The combinatorics of Bernstein functions
Author:
Thomas J. Haines
Journal:
Trans. Amer. Math. Soc. 353 (2001), 12511278
MSC (2000):
Primary 20C08; Secondary 14G35
Published electronically:
November 8, 2000
MathSciNet review:
1804418
Fulltext PDF Free Access
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Abstract: A construction of Bernstein associates to each cocharacter of a split adic group an element in the center of the IwahoriHecke 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:
http://dx.doi.org/10.1090/S0002994700027161
PII:
S 00029947(00)027161
Received by editor(s):
July 12, 1999
Published electronically:
November 8, 2000
Article copyright:
© Copyright 2000
American Mathematical Society
