Functions on symmetric spaces and oscillator representation

Abstract

In this paper, we study the L 2 functions on U ( 2 n ) / O ( 2 n ) and Mp ( n , R ) . We relate them using the oscillator representation. We first study some isometries between various L 2 spaces using the compactification we defined in [H. He, An analytic compactification of the symplectic group, J. Differential Geom. 51 (1999) 375–399]. These isometries were first introduced by Betten and Ólafsson in [F. Betten, G. Ólafsson, Causal compactification and Hardy spaces for spaces of Hermitian type, Pacific J. Math. 200 (2) (2001) 273–312]. 1 1 I was informed by Prof. Ólafsson of his work shortly after I finished this paper. We then give a description of the matrix coefficients of the oscillator representation ω in terms of algebraic functions on U ( 2 n ) / O ( 2 n ) . The structure of L 2 ( U ( 2 n ) / O ( 2 n ) ) enables us to decompose the L 2 space of odd functions on Mp ( n , R ) into a finite orthogonal direct sum, from which an orthogonal basis for L 2 ( Mp ( n , R ) ) is obtained. In addition, our decomposition preserves both left and right Mp ( n , R ) -action. Using this, we define the signature of tempered genuine representations of Mp ( n , R ) . Our result implies that every genuine discrete series representation occurs as a subrepresentation in one and only one of ( ⊗ p ω ) ⊗ ( ⊗ 2 n + 1 − p ω ∗ ) for p with a fixed parity, generalizing some result in [M. Kashiwara, M. Vergne, On the Segal–Shale–Weil representations and harmonic polynomials, Invent. Math. 44 (1978) 1–47]. Consequently, we prove some results in the papers by Adams and Barbasch [J. Adams, D. Barbasch, Genuine representations of the metaplectic group, Compos. Math. 113 (1) (1998) 23–66] and by Móeglin [C. Móeglin, Correspondance de Howe pour les paires reductives duales: quelques calculs dans le cas archimédien, J. Funct. Anal. 85 (1) (1989) 1–85] without going through the details of the Langlands–Vogan parameter. In a weak sense, our paper also provides an analytic alternative to the Adams–Barbasch theorem on Howe duality [R. Howe, Transcending invariant theory, J. Amer. Math. Soc. 2 (1989) 535–552].