Category $\mcal O$ for polynomial toroidal algebras and its subalgebras

TL;DR

This paper investigates the structure of Category O for polynomial toroidal Lie algebras, classifies irreducible objects, and computes character formulas using tilting module theory.

Contribution

It provides a classification of irreducible objects, identifies the origin of costandard objects, and derives character formulas for modules in Category O.

Findings

Irreducible objects are unique quotients of standard modules.

Costandard objects arise from Shen-Larsson modules.

Character formulas are computed for irreducible and tilting modules.

Abstract

In this paper we study Category $\mcal O$ for the polynomial toroidal Lie algebras and its $S,H$ type subalgebras. We classify irreducible objects of category $\mcal O$ as unique irreducble quotient of standard modules. Surprisingly, costandard objects of category $\mcal O$ arrises from Shen-Larsson type modules. We determine necessary sufficient conditions for irreducibility of Shen-Larsson modules. Finally appeling structure of Shen-Larsson modules and Soergel Tilting module theory of \cite{Soe}, we compute charcter formulas for irreducible modules and indecomposable Tilting modules of category $\mcal O$.