Relationship between differential intertwining operators on the indefinite orthogonal and unitary groups

Abstract

In this thesis, we will analyze degenerate principal series representations realized as smooth induced representations for the indefinite orthogonal and unitary groups G=SO_0(2p,2q) and H=U(p,q). We will induce from smooth characters of maximal-parabolic subgroups, which will each depend on a continuous parameter and a discrete parameter. Each of the principal series representations has an associated differential intertwining operator that can be identified as the right action of an element of the universal enveloping algebra. These operators correspond to the Euclidean and Heisenberg wave operators, respectively, and because of the group-invariance of these operators, their kernels will be subrepresentations.

A key result in this thesis is to establish a connection between the kernels for Euclidean and Heisenberg kernels. We will present a family of integral operators that provide a map between the principal series in the two settings which acts as a projection map of K-finite spaces. The most important feature of these integral operators, is that for the continuous parameter p+q-2, they intertwine the action of the differential operators the principal series. In particular, these integral operators take the kernels of the Euclidean wave operator in the orthogonal setting to the kernel of the Heisenberg wave operator in the unitary setting.