I’ve seen this notation a few times, and I can’t find any documentation of its meaning or basic properties (I have no idea what to search). Given an algebraic group $G$, a right $G$-torsor $mathcal{X} to X$, and a left $G$-space $Y$, how is the space $mathcal{X} times^G Y$ defined, and what is it called? When is it a $G$-torsor? Does the side of the $G$-action on $mathcal{X}$ or $Y$ matter? Is the notation symmetric? Are the conditions I’m imposing necessary, or am I missing some?

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), that still doesn’t answer all of my questions, and leaves me wondering what kind of quotient I should be using.