Let $G$ be a reductive complex algebraic group and $X$ an affine $G$-variety. Let $X/\!/G$ denote the affine variety corresponding to the (finitely generated) algebra of $G$-invariant functions ${\scr O}(X)^G$. From the inclusion ${\scr O}(X)^G\subset{\scr O}(X)$ we obtain a canoical quotient morphism $\pi_X\colon X\to X/\!/G$. Let ${\scr D}(X)$ (resp.\ ${\scr D}^n(X)$, $n\in{\bf Z}^+)$ denote the algebraic differential operators (resp.\ of order at most $n$) on $X$, and then $P$ preserves $(\pi_X)^*{\scr O}(X/\!/G)={\scr O}(X)^G\subset{\scr O}(X)$, hence $P$ induces $(\pi_X)_*(P)\in{\scr D}(X/\!/G)$, and $(\pi_X)_*$ sends ${\scr D}^n(X)^G$ to ${\scr D}^n(X/\!/G)$ for each $n$.
Suppose that $Y$ is a smooth and connected affine variety. Then it is known that (1) ${\scr D}(Y)$ is finitely generated; in fact, ${\rm gr}\,{\scr D}(Y)$ is finitely generated, where gr is taken with respect to the filtration ${\scr D}^n(Y)\subset{\scr D}^{n+1}(Y)\subset\cdots\subset {\scr D}(Y)$. (2) ${\scr D}(Y)$ is a simple algebra.
If $Y$ is singular, all of the above properties may fail. However,
recent research has given optimisim that the above properties hold for
quotients:
Conjectures. Assume that our $G$-variety $X$ is
smooth and connected. Then (1) ${\rm gr}\,{\scr D}(X/\!/G)$ is
finitely generated (2) ${\scr D}(X/\!/G)$ is simple.
The conjectures lead to the following questions: (3) When is $(\pi_X)_*\colon {\scr D}(X)^G\to {\scr D}(X/\!/G)$ surjective? When is ${\rm gr}(\pi_X)_*\colon{\rm gr}\,{\scr D}(X)^G\to {\rm gr}\,{\scr D}(X/\!/G)$ surjective? (If ${\rm gr}(\pi_X)_*$ is surjective, it follows that ${\rm gr}\,{\scr D}(X/\!/G)$ is finitely generated.) (4) What are the kernels of $(\pi_X)_*$ and ${\rm gr}(\pi_X)_*$? There is a natural inclusion $({\scr D}(X){\germ g})^G\subset{\rm Ker}(\pi_X)_*$. Does equality hold? Here ${\germ g}={\rm Lie}(G)$ is considered as a Lie subalgebra of the vector fields on $X$.
The answers to the questions in (4) are unknown even in the case that $X={\germ g}$ with the adjoint representation. We discuss recent progress on the conjectures and questions. We also discuss the more complicated case of invariant differential operators acting on sections of $G$-vector bundles over $X$.