The Bessel Period of U(3) and U(2) involving a non-tempered representation

TL;DR

This paper investigates the Bessel period for a specific pair of unitary groups involving a non-tempered representation, revealing that the tempered condition is essential for the conjecture's formulation.

Contribution

It extends the Gross-Prasad conjecture to include non-tempered representations, showing the necessity of the tempered condition for the conjecture's validity.

Findings

The formula for the Bessel period is modified for non-tempered cases.

The central critical L-value differs from the tempered case.

Tempered condition is proven to be indispensable in the conjecture.

Abstract

In \cite{Ha}, Neal Harris has given a refined Gross-Prasad conjecture for unitary group as an analogue of Ichino and Ikeda's paper \cite{Ich} concerning special orthogonal groups. In his paper, he stated a conjecture under the assumption that the pair of given representations should be tempered. In this paper, we consider a specific pair involving a non-tempered one. In this case, an analogous formula still exists but the central critical $L$-value is slightly different with the one in the conjecture. As a corollary, this verifies that the tempered condition is indispensable in formulating the conjecture.