# Journal of the Ramanujan Mathematical Society

Volume 36, Issue 1, March 2021  pp. 73–84.

Representations of ax + b group and Dirichlet Series

Authors:  Hongyu He
Author institution:Department of Mathematics, Louisiana State University Baton Rouge, LA 70803, USA

Summary:  Let G be the ax + b group. There are essentially two irreducible infinite dimensional unitary representations of G, (μ, L{2}(R+)) and (μ{*}, L{2}(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 μ{d, 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 μ{d, f} over the Siegel set. We further derive an inequality which may potentially be used to study the Dirichlet series L(s, d).