Good Product Expansions for Tame Elements of p-Adic Groups

Abstract

ABSTRACT. We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

Topological Jordan decompositions

“…Let T be an E-split maximal torus in H (hence in G) such that x belongs to the apartment of T in B(H, E). By Lemma 2.6 of [2], we have that γ p n ss ∈ T(E) 0+ . Let K/E be a finite extension such that γ ss ∈ G(K).…”

“…By Lemma 2.9 of [2], δ ∈ G(E) y . By Lemma 2.8 of [2], for x ∈ (y, z) sufficiently close to y, we have that δ ∈ G(E) +…”

“…If necessary, we will write [2], there is some locally compact subfield F of F such that G, N, and γ are all defined over F . Thus γ is contained in a compact modulo N(F ) subgroup of G(F ), hence a fortiori a compact modulo N subgroup of G.…”

“…Remark 2.12. If F is an algebraic extension of a locally compact field, then p > 0 and, by Lemma 2.2 of [2], we have G = lim −→ G(F ), the limit taken over all locally compact subfields F of F over which G is defined. For such a subfield,…”

“…In [2,Section 5], as preparation for the (positive-depth) character computations of [3], Adler and the author define the notion of a normal approximation of an element of a reductive p-adic group, a refinement of the topological Jordan decomposition. However, for the results of that paper, one needs notions of absolute semisimplicity and topological unipotence that make sense over a discretely valued field F which is not necessarily locally compact.…”