Suppose you have a linear reductive group $G$ acting on an algebraic variety $X$. Let $P\leq G$ be an algebraic subgroup and let $Y \subseteq X$ be a closed subvariety, invariant under $P$. Then a number of texts define the construction of the so called "contracted" or "twisted" product give by $G \times^P Y := G \times Y /\sim \quad \quad$ where $(gp, y) \sim (g, py)$ for all $p \in P$.
My question is this: What is the algebraic interpretation of this geometric space and how should I be thinking about this scheme? In the case that it is affine, is there simple description of its coordinate ring?
Also does anyone have any references for a good exposition of these concepts? If there were a more categorical description of this object that would also be amazing (i.e. a definition given by diagrams etc.).
Thanks! Greatly Appreciated!