Functions on Symmetric Spaces and Oscillator Representation

In this paper, we study the L functions on U(2n)/O(2n) and Mp(n,R). We relate them using the oscillator representation. We first study some isometries between various L spaces using the compactification we defined in [6]. These isometries were first introduced by Betten-Ólafsson in [3] . We then give a description of the matrix coefficients of the oscillator representation ω in terms of algebraic functions on U(2n)/O(2n). The structure of L(U(2n)/O(2n)) enables us to decompose the L space of odd functions on Mp(n,R) into a finite orthogonal direct sum, from which an orthogonal basis for L(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 (⊗ω)⊗ (⊗2n+1−pω∗) for p with a fixed parity, generalizing some result in [15]. Consequently, we prove some results in the papers by Adam-Barbasch [2] and by Moeglin [17] without going through the details of the Langlands-Vogan parameter. In a weak sense, our paper also provides an analytic alternative to the Adam-Barbasch Theorem on Howe duality ( [14]).