Vol. 84, No. 2, 1979

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On the reduction of certain degenerate principal series representations of SP(n, C)

Thomas Farmer

Vol. 84 (1979), No. 2, 291–303

This paper has its origins in the problem of proving irreducibility or reducibility for principal series representations of certain noncompact, complex, semi-simple groups by Fourier-analytic methods; for example, the abelian methods of Gelfand-Naimark for Sl(n,C), and the non commutative (nilpotent) methods of K. Gross for Sp(n,C). As is well-known, principal series representations are induced from unitary characters of a parabolic subgroup, the series being termed “nondegenerate” if the parabolic is minimal (i.e., the Borel subgroup) and otherwise “degenerate”. Here we consider degenerate principal series for Sp(n,C) corresponding to maximal parabolic subgroups (more general than the situation studied by Gross) and reduce them with respect to the “opposite” parabolic. Let n1 denote the complex dimension of the isotropic subspace corresponding to the maximal parabolic, let 0 < n1 < n, and n0 = n n1. The resulting reduction is described in terms of the natural representation of the complex orthogonal group O(n1,C) acting on the space L2(Cn1×n0) and the tensor product of n1 copies of the oscillator representation of Sp(n0,C). In the terminology introduced by R. Howe, this harmonic analysis reduces to the theory of a “dual reductive pair”, and any further resolution of the question of irreducibility by these methods will depend upon the study of the oscillator representations for such a dual reductive pair.

Mathematical Subject Classification 2000
Primary: 22E45
Received: 25 October 1978
Published: 1 October 1979
Thomas Farmer