## The adjoint representation in rings of functions

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- by Eric Sommers and Peter Trapa PDF
- Represent. Theory
**1**(1997), 182-189 Request permission

## Abstract:

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 of $W$ 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 $\mathfrak {g}:=\operatorname {Lie}(G)$ 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.

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## Additional Information

**Eric Sommers**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: esommers@math.mit.edu
**Peter Trapa**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: ptrapa@math.mit.edu
- Received by editor(s): April 28, 1997
- Received by editor(s) in revised form: May 31, 1997
- Published electronically: July 10, 1997
- Additional Notes: Supported in part by the National Science Foundation
- © Copyright 1997 American Mathematical Society
- Journal: Represent. Theory
**1**(1997), 182-189 - MSC (1991): Primary 22E46, 05E99
- DOI: https://doi.org/10.1090/S1088-4165-97-00029-0
- MathSciNet review: 1457243