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

Eva Viehmann

• The aim of this article is to define and study a new invariant of elements of loop groups that is invariant under s -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 s -conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic s -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