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Characteristic varieties of character sheaves. (English) Zbl 0683.22012
The following characterization of character sheaves is proved when the group \({\mathbb{G}}\) is defined over the complex number field \({\mathbb{C}}:\) let \({\mathbb{G}}\) be a connected complex reductive group. A \({\mathbb{G}}\)-equivariant irreducible perverse sheaf \({\mathcal F}\) on \({\mathbb{G}}\) is a character sheaf if and only if the characteristic variety of \({\mathcal F}\) lies in \({\mathbb{G}}\times {\mathcal N}\). Here \({\mathcal N}\) is the nilpotent cone in the Lie algebra of \({\mathbb{G}}\). Another characterization is also proved. Let \({\mathbb{G}}\) be a connected reductive group defined over an algebraically closed field, \(N\subset {\mathbb{G}}^ a \)maximal unipotent subgroup and \(\pi: {\mathbb{G}}\to {\mathbb{G}}/N\) the projection. Then an irreducible \({\mathbb{G}}\)-equivariant perverse sheaf \({\mathcal F}\) is a tame character sheaf if and only if \(\pi_*({\mathcal F})\) is constructible with respect to the Bruhat cells and tame.
Reviewer: M.Muro

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
14L10 Group varieties
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
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