The conservation relation for cuspidal representations

Abstract

Let π ∈ Cusp (U(Vn)) be a smooth cuspidal irreducible representation of a unitary group U(Vn) of dimension n over a non-Archimedean locally compact field. Let W± m the two isomorphism classes of Hermitian spaces of dimension m, and the denote by τ + ∈ Cusp U(W+m+) and τ− ∈ Cusp 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 [M. Harris, S. S. Kudla, J. Sweet, Theta dichotomy for unitary groups, Journal of the AMS, vol. 9, pp. 941-1004 (1996), Speculations 7.5 and 7.6]. We prove similar equalities for the other dual pairs of type I : the symplecticorthogonal dual pairs and the quaternionic dual pairs.