Some Results on Homogeneity Results for GLn

Abstract

Let K be a nonarchimedian local field of characteristic zero with a ring of integers R and prime ideal p. Let G be a connected reductive algebraic group defined over K with Lie algebra g. In one of DeBacker's papers, he established a range of validity for the Harish-Chandra–Howe local expansion for characters of admissible irreducible representations of G under some conditions, and he established an analogous homogeneity result on the Lie algebra of G, again with some restrictions. These restrictions are, essentially, restrictions on the characteristic of the residue field k of K. While the hope of removing the characteristic restriction for G in general is not possible, in the case where G =GLn, we still have hope. Our primary goal here is to move towards a proof of a special type of case for a certain key step which plays a prominent role for the homogeneity result of GLn without characteristic restriction. Finally, In the end, I will also provide a full written proof for the homogeneity result of GL3.