## The adjoint representation in rings of functions

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- by Eric Sommers and Peter Trapa
- Represent. Theory
**1**(1997), 182-189 - DOI: https://doi.org/10.1090/S1088-4165-97-00029-0
- Published electronically: July 10, 1997
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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 $\mathfrak {g}:=\operatorname {Lie}(G)$ 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.

## References

- Dean Alvis and George Lusztig,
*On Springer’s correspondence for simple groups of type $E_{n}$ $(n=6,\,7,\,8)$*, Math. Proc. Cambridge Philos. Soc.**92**(1982), no. 1, 65–78. With an appendix by N. Spaltenstein. MR**662961**, DOI 10.1017/S0305004100059703 - Walter Borho and Hanspeter Kraft,
*Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen*, Comment. Math. Helv.**54**(1979), no. 1, 61–104 (German, with English summary). MR**522032**, DOI 10.1007/BF02566256 - Bram Broer,
*Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety*, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 1–19. MR**1327529**, DOI 10.1007/978-1-4612-0261-5_{1} - Hans Grauert and Oswald Riemenschneider,
*Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen*, Invent. Math.**11**(1970), 263–292 (German). MR**302938**, DOI 10.1007/BF01403182 - Wim H. Hesselink,
*Polarizations in the classical groups*, Math. Z.**160**(1978), no. 3, 217–234. MR**480765**, DOI 10.1007/BF01237035 - Wim H. Hesselink,
*Characters of the nullcone*, Math. Ann.**252**(1980), no. 3, 179–182. MR**593631**, DOI 10.1007/BF01420081 - Bertram Kostant,
*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327–404. MR**158024**, DOI 10.2307/2373130 - Hanspeter Kraft and Claudio Procesi,
*Closures of conjugacy classes of matrices are normal*, Invent. Math.**53**(1979), no. 3, 227–247. MR**549399**, DOI 10.1007/BF01389764 - Hanspeter Kraft and Claudio Procesi,
*On the geometry of conjugacy classes in classical groups*, Comment. Math. Helv.**57**(1982), no. 4, 539–602. MR**694606**, DOI 10.1007/BF02565876 - G. I. Lehrer and T. Shoji,
*On flag varieties, hyperplane complements and Springer representations of Weyl groups*, J. Austral. Math. Soc. Ser. A**49**(1990), no. 3, 449–485. MR**1074514**, DOI 10.1017/S1446788700032444 - George Lusztig,
*Singularities, character formulas, and a $q$-analog of weight multiplicities*, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR**737932** - G. Lusztig and N. Spaltenstein,
*Induced unipotent classes*, J. London Math. Soc. (2)**19**(1979), no. 1, 41–52. MR**527733**, DOI 10.1112/jlms/s2-19.1.41 - William M. McGovern,
*Rings of regular functions on nilpotent orbits and their covers*, Invent. Math.**97**(1989), no. 1, 209–217. MR**999319**, DOI 10.1007/BF01850661 - Peter Orlik and Louis Solomon,
*Combinatorics and topology of complements of hyperplanes*, Invent. Math.**56**(1980), no. 2, 167–189. MR**558866**, DOI 10.1007/BF01392549 - Peter Orlik and Louis Solomon,
*Coxeter arrangements*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 269–291. MR**713255**, DOI 10.1090/pspum/040.2/713255 - E. Sommers,
*A family of affine Weyl group representations,*submitted to Transformation Groups. - N. Spaltenstein,
*On the reflection representation in Springer’s theory*, Comment. Math. Helv.**66**(1991), no. 4, 618–636. MR**1129801**, DOI 10.1007/BF02566669

## Bibliographic 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