On a Gelfand-Tsetlin representation of $\mathfrak{sl}_3$ in the space of sections of a local system with two monodromy parameters

TL;DR

This paper constructs a Gelfand-Tsetlin representation of the Lie algebra rak{sl}_3 in a space of sections of a local system with two monodromy parameters, analyzing its structure on a specific flag variety subset.

Contribution

It introduces a novel Gelfand-Tsetlin representation of rak{sl}_3 in a geometric setting involving local systems with monodromy parameters.

Findings

Representation explicitly constructed and analyzed.

Structure of the representation characterized.

Connections to flag variety geometry established.

Abstract

We construct a Gelfand-Tsetlin representation of $\mathfrak{sl}_3$ in the space of sections of a local system. The local system lives on an open part of the flag variety given by the intersection of three translates of the big cell and has two complex monodromy parameters. We analyze the structure of this representation.