On a Gelfand-Tsetlin representation of $\mathfrak{sl}_3$ in the space of sections of a local system with two monodromy parameters
Claude Eicher
TL;DR
This work constructs a Gelfand–Tsetlin realization of a two-parameter $\mathfrak{sl}_3$-module in the space of sections of a local system on a carefully chosen open subset of the flag variety. The authors define the local system $\mathcal{L}_{\mu_1,\mu_2}$ with monodromy parameters $(\mu_1,\mu_2)$ and build a GT basis $\{w_{k,l,m}\}$, deriving explicit $\mathfrak{sl}_3$-actions and demonstrating that $M_{\mu_1,\mu_2}$ is a GT module with a central character zero; a dual module $M^{\vee}_{\mu_1,\mu_2}$ is constructed and analyzed. A sharp simplicity criterion is established: $M_{\mu_1,\mu_2}$ and $M^{\vee}_{\mu_1,\mu_2}$ are simple iff $\mu_1,\mu_2\notin\mathbb{Z}$, while integral cases yield explicit submodules and cyclic generators; the subquotients are then classified and identified with relaxed Verma modules $\mathop{\mathrm{R}}_{\mathfrak{p}_1,\lambda,\mu}$ and their simples $\mathop{\mathrm{L}}_{\mathfrak{p}_1,\lambda,\mu}$. This builds a bridge between local-system sections on the flag variety and the relaxed highest-weight category for $\mathfrak{sl}_3$, enriching the GT representation theory with two monodromy parameters and detailed subquotient structures.
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.