The combinatorics of Bernstein functions

Author:
Thomas J. Haines

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1251-1278

MSC (2000):
Primary 20C08; Secondary 14G35

Published electronically:
November 8, 2000

MathSciNet review:
1804418

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Abstract | References | Similar Articles | Additional Information

A construction of Bernstein associates to each cocharacter of a split -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 ; 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 ``-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.

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Additional Information

**Thomas J. Haines**

Affiliation:
University of Toronto, Department of Mathematics, 100 St. George Street, Toronto, Ontario, Canada M5S 1A1

Email:
haines@math.toronto.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02716-1

Received by editor(s):
July 12, 1999

Published electronically:
November 8, 2000

Article copyright:
© Copyright 2000
American Mathematical Society