On existence of double coset varieties

TL;DR

This paper investigates the conditions under which double coset varieties exist, providing examples of non-existence in algebraic categories and establishing existence in the broader constructible space category.

Contribution

It demonstrates that double coset varieties may not always exist as algebraic varieties and identifies conditions for their existence as constructible spaces.

Findings

Double coset varieties do not always exist as algebraic varieties.

Existence of double coset varieties as constructible spaces is possible under general conditions.

Examples show limitations of algebraic structures for double coset varieties.

Abstract

Let $G$ be a complex affine algebraic group and $H, F \subset G$ be closed subgroups. The homogeneous space $G / H$ can be equipped with structure of a smooth quasiprojective variety. The situation is different for double coset varieties $\dcosets{F}{G}{H}$. In this paper we give examples showing that the variety $\dcosets{F}{G}{H}$ does not necessarily exist. We also address the question of existence of $\dcosets{F}{G}{H}$ in the category of constructible spaces and show that under sufficiently general assumptions $\dcosets{F}{G}{H}$ does exist as a constructible space.