Truncations of level $1$ of elements in the loop group of a reductive group

Eva Viehmann

doi:https://doi.org/10.4007/annals.2014.179.3.3

Abstract

The aim of this article is to define and study a new invariant of elements of loop groups that is invariant under $\sigma$-conjugation by a hyperspecial maximal open subgroup and that we call the truncation of level 1. We classify truncations of level 1 and describe their specialization behavior. Furthermore, we prove group-theoretic conditions for the set of $\sigma$-conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic $\sigma$-conjugacy class in any given truncation stratum. In the last section we relate our invariant to the Ekedahl-Oort stratification of the Siegel moduli space and to generalizations to other PEL Shimura varieties.

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Truncations of level $1$ of elements in the loop group of a reductive group

Abstract

Authors