The multiplicity problems for the unitary Ginzburg-Rallis 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.