The conservation relation for cuspidal representations

Abstract

Abstract. -Let π ∈ Cusp (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 ± 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 [HKS, 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.