Eigenfunctions of the Weil representation of unitary groups of one variable

Abstract

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the p-adic case when p is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke L-functions in separate papers. We also recover Moen’s results on when a local theta lifting from U(1) to itself does not vanish. 0. Introduction and Notation Let E/F be a quadratic extension of local fields. If (V, ( , )) is an Hermitian space over E, and (W, 〈 , 〉) is a skew-Hermitian space over E, the unitary groups G = G(W ) and G′ = G(V ) form a reductive dual pair in Sp(W), where W = V ⊗W has the symplectic form 12 trE/F ( , ) ⊗ 〈 , 〉 over F . According to a well-known result, this dual pair splits in the metaplectic cover of Sp(W), and thus has a Weil representation ω. We consider the very special case in which dimE V = dimEW = 1. In this case, G ∼= G′ = U(1) = E is compact and abelian, where E is the kernel of the norm map N : E∗ −→ F ∗. So irreducible representations of G are just characters, and the Weil representation has a direct sum decomposition: