Invariant differential operators and Frobenius decomposition of a \(G\)-variety (Q2718660)

Let \(G\) denote a connected, reductive algebraic group. Suppose that \(G\) acts on a smooth, affine complex variety \(M.\) Let \({\mathcal D}^G(M)\) denote the \(G\)-invariant algebraic differential operators on \(M.\) Let \(\mathbb C[M]\) denote the coordinate ring of \(M.\) Decompose \(\mathbb C[M]\) into \(G\)-isotypical components. Then it is shown that the occurring components are irreducible pairwaise non-equivalent \({\mathcal D}^G(M)\)-modules, with a central character and uniquely determined by this character. As an application it is shown that the \(G\)-action on \(M\) is multiplicity free provided that the quotient of the moment map is finite. Moreover, there is an analogous decomposition for real forms. Furthermore, by some singular examples it is shown that the corresponding results for singular varieties are not true.