The combinatorics of Bernstein functions

Haines, Thomas

doi:10.1090/S0002-9947-00-02716-1

The combinatorics of Bernstein functions
HTML articles powered by AMS MathViewer

by Thomas J. Haines PDF

Trans. Amer. Math. Soc. 353 (2001), 1251-1278 Request permission

Abstract:

A construction of Bernstein associates to each cocharacter of a split $p$-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz’ conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of $Gl_n$; one example can be used to verify Kottwitz’ conjecture for a special class of Shimura varieties (the “Drinfeld case”). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the “$\mu$-admissible” set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.

References

I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
J.-F. Dat, Caractères à valeurs dans le centre de Bernstein, J. Reine Angew. Math. 508 (1999), 61–83 (French, with English summary). MR 1676870, DOI 10.1515/crll.1999.031
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
T. Haines, Test Functions for Shimura Varieties: The Drinfeld Case, preprint (1998), to appear in Duke Math. Journal.
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
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
Shin-ichi Kato, Spherical functions and a $q$-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), no. 3, 461–468. MR 662602, DOI 10.1007/BF01389223
Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI 10.1090/S0894-0347-1992-1124982-1
R. Kottwitz and M. Rapoport, Minuscule Alcoves for $Gl_n$ and $GSp_{2n}$, preprint 1998, to appear in Manuscripta Mathematica.
M. Rapoport, On the bad reduction of Shimura varieties, Automorphic forms, Shimura varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 253–321. MR 1044832
M. Rapoport, Letter to Waldspurger, January/February 1989.
J.L. Waldspurger, Letter to Rapoport, February 1989.