Available in electronic format
Available in print format
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
     

On the restriction of Deligne-Lusztig characters

Author(s): Mark Reeder
Journal: J. Amer. Math. Soc. 20 (2007), 573-602.
MSC (2000): Primary 20C33
Posted: July 14, 2006
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study the multiplicities of Deligne-Lusztig characters upon restriction from a finite reductive group to a finite reductive subgroup. The result is a qualitative formula for the growth of multiplicities in terms of complexity. For restrictions from $ SO_{2n+1}$ to $ SO_{2n}$ we get exact multiplicities.


References:

1.
D. Akhiezer and D. Panyushev, Multiplicities in the branching rules and the complexity of homogeneous spaces, Moscow Math. Jour., 2 no. 1, (2002) pp. 17-33. MR 1900582 (2003d:14057)

2.
E. Bannai, N. Kawanaka, and S.Y. Song, The character table of the Hecke algebra $ \mathcal{H}(GL_{2n}(\mathbf F_q),Sp_{2n}(\mathbf F_q))$, J. Algebra, 129 (1990), no. 2, pp. 320-366. MR 1040942 (91d:20052)

3.
M. Beynon and N. Spaltenstein, Tables of Green Polynomials for exceptional groups, Warwick computer science centre report no. 23 (1986).

4.
A. Borel, Linear algebraic groups, second enlarged edition, Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)

5.
R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, John Wiley & Sons, Ltd., Chichester, 1993. MR 1266626 (94k:20020)

6.
-, Semisimple conjugacy classes and classes in the Weyl group, Jour. of Alg., 260 (2003) pp. 99-110. MR 1973577 (2004b:20071)

7.
S. DeBacker and M. Reeder, Depth-Zero Supercuspidal $ L$-packets and their Stability, preprint 2004.

8.
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math., 103 (1976) pp. 103-161. MR 0393266 (52:14076)

9.
M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE - A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Engrg. Comm. Comput., 7 (1996) pp. 175-210. MR 1486215 (99m:20017)

10.
B. Gross, On the centralizer of a regular, semi-simple, stable conjugacy class, Electronic J. of Representation Theory (2005). MR 2133761 (2006a:20089)

11.
B. Gross and D. Prasad, On the decomposition of a representation of $ SO_n$ when restricted to $ SO_{n-1}$, Canad. J. Math., 44 (1992), pp. 974-1002. MR 1186476 (93j:22031)

12.
B. Gross and M. Reeder, From Laplace to Langlands via representations of orthogonal groups, Bull. Amer. Math. Soc., 43 (2006), pp. 163-205.

13.
T. Hagedorn, Multiplicities in Restricted Representations of $ GL_n(\mathbf F_q)$, $ U_n(\mathbf F_{q^2})$, $ SO_n(\mathbf F_q)$, Ph.D. thesis, Harvard University (1994).

14.
D. Kazhdan, Proof of Springer's hypothesis, Israel J. Math., 28 (1977), pp. 272-286. MR 0486181 (58:5959)

15.
D. Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math., 16 (1972), pp. 1-5. MR 0294351 (45:3421)

16.
G. Lusztig, Green functions and character sheaves, Ann. Math., 131 (1990), pp. 355-408.MR 1043271 (91c:20054)

17.
-, Symmetric spaces over a finite field, Grothendieck Festschrift, III Birkhäuser(1990), pp. 57-81.MR 1106911 (92e:20034)

18.
T. Shoji, On the Green polynomials of a Chevalley group of type $ F_4$, Comm. in Algebra, 10 (1982), pp. 505-543.MR 0647835 (83d:20030)

19.
-, On the Green polynomials of classical groups, Invent. Math., 74 (1983), pp. 239-267.MR 0723216 (85f:20032)

20.
-, Green functions of reductive groups over a finite field, Proc. Symp. Pure Math., (47), Amer. Math. Soc. (1987), pp. 289-301.MR 0933366 (88m:20014)

21.
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36 (1976), pp. 173-207.MR 0442103 (56:491)

22.
-, A construction of representations of Weyl groups, Invent. Math., 44 (1978), pp. 279-293. MR 0491988 (58:11154)

23.
-, A purity result for fixed point varieties in flag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 31 (1984), pp. 271-282.MR 0763421 (86c:14034)

24.
T. A. Springer and R. Steinberg, Conjugacy Classes, Seminar in algebraic groups and related finite groups, Lecture Notes in Math., 131 (1970), pp. 167-266.MR 0268192 (42:3091)

25.
R. Steinberg, On the desingularization of the unipotent variety, Invent. Math., 36 (1976), pp. 209-224. MR 0430094 (55:3101)

26.
B. Srinivasan, Green polynomials for finite classical groups, Comm. in Algebra, 5 (1976), pp. 1241-1259. MR 0498889 (58:16905)

27.
E. Thoma, Die Einschränkung der Charaktere von GL(n,q) auf GL(n-1,q), Math. Z., 119 (1971), pp. 321-338. MR 0288190 (44:5388)


Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 20C33

Retrieve articles in all Journals with MSC (2000): 20C33


Additional Information:

Mark Reeder
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: reederma@bc.edu

DOI: 10.1090/S0894-0347-06-00540-6
PII: S 0894-0347(06)00540-6
Received by editor(s): June 17, 2005
Posted: July 14, 2006
Additional Notes: The author was supported by NSF grant DMS-0207231
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google