Affineness of some quotient dur sheaves of a super affine group

TL;DR

This paper proves that certain quotient sheaves of super affine groups are affine under specific conditions, extending classical results to the super algebraic setting and addressing a question posed by Brundan.

Contribution

It establishes conditions under which the quotient sheaf of a super affine group by a subgroup is affine, particularly when the associated affine group is reductive or pro-finite.

Findings

Quotient sheaves are affine when the associated affine group is reductive.

Quotient sheaves are affine when the associated affine group is pro-finite.

Provides a positive answer to Brundan's question in the algebraic case.

Abstract

We prove that given a super affine closed subgroup $H$ of a super affine group $G$ over a field $k$ of charctersitic $\mathrm{ch} k \ne 2$, the dur $k$-sheaf $G\tilde{\tilde{/}} H$ of right cosets is affine if the affine $k$-group $\bar{H}$ assocoiated to $H$ is (a) reductive or (b) pro-finite. Especially when $G$ is algebraic, the result in Case (a) gives rise to a positive answer to Brundan's question which was recently discussed by Zubkov \cite{Z}.