On a ${\mathbb C}^2$-valued integral index transform and bilateral hypergeometric series

TL;DR

This paper explores the spectral decomposition of hypergeometric differential operators related to tensor product representations of the universal cover of SL(2,R), aiming to find natural bases and inversion formulas.

Contribution

It provides new insights into the spectral analysis of hypergeometric operators on the line Re z=1/2, connecting to representation theory of SL(2,R).

Findings

Spectral decomposition of hypergeometric operators on the line Re z=1/2.

Identification of natural bases in generalized eigenspaces.

Development of variants of the inversion formula.

Abstract

We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.