The conservation relation for cuspidal representations

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi \in Cusp ({\rm U}(V_n))}$$\end{document} be a smooth cuspidal irreducible representation of a unitary group U(Vn) of dimension n over a non-Archimedean locally compact field. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W^\pm_m}$$\end{document} be the two isomorphism classes of Hermitian spaces of dimension m, and the denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau^+ \in Cusp ({\rm U}(W_{m^+}^+))}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau^- \in Cusp ({\rm U}(W_{m^-}^-))}$$\end{document} 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.