## Generalized exponents of small representations. I

HTML articles powered by AMS MathViewer

- by Bogdan Ion PDF
- Represent. Theory
**13**(2009), 401-426 Request permission

## Abstract:

This is the first paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. The main part of this paper illustrates the overall structure of the argument on root systems of type $A$ and discusses the relationship with the Lascoux-Schützenberger charge formula## References

- W. M. Beynon and G. Lusztig,
*Some numerical results on the characters of exceptional Weyl groups*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 3, 417–426. MR**503002**, DOI 10.1017/S0305004100055249 - Abraham Broer,
*The sum of generalized exponents and Chevalley’s restriction theorem for modules of covariants*, Indag. Math. (N.S.)**6**(1995), no. 4, 385–396. MR**1365182**, DOI 10.1016/0019-3577(96)81754-X - Ranee Kathryn Brylinski,
*Limits of weight spaces, Lusztig’s $q$-analogs, and fiberings of adjoint orbits*, J. Amer. Math. Soc.**2**(1989), no. 3, 517–533. MR**984511**, DOI 10.1090/S0894-0347-1989-0984511-X - Lynne M. Butler,
*Subgroup lattices and symmetric functions*, Mem. Amer. Math. Soc.**112**(1994), no. 539, vi+160. MR**1223236**, DOI 10.1090/memo/0539 - Ira M. Gessel,
*Multipartite $P$-partitions and inner products of skew Schur functions*, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR**777705**, DOI 10.1090/conm/034/777705 - E. A. Gutkin,
*Representations of the Weyl group in the space of vectors of zero weight*, Uspehi Mat. Nauk**28**(1973), no. 5 (173), 237–238 (Russian). MR**0409671** - Bogdan Ion,
*The Cherednik kernel and generalized exponents*, Int. Math. Res. Not.**36**(2004), 1869–1895. MR**2058356**, DOI 10.1155/S1073792804133485 - B. Ion, Generalized exponents of small representations. II. http://arxiv.org/abs/0904.2487
- B. Ion, Generalized exponents of small representations. III. In preparation.
- Bertram Kostant,
*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327–404. MR**158024**, DOI 10.2307/2373130 - Bertram Kostant,
*On Macdonald’s $\eta$-function formula, the Laplacian and generalized exponents*, Advances in Math.**20**(1976), no. 2, 179–212. MR**485661**, DOI 10.1016/0001-8708(76)90186-9 - Alain Lascoux and Marcel-Paul Schützenberger,
*Sur une conjecture de H. O. Foulkes*, C. R. Acad. Sci. Paris Sér. A-B**286**(1978), no. 7, A323–A324 (French, with English summary). MR**472993** - G. Lusztig,
*Irreducible representations of finite classical groups*, Invent. Math.**43**(1977), no. 2, 125–175. MR**463275**, DOI 10.1007/BF01390002 - 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** - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - Mark Reeder,
*Zero weight spaces and the Springer correspondence*, Indag. Math. (N.S.)**9**(1998), no. 3, 431–441. MR**1692153**, DOI 10.1016/S0019-3577(98)80010-4 - R. Steinberg,
*A geometric approach to the representations of the full linear group over a Galois field*, Trans. Amer. Math. Soc.**71**(1951), 274–282. MR**43784**, DOI 10.1090/S0002-9947-1951-0043784-0 - John R. Stembridge,
*First layer formulas for characters of $\textrm {SL}(n,\textbf {C})$*, Trans. Amer. Math. Soc.**299**(1987), no. 1, 319–350. MR**869415**, DOI 10.1090/S0002-9947-1987-0869415-X - John R. Stembridge,
*On the eigenvalues of representations of reflection groups and wreath products*, Pacific J. Math.**140**(1989), no. 2, 353–396. MR**1023791**, DOI 10.2140/pjm.1989.140.353 - John R. Stembridge,
*Graded multiplicities in the Macdonald kernel. I*, IMRP Int. Math. Res. Pap.**4**(2005), 183–236. MR**2199453**, DOI 10.1155/IMRP.2005.183 - David A. Vogan Jr.,
*Associated varieties and unipotent representations*, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR**1168491**

## Additional Information

**Bogdan Ion**- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260; Algebra and Number Theory Research Center, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei St., Bucharest, Romania
- MR Author ID: 645344
- Email: bion@pitt.edu
- Received by editor(s): April 18, 2009
- Received by editor(s) in revised form: June 26, 2009
- Published electronically: September 10, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**13**(2009), 401-426 - MSC (2000): Primary 17B10
- DOI: https://doi.org/10.1090/S1088-4165-09-00359-8
- MathSciNet review: 2540703