Characterizations of character sheaves for complex reductive algebraic groups
Let \(G\) be a connected complex reductive algebraic group. As the introduction explains, several definitions of character sheaves have been around. Here the situation is clarified (at least over the complex numbers). Identifying the tangent bundle of \(G\) with the cotangent bundle, we may think of the microsupport \(SS({\mathcal F})\) of a \(G\)- equivariant perverse sheaf \(\mathcal F\) on \(G\) as lying in the tangent bundle. The author now characterizes character sheaves by the property that their microsupport lies in the nullcone part of the tangent bundle. This criterion comes in five variations, depending on whether one looks at the elements of \(G\), conjugacy classes in \(G\), the set of semisimple elements of \(G\) \dots The characterization is also compared with Lusztig's original description of character sheaves, where the building block is the intersection cohomology complex of an irreducible \(G\)- equivariant local system on a \(G\)-orbit close to the zero section in the normal bundle of a semisimple conjugacy class. There is an appendix on microsupport of constructible sheaves in the presence of a group action.
connected complex reductive algebraic group
scientific article; zbMATH DE number 755404