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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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The combinatorics of Bernstein functions
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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.
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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:
  • 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