) Our proof of conjecture (Sur) is based on two general statements. The first one is well known, cf. [St,1.5]. Theorem 1. Let φ : X −→ Y be a morphism of affine algebraic varieties. Then the comorphism φ∗ : k[Y ] −→ k[X] is surjective if and only if φ is a closed embedding. The second statement gives a criterion of closedness of certain orbits (as a matter of fact for the proof of conjecture (Sur) we need only the implication (ii) ⇒ (i)).

Theorem 2. Let V be a direct sum of finitely dimensional G-modules V1 , . . ., Vs and vi ∈ Vi a nonzero weight vector of a weight λi ∈ X. Then the following properties of the vector v = v1 + . . .+ vs ∈ V are equivalent: (i) G·v is closed in V ; (ii) 0 is an interior point of the convex envelope conv{λ1 , . . . , λs } of the weights λ1 , . . . , λs in X ⊗Z Q; (iii) T ·v is closed in V . Proof. (ii) ⇔ (iii). This is well known, cf. [PV, 6.11] or [P1].

(iii) ⇒ (i). We can assume, according to the Lefschetz principle, that k = C, cf. for instance [Si,VI,§6]. Assume that T ·v is closed in V . Let K be a maximal connected compact subgroup of G. Fix in V a K-invariant Hermitian inner product ( , ), such that Vi and Vj are orthogonal if i 6= j. Since T ·v is closed, the function v 7−→ (v, v) attains its minimum on T ·v. Since t·vi , like vi , is also a nonzero weight vector of the weight λi , changing the notation we can assume that this minimum is attained at v. Therefore, (t · v, v) = 0. (5.1) Since gα·(Vi )λi ⊆ (Vi )α+λi for any root α, it follows from orthogonality of the weight spaces of different weights and pairwise orthogonality of the subspaces Vi that (gα ·v, v) = 0.

(5.2)

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS

5

Since g = t ⊕ (⊕gα ), it follows from (5.1) and (5.2) that α

(g·v, v) = 0.

(5.3)

Now closedness of G·v in V follows from (5.3) and the Kempf–Ness criterion, [KN] (cf. also [PV, 6.12]). (i) ⇒ (ii). Assume that G·v is closed in V . If (ii) is not satisfied, there is a linear function l ∈ (X ⊗Z Q)∗ such that all the numbers ni := l(λi ) are nonnegative and at least one of them, say n1 , is positive. Multiplying l by an appropriate positive integer one can assume that l(χ) ∈ Z for any χ ∈ X. Hence there is a homomorphsm µ : k∗ −→ T such that µ(t) · v = tn1 v1 + . . . + tns vs . Therefore the closure of T ·v in V contains the vector u := u1 + . . . + us , where ui = vi if ni = 0 and ui = 0 if ni > 0. Since G·v is closed, this closure is contained in G·v. On the other hand, u∈ / G·v, since u1 = 0, – a contradiction. ¤ Remark. The proof given here heavily depends on the fact that the ground filed has characteristic zero. In fact Theorem 2 is valid over ground filed of arbitrary characteristic. The characteristic free proof of it is given in our article [P2]. We will use this fact in a forthcoming publication whether the main results of this preprint will be proved over ground field of arbitrary characteristic. (6) Now we can give a proof of conjecture (Sur). Theorem 3. Homomorphism (4.1) is surjective. Proof. Consider the action of G on itself by the left translations. Since k-algebra S is finitely generated, there is an irreducible affine algebraic variety X endowed with a regular action of G and a dominant equivariant morphism α : G −→ X, such that α∗ (k[X]) = S. Let x = α(e). The orbit G · x is open and dense in X. Let π : G −→ G/U be the canonical morphism. Then π ∗ (k[G/U ]) = S, cf. (3.1) and [PV, 3.1]. Hence, there is an equivariant morphism ι : G/U −→ X such that α = ι ◦ π.

(6.1)

Since U has no nontrivial characters, G/U is qusiaffine, cf. [PV, 3.7], and therefore k(G/U ) is the field of fractions of k[G/U ]. Since k(X) is the field of fractions of k[X], and, by (6.1), ι∗ : k[X] −→ k[G/U ] is an isomorphism, it follows from here that ι is a birational isomorphism. Since ι is equivariant, this means that ι is an open embedding and its image is G·x. It follows from (6.1) that Gx = U . Therefore, if y = w0 ·x and β : G −→ X, g 7→ g·y, then Gy = U − and β ∗ (k[X]) = S − . Consider the morphism γ = α × β : G −→ X × X. We canonically identify the functions on each of the factors of X × X with the functions on X × X. Then τ : k[X] ⊗k k[X] −→ k[X × X], f ⊗ h 7→ f h,

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VLADIMIR L. POPOV

is an equivariant isomorphism. Now it follows directly from the definitions of α, β, γ, τ and µ that the composition of the homomorphisms (α∗ )−1 ⊗(β ∗ )−1

τ

γ∗

S ⊗k S − −−−−−−−−−−→ k[X] ⊗k k[X] − → k[X × X] −→ k[G].

(6.2)

V = R(λ1 ) ⊕ . . . ⊕ R(λs ), v = (v1 , . . . , vs ),

(6.3)

coincides with µ. Therefore surjectivity of µ is equivalent to surjectivity of γ ∗ . By Theorem 1, this means that the problem is reduced to proving that γ is a closed embedding. Let z = (x, y) ∈ X × X. Then Gz = Gx ∩ Gy = U ∩ U − = {e}. Therefore γ is an equivariant isomorphism of G and G·z. Hence, it remains to prove that G·z is closed in X × X. Let {λ1 , . . . , λs } be a set of generators of the monoid P++ and vi a highest vector of R(λi ) with respect to B. Let and G·v be the closure of G·v in V . Restricting the functions to G·v, we identify k[G·v] with a subalgebra of k[G·v]. Since U ⊆ Gvi for each i, we have U ⊆ Gv . Therefore, if δ : G −→ G·v, g 7→ g·v, then δ ∗ (k[G·v]) ⊆ S. Since the submodule R(λi )∗ = R(λ∗i ) of V ∗ = R(λ1 )∗ ⊕ . . . ⊕ R(λs )∗ is simple and the projection of G·v to R(λi ) is nonzero, the restriction of functions to G·v defines an embedding of R(λi )∗ into k[G·v]. Hence, δ ∗ (k[G·v]) contains Sλi for all i. It follows from here and (3.3) that δ ∗ (k[G·v]) = δ ∗ (k[G · v]) = S. Therefore one can take X = G·v and x = v. Since λ∗1 , . . ., λ∗s is a set of generators of P++ as well, the similar arguments show that if ui is a highest vector of R(λ∗i ) with respect to B and u = (u1 , . . . , us ) ∈ V ∗ = R(λ∗1 ) ⊕ . . . ⊕ R(λ∗s ), then one can take X to be the closure G·u of G·u in V ∗ , x = u and y = w0 ·u = (w0 ·u1 , . . . , w0 ·us ). Hence, under these identifications, X × X = G·v × G·u ⊂ V ⊕ V ∗ = R(λ1 ) ⊕ . . . ⊕ R(λs ) ⊕ R(λ∗1 ) ⊕ · · · ⊕ R(λ∗s ) and z = (v1 , . . . , vs , w0 ·u1 , . . . , w0 ·us ). Since the weight of vi is λi and the weight of w0 ·ui is w0 ·λ∗i = −λi , the convex envelope of the weights of the projections of z to the irreducible direct summands of V ⊕ V ∗ is C := conv{λ1 , . . . , λs , −λ1 , . . . , −λs }. Clearly 0 ∈ C. Since −C = C, 0 is an interior point of C. Whence, by Theorem 2, the orbit G·v is closed in V ⊕ V ∗ , and therefore closed in X × X as well. ¤ (7) Now we pass to a proof of conjecture (Ker).

Theorem 4. The kernel of homomorphism (4.1) is generated by G-invariants. Proof. We use the results obtained in the course of proof of Theorem 3 and keep the notation introduced therein. Since the composition of homomorphisms in (6.2) coincides with µ, the statement of Theorem 4 is equivalent to the following one: The kernel of homomorphism k[X × X] −→ k[G·z], f 7→ f |G·z , is generated by its intersection with k[X × X]G . In turn, this statement follows from a general Theorem 5 proved below. ¤

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS

7

Theorem 5. Let Y be an affine algebraic variety endowed with a regular action of a reductive algebraic group H. Let y ∈ Y be a point such that (i) H ·y is closed in Y ; (ii) Hy is trivial. Then the kernel of homomorphism k[Y ] −→ k[H ·y], f 7→ f |H·y , is generated by its intersection with k[Y ]H . Proof. There is a closed equivariant embedding of Y in a linear action, cf. [PV, 1.2], therefore one can assume that Y is an H-module. Since Y is smooth, one can consider the Luna stratifications of Y , [L], (cf. also [PV, 6.9]). It follows from (i) and (ii) that y lies in the principal stratum of this stratification. Therefore the canonical morphism πY //H : Y −→ Y //H =: Spec k[Y ]H is smooth at any point of H ·y, [L, Cor. 6] (cf. also [PV, Th. 6.11]). Let ξ = πY //H (y). Since each fibre of πY //H contains a unique closed orbit which lies in the closure of any orbit contained in this fiber, cf. [PV, 4.4], it follows from (i) and (ii) that (7.1) πY−1 //H (ξ) = H ·y. Therefore πY //H is a smooth morphism at any point of πY−1 //H (ξ). Hence, the fiber −1 πY //H (ξ) is reduced, cf. [R, Lemma 2.3]. This means that the ideal of functions in ∗ k[Y ] vanishing on πY−1 //H (ξ) is generated by πY //H (mξ ), where mξ is the ideal of ξ in k[Y //H]. Since πY∗ //H (k[Y //H]) = k[Y ]H , this completes the proof because of (7.1). ¤ (8) Now we want to give an explicit description of the generators of ker µ. By Theorem 5, this is reduced to giving an explicit description of the algebra (ker µ)G . By (3.3) and (3.6), (S ⊗k S − )G =

⊕

λ,γ∈P++

(Sλ ⊗k Sγ− )G .

(8.1)

Since Sλ ⊗k Sγ− ' (Sλ∗ )∗ ⊗k Sγ− ' Hom(Sλ∗ , Sγ− ), it follows from the Schur Lemma that ½ 0, if γ 6= λ∗ , (8.2) dim(Sλ ⊗k Sγ− )G = 1, if γ = λ∗ , and therefore, by (8.1) and (8.2), (S ⊗k S − )G =

⊕ (Sλ ⊗k Sλ−∗ )G .

λ∈P++

(8.3)

By (3.3) and (3.6), the decomposition (8.3) is a structure of a P++ -graded algebra on (S ⊗k S − )G , i.e. − G (Sλ ⊗k Sλ−∗ )G ·(Sγ ⊗k Sγ−∗ )G = (Sλ+γ ⊗k S(λ+γ) ∗) .

(9) First we describe the structure of (ker µ)G as a vector space.

(8.4)

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VLADIMIR L. POPOV

Proposition 1. (a) Any finite sum X f= (fλ − fλ (e, e)), fλ ∈ (Sλ ⊗k Sλ−∗ )G ,

(9.1)

λ

lies in (ker µ)G . (b) Conversely, any element f ∈ (ker µ)G can be written in a unique way in the form (9.1). Proof. (a) We consider (S ⊗k S − )G in a natural way as a subalgebra of k[G × G]. Since µ : (S ⊗k S − )G −→ k[G]G = k, we have µ(f ) = f (e, e) for any f ∈ (S ⊗k S − )G , and the claim follows. (b) By (8.3), there is a unique decomposition X fλ , fλ ∈ (Sλ ⊗k Sλ−∗ )G , f=

(9.2)

λ

whence (9.1), since 0 = µ(f ) = f (e, e) = from uniqueness of (9.2). ¤

P

λ

f (e, e). Uniqueness of (9.1) follows

(10) To describe the algebra structure of (ker µ)G we need a few adittional facts. Lemma 1. Let V and W be two simple G-modules and P (V, W ) the vector space of all invariant bilinear pairings h , i : V × W −→ k, (v, w) 7→ hv, wi. Then: (i) dim P (V, W ) = 0 or 1, and the latter case happens if and only if W ∼ = V ∗. (ii) If {vi } and {wi } are the dualPbases of V and W with respect to a nonzero pairing h , i ∈ P (V, W ), then i (vi ⊗wi ) is a nonzero element of (V ⊗k W )G which does not depend on the choice of bases {vi } and {wi }. (iii) Let W ∼ = V ∗ and v + ∈ V, w− ∈ W be respectively a highest and a lowest weight vectors with respect to B. Then hv + , w− i 6= 0 for any nonzero pairing h , i ∈ P (V, W ). For any nonzero constant α ∈ k, there is a unique pairing h , i ∈ P (V, W ) such that hv + , w− i = α. Proof. (i) The G-module of all bilinear pairings V × W −→ k is isomorphic to the G-module Hom(V, W ∗ ), whence the claim by the Schur Lemma. (ii) A nonzero P an isomorphism of G-modules ε : V ⊗kPW −→ P pairing h , i defines (e ⊗ f ) → 7 {v → 7 Hom(V, V ), i i (vi ⊗ i hv, fi iei }. It follows from here that ε( i i wi )) = IdV , whence the claim.

(iii) Let {vi }, v1 = v + , be a basis of V consisting of the weight vectors, and {wi } the dual basis of W with respect to h , i. The linear span V 0 of {vi | i > 1} 0 0 0 0 − is U − -invariant and u · vP 1 = v1 + v , v = v (u) ∈ V , for any u ∈ U . Hence P −1 0 (u · w )( α v ) = w ( α (u · v )) = w (α v + an element in V ) = α1 = 1 i 1 1 1 i i i i i P1 w1 ( i αi vi ). Therefore, u·w1 = w1 , hence w− = λw1 for a certain nonzero λ ∈ k. Whence, hv + , w− i = λ 6= 0. The second statement follows from here and (i). ¤

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS

9

Theorem 6. µ : (Sλ ⊗k Sλ−∗ )G −→ k[G]G = k is an isomorphism.

Proof. By (8.2), one has to show that there is f ∈ (Sλ ⊗k Sλ−∗ )G such that µ(f ) = f (e, e) 6= 0. Let v + and v− be a highest weight vector and a lowest weight vector of Sλ−∗ and Sλ and respectively. Consider the morphisms α : G −→ Sλ−∗ , g 7→ g ·v + , and ' ' β : G −→ Sλ , g 7→ g · v− . Then α∗ : (Sλ−∗ )∗ −→ Sλ , β ∗ : (Sλ )∗ −→ Sλ−∗ and − ] ⊗k k[Sλ ] −→ k[G × G] = k[G] ⊗k k[G] therefore (α × β)∗ : k[Sλ−∗ × Sλ ] = k[Sλ∗ − ∗ ∗ ∗ induces (α × β) ((Sλ∗ ) ⊗k (Sλ ) ) = Sλ ⊗k Sλ−∗ . Fix a nonzero invariant bilinear pairing h , i : (Sλ−∗ )∗ ⊗k (Sλ )∗ −→ k. Let {pi } and {qi } be the dual bases of (Sλ−∗ )∗ and (Sλ )∗ with respect to h , i. It follows P from Lemma 1, (ii), that h := i (pi ⊗ qi ) ∈ ((Sλ−∗ )∗ ⊗k (Sλ )∗ )G , h 6= 0. Therefore, f := (α × β)∗ (h) ∈ (Sλ ⊗k Sλ−∗ )G , f 6= 0. The mapping Sλ−∗ × Sλ −→ k, (a, b) 7→ h(a, b), is a nonzero invariant bilinear pairing. Since f (e, e) = (α × β)∗ (h)(e, e) = h(α(e), β(e)) = h(v + , v− ), the claim follows from Lemma 1, (iii). ¤ (11) It follows from Proposition 2 that for any λ ∈ P++ there is a unique element hλ ∈ (Sλ ⊗k Sλ−∗ )G such that µ(hλ ) = hλ (e, e) = 1. This element hλ can be described in either of two ways: (a) Fix a nonzero invariant pairing h, i : Sλ × Sλ−∗ −→ k. Let {pi } and {qi } be a pair of dual bases of Sλ and Sλ−∗ with respect to h, i. Then hλ = (

X i

pi ⊗ qi )/(

X

pi (e)qi (e)).

i

(b) Let {Xi } and {Xi∗ } be a pair of dual bases of the vector space g with respect − to the Killing form. The elements Xi and Xi∗ act on Sλ and Sλ∗ linearly . Consider the linear operator X (Xi ⊗ Xi∗ + Xi∗ ⊗ Xi ) ∆= i

Sλ ⊗k Sλ−∗ .

on Let h | i be the inner product on the lattice of weights of G induced by the Killing form, ρ half the sum of all positive roots and c(λ) = −hλ + 2ρ|λi − hλ∗ + 2ρ|λ∗ i. Proposition 2. Let f ∈ Sλ ⊗k Sλ−∗ .Then the following properties are equivalent: (i) f = hλ ; (ii) ∆·f = c(λ)f, f (e, e) = 1. Proof. Let Ω ∈ U(g) be the Casimir operator given by the formula Ω=

X i

Xi Xi∗ ,

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VLADIMIR L. POPOV

If λ ∈ P++ , then Ω acts on a simple U(g)-module with the highest weight λ as scalar multiplication by hλ + 2ρ|λi, cf. [Bo]. Therefore for any s ∈ Sλ , t ∈ Sλ−∗ one has the equality X ((Xi Xi∗ ·s) ⊗ t + (Xi∗ ·s) ⊗ (Xi · t)+ Ω · (s ⊗ t) = i

+ (Xi ·s) ⊗ (Xi∗ ·t) + s ⊗ (Xi Xi∗ ·t)) = −c(λ)(s ⊗ t) + ∆·(s ⊗ t).

(11.1)

Since hλ + 2ρ|λi > 0 if λ 6= 0, the kernel of Ω in any U(g)-module V is V G . The claim follows from here, (8.2) and (11.1). ¤ (12) Now we can describe the algebra structure of (ker µ)G and indicate the generators of the ideal (ker µ). Theorem 7. Let λ1 , . . . , λs be a set of generators of the monoid P++ . Then (ker µ)G coincides with k[hλ1 − 1, . . ., hλs − 1]+ , the augmentation ideal of k[hλ1 − 1, . . . , hλs − 1]. G G Proof. Since hλi − 1 ∈ (ker µ) P , one has (ker µ) ⊇ k[hλ1 − 1, . . ., hλs − 1]+ . Let λ ∈Q P++ . Then Q λ = i ni λi for some integers ni > 0. It follows from (8.4) that hλ = i hnλii = i ((hλi − 1) + 1)ni . Therefore, hλ − 1 ∈ k[hλ1 − 1, . . ., hλs − 1]+ . Hence, by (8.2) and Proposition 1, (ker µ)G ⊆ k[hλ1 − 1, . . ., hλs − 1]+ . ¤

Remark. The proofs of Theorem 3, Theorem 4, Theorem 5, Theorem 6, Theorem 7 given here use the assumption that the ground field has characteristic zero. However these statements are true over the ground field of arbitrary characteristic. The proofs will given in a forthcoming publication. (13) Now we shall indicate the explicit presentation of the algebras S and S − by the generators and relations. For the first time such presentation was obtained (but not published) by B. Kostant; his result and proof appears in [LT] . We just present it here in a slightly different form than in [LT]. We use the results obtained in the course of the proof of Theorem 3 and keep the notation introduced therein. The decomposition (6.3) induced the natural Ns -grading of k[V ]. Let ei = (0, . . ., 0, 1, 0, . . ., 0) ∈ Ns . It follows from (6.3) that ↑

i

k[V ]ei +ej '

½

R(λi )∗ ⊗k R(λj )∗ , 2

∗

S R(λi ) ,

if i 6= j;

if i = j.

(13.1)

By (13.1), for all 1 6 i 6 j 6 s the G-module k[V ]ei +ej contains a unique simple submodule with the highest weight λ∗i + λ∗j , the Cartan component of k[V ]ei +ej . Denote by Qi,j the G-invariant direct complement in k[V ]ei +ej to this submodule. Let J be the ideal in k[V ] generated by all Qi,j , 1 6 i 6 j 6 s. Since V U is B-stable, G·V U is closed in V . It is shown in [Br, 4.1]1 that J is the ideal of G·V U in k[V ]. 1 The part of Theorem in [Br, 4.1] concerning the ideal I contains a mistake but the part concerning the ideal J is correct.

GENERATORS AND RELATIONS OF THE AFFINE COORDINATE RINGS

11

Theorem 8. Assume that λ1 , . . ., λs is a free system of generators of P++ (i.e. each element of P++ uniquelly expressed as a linear combination of these generators with the nonnegative integer coefficients). Then S is G-isomorphic to k[V ]/J. Proof. The condition on λ1 , . . ., λs is equivalent to the equality s = dim T . Since Gv = U , we have Tv = {e}. Therefore, dim T · v = dim T = s = dim V U . Since dim T ·v ⊂ V U and G·V U is closed, it follows from here that X = G·v = G·V U . ¤ Remark. The presentation given in Section (13) heavily depends on the assumption that the ground field has characteristic zero. In a forthcoming publication we will give a characteristic free presentation. It is related to the theory of standard monomials.

References [Bo] N. Bourbaki, Groupes et alg` ebres de Lie., Chap.VI et VIII, Hermann. ´ Norm. [Br] M. Brion, Repres´ entations exceptionnelles des groupes semi-simples, Ann. scien. Ec. Sup. 4e s´ erie 18 (1985), 345-387. [F] D. Flath, Coherent tensor operators, Contemporary Mathematics. [FT] D. E. Flath, J. Towber, Generators and relations for the affine rings of the classical groups, Commun. in Algebra 20(10) (1992), 2877-2902. [KP] V. G. Kac, D. H. Peterson, Regular functions on certain infinite-dimensional groups. [K] G. Kempf, Projective coordinate rings of abelian varieties, In: Algebraic Analysis, Geometry and Number Theory, Edited by J.I.Igusa: The John Hopkins Press (1989), 1989. [KN] G. R. Kempf, L. Ness, The length of vectors in representation spaces, Lecture Notes in Math.: Springer Verlag 732 (1979), 233-243. [LT] G. Lancaster, J. Towber, Representation-functors and flag-algebras for the classical groups, I. J. of Algebra 59 (1979), 16-38. [LB] H. Lange, C. Birkenhake, Complex Abelian Varieties, Grundlehren der mathematischen Wissenschaften: Springer Verlag 302 (1992). [L] D. Luna, Slices ´ etales, Bull. Soc. Math. France, Mem. 33 (1973), 81-105. [R] R. W. Richardson, An application of the Serre conjecture to semisimple algebraic groups, In: Algebra, Carbondale, 1980: Springer Verlag 1981 (1981), 141-151. [P1] V. L. Popov, Closed orbits of Borel subgroups, Matem. Sbornik 135(177), n.3 (1988), 385-402. English translation: Math. USSR Sbornik, 63, n.2 (1989), 375-392. [P2] V. L. Popov, On closedness of some orbits of algebraic groups, Funct. Anal. Appl. 31 (1997), no. 4, 76–79. [PV] V. L. Popov, E. B. Vinberg, Invariant Theory, In: Encyclopaedia of Math.Sci.: Algebraic Geometry IV. Springer Verlag 55 (1994), 123-284. [Si] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics: Springer Verlag 106 (1986). [St] R. Steinberg, Conjugacy Classes in Algebraic Groups, Lecture Notes in Math.: Springer Verlag 366 (1974). Department of Mathematics, Moscow State University MGIEM, Bol’sho svyatitel’ski ˇi per., 3/12, 109028 Moscow, RUSSIA E-mail address: [email protected]

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