Representations of ax + b group and Dirichlet Series

Document Type

Article

Publication Date

3-1-2021

Abstract

Let G be the ax + b group. There are essentially two irreducible infinite dimensional unitary representations of G, (μ, L2(R +)) and (μ*, L2(R +)). In this paper, we give various characterizations about smooth vectors of μ and their Mellin transforms. Let d be a linear sum of delta distributions supported on the positive integers Z +. We study the Mellin transform of the matrix coefficients μd, f (a) with f smooth. We express these Mellin transforms in terms of the Dirichlet series L(s, d). We determine a sufficient condition such that the generalized matrix coefficient μd, f is a locally integrable function and estimate the L2-norms of μd, f over the Siegel set. We further derive an inequality which may potentially be used to study the Dirichlet series L(s, d).

Publication Source (Journal or Book title)

Journal of the Ramanujan Mathematical Society

First Page

73

Last Page

84

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