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The adjoint representation in rings of functions


Authors: Eric Sommers and Peter Trapa
Journal: Represent. Theory 1 (1997), 182-189
MSC (1991): Primary 22E46, 05E99
DOI: https://doi.org/10.1090/S1088-4165-97-00029-0
Published electronically: July 10, 1997
MathSciNet review: 1457243
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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 $\ensuremath {\mathfrak g}:=Lie(G)$ 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.


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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

DOI: https://doi.org/10.1090/S1088-4165-97-00029-0
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
Article copyright: © Copyright 1997 American Mathematical Society