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

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.