Zero weight space for tori inside a division Algebra

Abstract

Let F be a non-Archimedean local eld of residue characteristic p. Consider a division algebra D over F of index `, where ` is an odd prime and ` 6= p. Let be an irreducible representation of D with trivial central character. Let K=F be any eld extension of degree ` in D. We compute the complex dimension of K - invariant vectors of as both and K vary.