Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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


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.
  • 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.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C08, 14G35
  • Retrieve articles in all journals with MSC (2000): 20C08, 14G35
Additional Information
  • Thomas J. Haines
  • Affiliation: University of Toronto, Department of Mathematics, 100 St. George Street, Toronto, Ontario, Canada M5S 1A1
  • MR Author ID: 659516
  • Email:
  • Received by editor(s): July 12, 1999
  • Published electronically: November 8, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 1251-1278
  • MSC (2000): Primary 20C08; Secondary 14G35
  • DOI:
  • MathSciNet review: 1804418