An inequality for local unitary Theta correspondence

Abstract

Given a representation π of a local unitary group G and another local unitary group H, either the Theta correspondence provides a representation θ H (π) of H or we set θ H (π)=0. If G is fixed and H varies in a Witt tower, a natural question is: for which H is θ H (π)≠0 ? For given dimension m there are exactly two isometry classes of unitary spaces that we denote H m ± . For ϵ∈{0,1} let us denote m ϵ ± (π) the minimal m of the same parity of ϵ such that θ H m ± (π)≠0, then we prove that m ϵ + (π)+m ϵ - (π)≥2n+2 where n is the dimension of π.