Deforming Representations of SL(2,R)

TL;DR

This paper introduces a new family of completely reducible representations of SL(2,R) that deform the classical principal series, linking them to invariant Hermitian forms and expanding understanding of representation structures.

Contribution

It constructs a continuous family of representations with the same composition factors as principal series but are fully reducible, revealing new deformation possibilities.

Findings

Established a new family of reducible representations $ ilde{ u}$

Connected these representations to invariant Hermitian forms

Expanded the understanding of representation deformations in SL(2,R)

Abstract

The spherical principal series representations $\pi(\nu)$ of SL(2,$\mathbb R$) is a family of infinite dimensional representations parametrized by $\nu\in\mathbb C$. The representation $\pi(\nu)$ is irreducible unless $\nu$ is an odd integer, in which case it is indecomposable. We find a new continuous family of representations $\Pi(\nu)$ such that $\pi(\nu)$ and $\Pi(\nu)$ have the same composition factors, and $\Pi(\nu)$ is completely reducible, for all $\nu$. We also describe a connection between this construction and families of invariant Hermitian forms on the representations.