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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.

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Journal of Functional Analysis

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