A generic $H$-invariant in a multiplicity-free $G$-action

Abstract

For a prehomogeneous action of a reductive group G on a vector space V, we construct a formal power series L that is shown to have a nonzero projection to every G-isotypic component of P(V). When (G, V) is multiplicity-free, these Fourier components of the function L provide all the H-invariants in P(V) for some spherical subgroup H of G. Three interesting examples are presented. I. MAIN RESULT Let G be a connected reductive complex algebraic group, G D CV . Suppose that G acts on a complex vector space V prehomogeneously, i.e., G has a Zariski open (hence dense) orbit in V. From [Se], G acts on V* prehomogeneously. Fix 0 $& q in the open G-orbit in V*. Consider L = en = >n>0 q/n! as a formal power series on V. Let P(V) = > P(V), afEG be the G-isotypic decomposition of P( V), the algebra of polynomial functions on V. Theorem 1. L has a nonzero projection to all P(V)q such that P(V), :$ 0. Let H be the stabilizer in G of 0. By definition, b is an H-invariant, as is the projection of L = en to P(V)q . Let L ZLa aEG be the G-isotypic decomposition of L, thus we have 0 :$ La e P(V)q n P(V)H, if P(V)a $O. Now suppose the action of G on V is multiplicity-free, namely, the natural action of G on P(V) is multiplicity-free as a representation of G. From [Se, VK], G acts on V (V*, resp.) prehomogeneously. Let q, H, L be defined as in above. In this case, since H C G is spherical (see [HU] for example), i.e., dimc ,H < 1 for any irreducible finite-dimensional representation of G, and Received by the editors October 23, 1990 and, in revised form, December 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20G20, 15A72. Q) 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page