Representation of ax+b group and Dirichlet Series

TL;DR

This paper explores the representation theory of the ax+b group, characterizes smooth vectors, and connects Mellin transforms of matrix coefficients to Dirichlet series, providing conditions for integrability and estimates for their norms.

Contribution

It introduces new characterizations of smooth vectors in the ax+b group representations and links Mellin transforms of matrix coefficients to Dirichlet series with integrability conditions.

Findings

Characterization of smooth vectors for the ax+b group representations.

Expression of Mellin transforms of matrix coefficients in terms of Dirichlet series.

Conditions for local integrability and estimates of L^2 norms of matrix coefficients.

Abstract

Let $G$ be the $ax+b$ group. There are essentially two irreducible infinite dimensional unitary representations of $G$, $(\mu, L^2(\mathbb R^+))$ and $(\mu^*, L^2(\mathbb R^+))$. In this paper, we give various characterizations about smooth vectors of $\mu$ and their Mellin transforms. Let $\f d$ be a linear sum of delta distributions supported on the the positive integers $\mathbb Z^+$. We study the Mellin transform of the matrix coefficients $\mu_{ \f d, f}(a)$ with $f$ smooth. We express these Mellin transforms in terms of the Dirichlet series $L(s, \f d)$. We determine a sufficient condition such that the generalized matrix coefficient $\mu_{\f d, f}$ is a locally integrable function and estimate the $L^2$-norms of $\mu_{\f d, f}$ over the Siegel set. We further derive an inequality which may potentially be used to study the Dirichlet series $L(s, \f d)$.