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The combinatorics of Bernstein functions. (English) Zbl 0962.14018
Author’s 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 functions associated to a minuscule cocharacter (the case of 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 case of Shimura varieties (the “Drinfeld case”). In addition, we prove general facts concerning the support of Bernstein functions, and concerning some important set called “\(\mu\)-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 bad reduction”.

MSC:
14G35 Modular and Shimura varieties
22E35 Analysis on \(p\)-adic Lie groups
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