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A New Bound for the Uniform Admissibility Theorem

In "All p-adic reductive groups are tame" Bernstein proved that for a reductive group G over a local non-archimedean field F and a compact open subgroup K of G there exists a uniform bound N(G,K) such that for every irreducible, smooth, and admissible representation V of G the dimension of the subspace of K-fixed vectors in V is bounded by N(G,K). In this note I repeat the proof of Bernstein and give my proof to one of the two main lemmas. The new proof of this lemma gives a new, sharper bound for the constant N(G,K).