## The combinatorics of Bernstein functions

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- 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.

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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
- MR Author ID: 659516
- Email: haines@math.toronto.edu
- 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: https://doi.org/10.1090/S0002-9947-00-02716-1
- MathSciNet review: 1804418