Special function models of indecomposable $sl(2)$ representations: The Laguerre Case

TL;DR

This paper explores indecomposable $sl(2)$ representations linked to orthogonal polynomials, constructing generating functions, analyzing the Casimir action, and relating to differential equations, zeroes, and recurrence relations.

Contribution

It introduces a new class of indecomposable $sl(2)$ representations connected to orthogonal polynomials, with explicit constructions and differential equation relations.

Findings

Connections between indecomposable $sl(2)$ reps and orthogonal polynomials

Explicit generating functions and Casimir actions derived

Relations to differential equations and polynomial zeroes established

Abstract

In this paper, we point out connections between certain types of indecomposable representations of $sl(2)$ and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras $sl(2)$ act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.