The Adjoint Representation in Rings of Functions

Let G be a connected, simple Lie group of rank n defined over the complex numbers. To a parabolic subgroup P in G of semisimple rank r, one can associate n−r positive integers coming from the theory of hyperplane arrangements (see P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189; Coxeter arrangements, in Proc. of Symposia in Pure Math., Vol. 40 (1983) Part 2, 269-291). In the case r=0, these numbers are just the usual exponents of the Weyl group W of G. These n−r numbers are called coexponents. Spaltenstein and Lehrer-Shoji have proven the observation of Spaltenstein that the degrees in which the reflection representation ofW occurs in a Springer representation associated to P are exactly (twice) the coexponents (see N. Spaltenstein, On the reflection representation in Springer’s theory, Comment. Math. Helv. 66 (1991), 618-636 and G. I. Lehrer and T. Shoji, On flag varieties, hyperplane complements and Springer representations of Weyl groups, J. Austral. Math. Soc. (Series A) 49 (1990), 449-485). On the other hand, Kostant has shown that the degrees in which the adjoint representation of G occurs in the regular functions on the variety of regular nilpotents in g := Lie(G) are the usual exponents (see B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404). In this paper, we extend Kostant’s result to Richardson orbits (or orbit covers) and we get a statement which is dual to Spaltenstein’s. We will show that the degrees in which the adjoint representation of G occurs in the regular functions on an orbit cover of a Richardson orbit associated to P are also the coexponents.