Unitary representations and theta correspondence for type I classical groups

In this paper, we discuss the positivity of the Hermitian form ð; Þp introduced by Li in Invent. Math. 27 (1989) 237–255. Let ðG1;G2Þ be a type I dual pair with G1 the smaller group. Let p be an irreducible unitary representation in the semistable range of yðMG1;MG2Þ (see Communications in Contemporary Mathematics, Vol. 2, 2000, pp. 255–283). We prove that the invariant Hermitian form ð; Þp is positive semidefinite under certain restrictions on the size of G2 and a mild growth condition on the matrix coefficients of p: Therefore, if ð; Þp does not vanish, yðMG1;MG2ÞðpÞ 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. r 2003 Elsevier Science (USA). All rights reserved.