The multiplicity problems for the unitary Ginzburg-Rallis models

TL;DR

This paper investigates the local multiplicity problems for Ginzburg-Rallis models in the context of unitary and unitary similitude groups, establishing multiplicity formulas for tempered representations and their sums over Vogan L-packets.

Contribution

It proves new multiplicity formulas for tempered representations in both unitary and similitude cases, linking them to Vogan L-packets.

Findings

Multiplicity sum equals 1 for unitary similitude groups.

Multiplicity sum equals 2 for unitary groups.

Established local trace formula for the models.

Abstract

We consider the local multiplicity problems of the analogy of the Ginzburg-Rallis model for the unitary group and the unitary similitude group cases. For the unitary similitude group case, by proving a local trace formula for the model, we are able to prove a multiplicity formula for all tempered representations, which implies that the summation of the multiplicities is equal to $1$ over every tempered local Vogan $L$-packet. For the unitary group case, we also prove a multiplicity formula for all tempered representations which implies that the summation of the multiplicities is equal to $2$ over every tempered local Vogan $L$-packet.