Invariants for the action of a finite group on the enveloping algebra of a semisimple Lie algebra
Let \(G\) be a simply connected semisimple group and \(U\) the enveloping algebra of its Lie algebra (over \(\mathbb{C}\)). The object of this article is to study, for any finite subgroup \(H\) of \(G\), the algebra \(R=U^H\) of invariants under the natural action of \(H\) on \(U\), in particular the finite-dimensional irreducible \(R\)-modules and their annihilators. By means of the standard Morita context between \(R\) and the skew group ring \(S=U*H\), every (finite-dimensional, irreducible) \(R\)-module \(V\) can be written as the submodule \(W^H\) of \(H\)-invariants of some (finite-dimensional, irreducible) \(S\)-module \(W\). Thus it suffices to describe the finite-dimensional irreducible \(S\)-modules \(W\). These are shown to be of the form \(W=M\otimes_\mathbb{C} X\), where \(M\) is an irreducible \(U\)-module and \(X\) is an irreducible representation of \(H\). Especially complete results are obtained for \(G=\text{SL}_2\), for which the finite subgroups \(H\) are known. In each case, the McKay correspondence is used to explicitly calculate the number of non-isomorphic irreducible \(R\)-modules of a given dimension. It turns out that this number is always independent of the chosen dimension. Much of this article is motivated by and elaborates on general results of \textit{H. Kraft} and \textit{L. W. Small} [Math. Res. Lett. 1, No. 3, 297-307 (1994; see the following review Zbl 0849.16036)].
scientific article; zbMATH DE number 883793
scientific article