I've seen a superscript on $\times$ a few times now in the setting of $G$-spaces, and I can't find any documentation of its definition or basic properties, as I have no idea what to search.
Given an algebraic group $G$, a right $G$-torsor $X \to S$, and a left $G$-space $Y$, how is the space $X \times^G Y$ defined, and what is it called?
An example of its use (from Richarz, "A new approach to the geometric Satake equivalence"; note that $\mathrm{Gr}_G$ is a right $fpqc$-quotient $\mathrm{Gr}_G := LG/L^+G$):
Consider the following diagram of ind-schemes $$ \mathrm{Gr}_G \times \mathrm{Gr}_G \overset{p}{\leftarrow} LG \times \mathrm{Gr}_G \overset{q}{\to} LG \times^{L^+G} \mathrm{Gr}_G \overset{m}{\to} \mathrm{Gr}_G. $$ Here $p$ (resp. $q$) is a right $L^+G$-torsor with respect to the $L^+G$-action on the left factor (resp. the diagonal action). The $LG$-action on $\mathrm{Gr}_G$ factors through $q$, giving rise to the morphism $m$.
If I want to take this as a definition (quotient a Cartesian product of two $G$-sets by the diagonal action), I am still not sure what kind of quotient I should be using, and I have some further questions I'd rather not probe alone if I don't have to.
When is $X \times^G Y$ a $G$-torsor, an $fpqc$ sheaf, an algebraic space, an ind-scheme, etc.? Does the side of the $G$-action on $X$ or $Y$ matter for the definition? Is the notation symmetric in $X$ and $Y$? Is it necessary that $X$ be a torsor? Are there other necessary conditions to define the space?
Edit: after some random browsing here and on Overflow, I have found two possible names for the object and a possible definition. It is referred to as either a twisted product or a contracted product. I want to point out that none of the questions linked below have answers (outside of some fairly brief comments) or any references, so I would greatly appreciate a basic reference. I still can't seem to find any.
What is the algebraic interpretation of a contracted product? (MSE)
Contracted product of torsors (MO)
Question 3 includes a definition, and asks some of the same questions I'm interested in, but imposes some more conditions on the spaces involved than I thought would be necessary.