Archimedean zeta integrals on U(2,1)U(2,1)

It is known that for a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L  -functions, which have applications to arithmetic. In this paper we explicitly calculate archimedean zeta integrals of this type for U(2,1)U(2,1). In particular we compute the matrix coefficients of Weil representations using joint harmonics in the Fock model, and those of discrete series using Schmid operators.