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

In this paper we study Category $\text{mcal}O$ for the polynomial toroidal Lie algebras and its $S,H$ type subalgebras. We classify irreducible objects of category $\text{mcal}O$ as unique irreducble quotient of standard modules. Surprisingly, costandard objects of category $\text{mcal}O$ 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 $\text{mcal}O$.