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 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.