Unitary representations and theta correspondence for type I classical groups

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 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.