Donkin-Koppinen filtration for general linear supergroup

TL;DR

This paper generalizes Donkin-Koppinen filtrations to the coordinate superalgebras of general linear supergroups, establishing a new filtration structure and applying it to invariants of (co)adjoint actions.

Contribution

It introduces a new filtration for coordinate superalgebras of general linear supergroups, extending known results to infinite weight sets and connecting to highest weight category theory.

Findings

Established a decreasing filtration for coordinate superalgebras of GL(m|n).

Connected the filtration to highest weight category structures.

Applied results to describe invariants of (co)adjoint actions.

Abstract

We consider a generalization of Donkin-Koppinen filtrations for coordinate superalgebras of general linear supergroups. More precisely, if $G=GL(m|n)$ is a general linear supergroup of (super)degree $(m|n)$, then its coordinate superalgebra $K[G]$ is a natural $G\times G$-supermodule. For every finitely generated ideal $\Gamma\subseteq \Lambda\times\Lambda$, the largest subsupermodule $O_{\Gamma}(K[G])$ of $K[G]$, which has all composition factors of the form $L(\lambda)\otimes L(\mu)$ where $(\lambda, \mu)\in\Gamma$, has a decreasing filtration $O_{\Gamma}(K[G])=V_0\supseteq V_1\supseteq...$ such that $\bigcap_{t\geq 0}V_t=0$ and $V_t/V_{t+1}\simeq V_-(\lambda_t)^*\otimes H_-^0(\lambda_t)$ for each $t\geq 0$. Here $H_-^0(\lambda)$ is a costandard $G$-supermodule, and $V_-(\lambda)$ is a standard $G$-supermodule, both of highest weight $\lambda\in\Lambda$ (see \cite{z}). We deduce the existence of such a filtration from more general facts about standard and costandard filtrations in certain highest weight categories which will be proved in Section 4. Until now, analogous results were known only for highest weight categories with finite sets of weights. We believe that the reader will find the results of Section 4 interesting on its own. Finally, we apply our main result to describe invariants of (co)adjoint action of $G$.