Two Topics on Local Theta Correspondence

Abstract

Theta correspondence is a tool to study representations of classical groups. In my thesis, I will discuss two topics both devote to understand the relationship between theta correspondence and the general theories of representations of real reductive groups.

In the first topic, we study the composition of theta lifting and transfer of K-types (certain Zuckerman functor). We show that the transfer of the theta lift of a one-dimensional representation is determined by its K-spectrum. As a consequence, we show that transfer of K-type and theta lifting are "compatible" in these cases. Moreover, we extend the result to theta lifts of unitary highest weight modules.

In the second topic, we developed a geometric framework to study the invariants of theta lifts. Under this framework, we derived the formula on isotropic representations of theta lifts of finite dimensional unitary representations and unitary highest weight modules. These formulae imply that taking associated cycle and theta lifting are compatible for unitary highest weight modules in certain ranges (including the stable range). At the same time, we also get some non-compatible cases outside the stable range. Furthermore, we show that theta lifts of certain highest weight modules is isomorphic to the spaces of global sections of algebraic vector bundles on nilpotent orbits as K-modules.