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Test functions for Shimura varieties: the Drinfeld case. (English) Zbl 1014.20002
Let \((G,X,K)\) be a Shimura datum with reflex field \(E\), and let \(\mathfrak p\) be a prime ideal of \(E\) lying over a prime number \(p\) such that \(E_{\mathfrak p}\) is an unramified extension of \(\mathbb{Q}_p\). Let \(F\) be an unramified extension containing \(E_{\mathfrak p}\), and assume that \(G\) is quasi-split over \(F\). If \(\mu\) is a miniscule cocharacter of the group \(G_{\overline E_{\mathfrak p}}\), then the associated Bernstein function \(z_\mu\) belongs to the center of the Iwahori-Hecke algebra of the \(p\)-adic group \(G(F)\). According to a conjecture of Kottwitz, the Bernstein function \(z_\mu\) plays an important role in the bad reduction of a Shimura variety with Iwahori level structure. More precisely, the function trace of Frobenius on nearby cycles is conjecturally expressible in terms of \(z_\mu\). In this paper the author obtains an explicit formula for \(z_\mu\) in terms of the standard basis for the Iwahori-Hecke algebra. He also uses this formula to prove Kottwitz’s conjecture for a particular Shimura variety attached to \(G=\text{GU}(1,d-1)\) and \(\mu=(1,0^{d-1})\), known as the Drinfeld case.

20C08 Hecke algebras and their representations
11G18 Arithmetic aspects of modular and Shimura varieties
Full Text: DOI
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