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.
