Restrictions of certain degenerate principal series of the universal covering of the symplectic group

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Let S̃p(n,ℝ) be the universal covering of the symplectic group. In this paper, we study the restrictions of the degenerate unitary principal series I(ε, t) of fSp(n,ℝ) onto S̃p(p,ℝ) S̃p(n - p,ℝ). We prove that if n ≥ 2p, I(ε, t)| S̃p(p,ℝ)fSp(n-p,ℝ) is unitarily equivalent to an L2 -space of sections of a homogeneous line bundle L 2(S̃p(n - p,ℝ) ̃ GL(n-2p)N ℂε,t+p) (see Theorem 1.1). We further study the restriction of complementary series C(ε, t) onto Ũ(n - p)S̃p(p,ℝ). We prove that this restriction is unitarily equivalent to I(ε, t)|Ũ (n-p)S̃p(p,ℝ) for t ̃ ε ℝ. Our results suggest that the direct integral decomposition of C(ε, t)| S̃p(p,ℝ)S̃p(n-p,ℝ) will produce certain complementary series for S̃p(n - p,ℝ) ([He09]). © 2010 Heldermann Verlag.

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Journal of Lie Theory

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