Theta dichotomy for unitary groups

Abstract

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of arithmetic interest [12]. This conjectural picture is based, to some extent, on the now classic work of Tunnell [47] and Waldspurger [49] as well as on the case of the triple product L-function [13, 34, 17]. In all of these examples, the local root number detects the existence of a certain type of invariant linear functional. It is possible to set up analogous conjectures for the isometry groups of Hermitian or quaternion-hermitian forms. It turns out that root numbers also play a role in the local and global theta correspondence. Roughly speaking, certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the ‘same size’, e.g., for dual pairs of the form (Sp(n), O(2n+1)) and (U(n), U(n)). As will be seen, the theta correspondence for such pairs is connected with a certain induced representation In(s, χ) at the point s = 0 on the unitary axis. By contrast, the correspondence for the pairs (Sp(n), O(2n)) and (Sp(n), O(2n+ 2)), discussed by Prasad [35], is connected to the behavior of a similar induced representation at the points ± 12 , in which case no epsilon factor arises. In this paper we consider the local theta correspondence for unitary groups in the non-archimedean case. Let F be a non-archimedean local field of characteristic not equal to 2, and let E be a quadratic extension of F . For further notation, see the notation section at the end of this introduction. Let V and W be E vector spaces of dimensions m and n equipped, respectively, with a Hermitian form ( , ) : V × V −→ E and a skew-Hermitian form 〈 , 〉 : W ×W −→ E. Then the isometry groups G(V ) and G(W ) of the spaces V and W form a dual reductive pair in the symplectic group Sp(W), where W = V ⊗E W , viewed as an F vector space of dimension 2mn and equipped with the symplectic form