Discrete Components of Some Complementary Series

TL;DR

This paper investigates the structure of complementary series representations of SO(n,1), demonstrating conditions under which they contain discretely embedded representations of smaller orthogonal groups, with implications for cohomological representations.

Contribution

It establishes new criteria for the presence of discretely embedded complementary series within larger groups and explores their cohomological implications.

Findings

Complementary series of SO(n,1) contain discretely embedded series of SO(m,1) under specific conditions.

Cohomological representations of SO(n,1) of degree less than m/2 contain discretely embedded cohomological representations of SO(m,1).

The results connect the reducibility points and representation types across different orthogonal groups.

Abstract

We show that complementary series representations of SO(n,1) contain discretely complementary series of SO(m,1) provided the continuous parameter is sufficiently close to the first point of reducibility and the representation of the compact part of the Levi is a sufficiently small fundamental representation. We prove as a consequence that cohomological representations of SO(n,1) of degree $i$ less than m/2 contain discretely, cohomogical representations of SO(m,1).