Document Type
Article
Publication Date
4-1-2003
Abstract
In this paper, we discuss the positivity of the Hermitian form (,)π introduced by Li in Invent. Math. 27 (1989) 237-255. Let (G1, G2) be a type I dual pair with G1 the smaller group. Let π be an irreducible unitary representation in the semistable range of θ(MG1, MG2) (see Communications in Contemporary Mathematics, Vol. 2, 2000, pp. 255-283). We prove that the invariant Hermitian form (,)π is positive semidefinite under certain restrictions on the size of G2 and a mild growth condition on the matrix coefficients of π. Therefore, if (,)π does not vanish, θ(MG1, MG2) (π) is unitary. Theta correspondence over ℝ was established by Howe in (J. Amer. Math. Soc. 2 (1989) 535-552). Li showed that theta correspondence preserves unitarity for dual pairs in stable range. Our results generalize the results of Li for type I classical groups (Invent. Math. 27 (1989) 237). The main result in this paper can be used to construct irreducible unitary representations of classical groups of type I. © 2003 Elsevier Science (USA). All rights reserved.
Publication Source (Journal or Book title)
Journal of Functional Analysis
First Page
92
Last Page
121
Recommended Citation
He, H. (2003). Unitary representations and theta correspondence for type I classical groups. Journal of Functional Analysis, 199 (1), 92-121. https://doi.org/10.1016/s0022-1236(02)00170-2
