Functions on symmetric spaces and oscillator representation
In this paper, we study the L2 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 L2 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].11I 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 L2 (U (2 n) / O (2 n)) enables us to decompose the L2 space of odd functions on Mp (n, R) into a finite orthogonal direct sum, from which an orthogonal basis for L2 (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]. © 2006 Elsevier Inc. All rights reserved.
Publication Source (Journal or Book title)
Journal of Functional Analysis
He, H. (2007). Functions on symmetric spaces and oscillator representation. Journal of Functional Analysis, 244 (2), 536-564. https://doi.org/10.1016/j.jfa.2006.11.008