The conservation relation for cuspidal representations

Let \({\pi \in Cusp ({\rm U}(V_n))}\) be a smooth cuspidal irreducible representation of a unitary group U(V _n ) of dimension n over a non-Archimedean locally compact field. Let \({W^\pm_m}\) be the two isomorphism classes of Hermitian spaces of dimension m, and the denote by \({\tau^+ \in Cusp ({\rm U}(W_{m^+}^+))}\) and \({\tau^- \in Cusp ({\rm U}(W_{m^-}^-))}\) the first non-zero theta lifts of π. In this article we prove that m ⁺ + m ^? = 2n + 2, which was conjectured in Harris et al. (J AMS 9:941–1004, 1996, Speculations 7.5 and 7.6). We prove similar equalities for the other dual pairs of type I: the symplectic-orthogonal dual pairs and the quaternionic dual pairs.