On the composition structure of the twisted Verma modules for $\mathfrak{sl}(3,\mathbb{C})$

TL;DR

This paper investigates the detailed composition structure of twisted Verma modules for rak{sl}(3,\,b C), including explicit singular vectors and generators, using Fourier transform techniques on b D-modules over flag manifolds.

Contribution

It provides explicit descriptions of singular vectors and generators for twisted Verma modules of rak{sl}(3,b C) and its subalgebra, employing Fourier transform methods.

Findings

Explicit structure of singular vectors for rak{sl}(3,b C) and rak{sl}(2,b C)

Identification of generators of twisted Verma modules

Application of Fourier transform to b D-module realizations

Abstract

We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $\mathfrak{sl}(3, \mathbb{C})$, including the explicit structure of singular vectors for both $\mathfrak{sl}(3, \mathbb{C})$ and one of its Lie subalgebras $\mathfrak{sl}(2, \mathbb{C})$, and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $\mathrm{{D}}$-modules on the Schubert cells in the full flag manifold for $\mathrm{SL}(3, \mathbb{C})$.