Characterizations of character sheaves for complex reductive algebraic groups

Abstract

Character sheaves are geometric objects associated to a reductive algebraic group. They are closely related to irreducible characters of finite Chevalley groups. The existence of character sheaves was conjectured by G. Lusztig in the book Characters of Reductive Groups over Finite Fields (1984). There he described what one should expect character sheaves to be for complex reductive groups. A definition of character sheaves for an arbitrary characteristic, quite different from this description, was given by him later (1985). It is shown in this thesis, in particular, that the original (1984) Lusztig's suggestion for describing character sheaves is equivalent to the later definition. We use alternative characterizations of character sheaves for a connected complex reductive algebraic group G that are obtained in this thesis and also the one (conjectured by G. Lusztig and G. Laumon) that was obtained in 1988 by V. Ginsburg and independently by I. Mirkovic and K. Vilonen. These characterizations involve microsupport and microlocalization of a sheaf. A microsupport $SS({\cal F})$ of a complex ${\cal F}$ of sheaves on G is a closed conic subset of $T\sp*G = G\times{\bf g};$ it tells us about singularities of ${\cal F}$. A microlocalization $\mu\sb{M}{\cal F}$ of ${\cal F}$ along a submanifold M of G is a complex of sheaves on the conormal bundle $T\sbsp{M}{*}G$ which allows us to analyze ${\cal F}$ on the infinitesimally small neighbourhood of M. Theorem. Let G be a complex connected reductive algebraic group, ${\cal F}$ be a G-equivariant (for conjugation) irreducible perverse sheaf on G. Let ${\cal N}$ be the nilpotent cone of the Lie algebra g of G. S may denote either G or the set of all semisimple elements of G. The following are equivalent: (i) ${\cal F}$ is a character sheaf; (ii) $SS({\cal F})\vert\sb{S}\subset S\times{\cal N};$ (iii) supp $\mu\sb{\cal O}{\cal F}\subset{\cal O}\times{\cal N},$ where ${\cal O}\subset S$ is a conjugacy class; (iv) supp $\mu\sb{x}{\cal F}\subset{\cal N}, x\in S.$ Characterization (ii) when S = G is the one given by Ginsburg, Mirkovic and Vilonen. Characterization (iii) with S being the set of semisimple elements was conjectured by J. Bernstein (for arbitrary characteristic). This kind of characterizations essentially simplifies the development and our understanding of the theory. Currently there is still no such description for positive characteristic. An important intermediate result is the description of the microsupport of a $\doubc$-constructible sheaf ${\cal F}$ on a manifold X in terms of microlocalizations to the points of this manifold: $SS({\cal F}) = \overline{\bigcup\limits\sb{x\in X} {\rm supp} \mu\sb{x}{\cal F}}.$