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

Abstract

The aim of this article is to dene and study a new invariant of elements of loop groups that is invariant under σ- conjugation by a hyperspecial max- imal open subgroup and that we call the truncation of level 1. We classify truncations of level 1 and describe their specialization behavior. Further- more, we prove group-theoretic conditions for the set of σ-conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic σ-conjugacy class in any given truncation stratum. In the last section we relate our invariant to the Ekedahl-Oort stratication of the Siegel moduli space and to generalizations to other PEL Shimura varieties. © 2014 Department of Mathematics, Princeton University.