Symplectic Representations and Its Irreducible Decomposition under the Dual Pair(U(1),U(1,n))
We considered the problem of irreducible decomposition of the symplectic representation under the dual pair(U(1),U(1,n)).The main idea is to construct the intertwining operator of representation of SL(2,R)and U(1,n),and the corresponding representation space is decomposed into irreducible SL(2,R)×U(1,n)-module.The main technique used in this paper is the Fourier-Poisson transform,combining the Plancherel formula of this transform,we obtain the spectrum of the symplectic representation.
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