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O. Yakimova Mathematisches Institut, UniversitЁ zu KЁ at oln KЁ Germany oln, yakimova@mpim-bonn.mpg.de

On symmetric invariants of Z2 -degenerations
The author is supported by the RFFR Grant 05-01-00988. Suppose that G is a simply connected reductive algebraic group defined over an algebraically closed field K of characteristic zero. Set g = Lie G and let r denote the rank of this Lie algebra. Let be an involution of g. Then it defines a Z2 -grading g = g0 g1 , where g0 := g is a symmetric subalgebra of g. The corresponding Z2 -degeneration of g is the Lie algebra g = g0 g1 , where g1 is considered as a commutative ideal. It was conjectured by Panyushev that the algebra of symmetric invariants S (g)g is polynomial. The conjecture is true and we will present basic ideas of a proof. First we need some basic fact concerning semidirect rpoducts and their invariants. Let q be an arbitrary Lie algebra and V a space of a finite dimensional representation of q. Then one form a semidirect products q V , where V is a commutative ideal. It is possible to define a certain Lie algebra ^ := { q K(V ) | (x) qx for all x V } q over the field K(V ), which is a Lie algebra of rational sections for the bundle = {y , qy } V в q. We need a more precise description (and the existence proof ) of this Lie algebra. Let us a choose an arbitrary basis 1 , . . . , k , where k = dim q, of q. An element a1 1 + . . . + ak n k with ai K(V ) lies in ^ if and only if i=1 ai x·i = 0 for all x V . Here x·i is a linear q function of q given by x·i (q ) = (ad (q )x) for each q q. Choosing a basis {x1 , . . . , xp } of V we get p = dim V equations defining ^ over K(V ). Rank of the arising k в p matrix q M = (mij ) with mij = [i , xj ] is equal to the codimension of a generic Q orbit on V . Therefore ^ is a linear space (over K(V )) of dimension dim qx (with x V generic). q Take any x V and let ^(x) be the image of an obvious "evaluation" homomorphism q from ^ to qx . Set q Vns := {x V | ^(x) = qx }. q Note that if an orbit Qx V
is not of maximal dimension, then x Vns . ns

Lemma 1. The codimansion of V

in V

is at least two.

Set G0 := G . Since G is assumed to be simply connected, G0 is connected. Let L denote a generic stabiliser of the action of G0 on g1 and set l := Lie L. Since the action of G0 on g1 is self dual, the group L is reductive. Set := rk l. For each point y gi (or in g ) let g0,y and i g1,y denote the stabiliser of y in g0 and g1 , respectively. There is a symmetric decomposition gy = g0,y g1,y . Lemma 2. Let Y = g в {y } be a subset of g with y g . Then K[Y ] 0 1 Proof. This is a particular case of [2, Lemma 4]. Set F := K(g )G0 . Clearly F K(g)G . In our case of interest F = Quot K[g ]G0 . Over 1 1 the new field F the action of G0 on g is locally transitive. Therefore Lemma 2 yields the 1 following. Corollary 1. We have K[g ]g
K[g ] 1
G0

g1

K[g =

0,y

].

F F[l ]L and K(g)G K(g ) = = 1

G

0

K(l )L F(l )L . =

Theorem 1. For each Z2 -degeneration g of a simple Lie algebra g, the algebra S (g)g of symmetric invariants is a Polynomial algebra in r variables.

1


Sketch of a proof. Choose a set of homogeneous generators {H1 , . . . , H } F[l ]L . Due to Corollary 1 they can be regarded as rational functions on g . Let U g be the maximal 1 subset such that all Hi are regular on g в U . The complement g \ U is a union of G0 -invariant 0 1 divisors, i.e., a union of zero sets of G0 -invariant regular functions. Therefore, multiplying by dominator, we can make all generators H1 , . . . , H regular. New functions also form a set of generators of K[g ]g K[g ]G0 F over the field F. For that reason we do not change the 1 notation. For y g let y : K[g ] K[g в {y }] 0 be the restriction homomorphism. The algebra of invariants S (g)g is free if and only if there is a set of generators H1 , . . . , H such that y (Hi ) are algebraically independent for the elements y of a big open subset of g . 1 To verify this we need to check that each G0 -invariant divisors X g1 contains a point x with x (Hi ) being algebraically independent. Making use of Lemma 1, this question can be reduced to simpler question concerning a so called degeneration of l to g0,x with x (g )ns . 1 For several involution Theorem 1 was proved by Panyushev [1]. The last part of our proof is a case by case verification for the remaining ones: · (E6 , F4 ), (E7 , E6 K), (E8 , E7 sl2 ), (E6 , so10 K), (E7 , so12 sl2 ); · (sp
2n+2m

, sp2n sp2m ), with n

m; (so2n , gln ); (sl2n , sp2n ). References

[1] D. Panyushev, On the coadjoint representation of Z2 -contractions of reductive Lie algebras, Adv. Math., to appear. [2] E.B. Vinberg, O.S. Yakimova, Complete families of commuting functions for coisotropic Hamiltonian actions, arXiv:math.SG/0511498.

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