Deforming Representations of 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.