Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/~panyush/covar-j.pdf Дата изменения: Mon Dec 13 16:35:44 2004 Дата индексирования: Mon Oct 1 23:11:23 2012 Кодировка: Поисковые слова: m 63

Indag. Mathem., N.S., 13 (l), 125-129

March 25,2002

On covariants

of reductive

algebraic

groups

by Dmitri

I. Panyushev

Independent University e-mail:panyuslz@mccme.ru

of

Moscow,

Bol'shoi

Vlasevskiiper.

II, 1.21002 Moscow,

Russia

Communicated by Prof. T.A. Springer at the meeting of February 25,2002

1. AN

EVALUATION

MAP

Recently, Lehrer and Springer have proved the surjectivity of some natural map associated with covariants of a finite reflection group in a complex vector space, see [LS, Theorem A]. The aim of this note is to show that a similar statement is valid in a greater generality; namely, for an arbitrary action of a reductive algebraic group G on an affine variety X and for a sufficiently good Gorbit (e.g. closed) in X. We also demonstrate some invariant-theoretic applications of it. The ground field k is algebraically closed and of characteristic zero. Let X be an affine variety, with coordinate ring k[X], which is acted upon by a reductive algebraic group G. For any (finite-dimensional) G-module M, the space k[X] 18 M is being identified with the space of all polynomial morphisms from X to M, denoted by P(X, M). Under this identification, f @ m cf E k[X], m E M) determines the mapping that takes x E X tof(x)m E M. The group G acts on k[X] by (g.f)(x) =f(g-`.x). This yields a natural G-module structure on k[X] @ M. Furthermore, the subset of G-invariant elements, denoted by (k[X] 18 M)Gor P,(X, M), is nothing but the set of G-equivariant polynomial morphisms from X to M. Clearly, P,(X, M) IS a module over k[XIG, the module of covariunts (of type A4). For any x E X, there is the `evaluation' map:
E, : P,(X,M) + MGX 7

125


which is defined by E,(c$) = 4(x). Here G, c G is the stabiliser of x. Notice that the set Po(X, M) can be considered for an arbitrary G-variety X. The main result of this note is the following
Theorem 1. Suppose

codimG(G.x

X is afine, the \ G.x) > 2. Then cx is onto.

closure

G.x

is

normal,

and

Let us first recall a simple and well-known

result.

Lemma. Let Y be an arbitrary closed G-stable subvariety of X and (p : Y --+ Man arbitrary G-equivariant morphism to a G-module. Then $5 extends to a G-equivariant morphism `p : X -+ M. In other words, the mapping P,(X, M) + PG( Y, M) is onto. Proof. The comorphism

(p* : k[M] + k[ Y] is fully determined by its values on M* = k[M],. (C onversely, any linear mapping M* --f k[ Y] determines an algebra homomorphism k[M] -+ L[ Y].) Therefore, to construct a required extension, it suffices to find alinear mapping `p* such that the following diagram be commutative:

The existence of such cp* follows from the fact that in view of complete reducibility of G-representations, the restriction homomorphism I!@] + L[ Y] admits a G-equivariant section. q
Proof of Theorem 1. The condition on codimension implies that k[G.x] = k[G. x ] , and therefore Po(G.x,M) = PG(G.x, M). Furthermore, it is clear that the mapping E, : P,(G.x, M) -+ M Gxis bijective. It remains to apply __ the Lemma to Y = G.x. 0 Corollary.

If G is afinite group, then E, is onto for all x E X.

In case X = V is a G-module and G is a reflection group in V, this corollary gives the aforementioned result of [LS]. As is known, PG(x, M) is a finitely generated module over k[XIG, see [Kr]. If it is a free module, Theorem 1 admits a matrix interpretation (cf. [LS]). Namely, let ~1,. . . , uy be a basis for the module and ml,. . . , m, a basis for M. Then ui = cj A, @ mj, where A, are some functions on X (1 < i 5 r).
Theorem 2. rf PG(X, M) IS a f ree Ik[XIG-module and x E X satisfies the hypotheses of Theorem 1, then the rank of the r x n-matrix(Av(x)) equals dim MGx.

126


Example.

module are the In this and we

Let X = M = B be Lie algebra of G. of rank Y = rk g. Iffi, . . ,fy E k[glG are coordinates on 8, then one can take A, = situation, the codimension condition in arrive at the following conclusion: v E 8; then rank

Then Pc(B, 8) basic invariants af,/axj (1 5 i Theorem 1 is

is a free k[81Gand x1, . . . , x, < r; 1
If G.v is normalfor

= dim gGY.

Notice also that if G, is connected, then gGvis the centre of Lie algebra e;,.
2. APPLICATIONS.

Let X be an irreducible normal G-variety and let H be a a generic stabiliser for the G-action on X. (The latter always exists, if X is smooth.) Suppose the action is stable and XH is irreducible. The stability implies that H is reductive. From [Lu, Cor. 41 it then follows that k[XIG = k[XHINGcH), where the isomorphism is induced by the restriction. Write J for this k-algebra. More generally, for any G-module A4, restricting G-equivariant morphisms to H-fixed points yields a homomorphism of J-modules
PM : pG(x, M) -j %`G(H)(XH, MH)

Since G.XH is dense in X, we have PM is injective. Consider the J-module ?G(X, End M). Clearly, it is an associative k-algebra, since EndM is. Notice that PG(X, EndM) contains J as commutative subalgebra, because idM E EndM.
Proposition 3. The algebra PG(X, EndM) multiplicity free H-module.

is commutative ifand only ifMjH

is a

(Here `multiplicity M is at most 1.)
Proof.

free' means that the multiplicity

of any simple H-module

in

As above, we have the injective homomorphism
PEndM : PG(X, EndM)
+ ~N~(H)(~~, (EndWH) .

If n/rlH is multiplicity free, then (EndM)H is a commutative subalgebra of End M. Hence P,(X, EndM) is commutative as a subalgebra of the commutative algebra PNG(H)(XH, (EndM)H). Conversely, suppose P,(X,EndM) is commutative. Take x E XH such that G.x is closed and G, = H. By Theorem 1, we have e,(Pc(X, EndM)) = (EndMfH. Therefore, (EndM)H is commutative, and hence A4 is multiplicity free as H-module. q This generalises a recent result of Kirillov concerning commutativity of `family' algebras, see [Ki, Cor. 11. For yet another application of Theorem 1, consider the following situation. 127


Let Vbe a G-module and H a generic stabiliser for the G-action on V. Suppose H is reductive and W := NG(H)/H is finite. By [Lu], we then have G.x is closed for any x E VH and k[VIG N %[VHlw(= J). Moreover, the quotient map 7r : V --+ V//G is equidimensional, and W is a reflection group in VH, see [Pa]. It follows that J is a polynomial algebra and %[V] is a free J-module. Furthermore, any module of covariants ?G( V, M) is a free J-module of rank dim MH. Similarly, MH is a W-module, and Pw( VH, MH) is a free J-module of rank dim lMH. As above, we have the injective homomorphism of J-modules

As both J-modules are free and of the same rank, pM becomes an isomorphism over the fraction field of J. Using Theorem 1, we show that it suffices to invert one specific function, which is valid for all M. Let D E J be the discriminant of the reflection group W. Recall that the zero locus of D in VH/ W is the ramification divisor for the finite morphism VH --) VH/ W. For u E VH, we have D(u) f 0 if and only if W, = {l} if and only if G, = H. Proposition 4. For any G-module M, the homomorphism morphism after inverting D E J. pM becomes an iso-

Proof. Let 241,. . . , u, (resp. Ui, . . . , &) be a basis for the J-module pG( V, M) (resp. Pw( VH, MH)), where n = dim V H. Since pi is injective, Ui = cjJ;jiij for someAj E J. For any x E VH, we have two evaluation maps E, : Pw( VH, MH) -+ MH and eX : ?o( V, M) + MH. By Theorem 1, if D(x) # 0 then both eX and Z, are onto. This means that 0 det&j(x)) # 0 f or all such x. Thus, det&) becomes invertible in J[l/D]. It may happen that PM is already an isomorphism for some M. For instance, this is always the case, if M = V*, see [Vu]. Later, Broer [Br] obtained an elegant description of all such M for the adjoint representation of G, i.e., for V = 8. It might be interesting to extend Broer's results to the case of representations accosiated with periodic automorphisms of semisimple Lie algebras.
REFERENCES [Br] [Ki] [Kr] [LS] [Lu] Pal Broer, A. -The sum of generalized exponents and Chevalley's restriction theorem, Indag. Math. 6, 385-396 (1995). Kirillov, A.A. -Introduction to family algebras, Moscow Math. J. 1,49-63 (2001). Kraft, H. - `Geometrische Methoden in der Invariantentheorie'. Braunschweig-Wiesbaden, Vieweg & Sohn, 1984. Lehrer, G. and Springer, T.A. - A note concerning fixed points of parabolic subgroups of unitary reflection groups, Indag. Math. 10,549-553 (1999). Luna, D. -Adherences d'orbite et invariants, Invent. Math. 29,231-238 (1975). Panyushev, D. - On orbit spaces of finite and connected linear groups, Math. USSR-Izv. 20, 97-101 (1983).

128


[Vu]

Vust, Th. ~ Covariants de groupes algkbriques rtductifs, Thkse no. 1671, Universitk de Gentve, (1974).

(Received December 2001)

129