Let F be a local field with ring of integers O. Let G denote the general linear group and T the subgroup of diagonal matrices.
In a remarkable 1973, Roger Howe defined a Satake-type isomorphism for every character of T(O). If this character is trivial, we recover the usual Satake Isomorphism.
In this talk I will give an overview of Howe's construction and its applications to geometric representation theory.