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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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