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Register a new userConstructible sheaves on nilpotent cones in rather good characteristic
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We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $mathbb k$ under the assumption that the characteristic $ell$ of $mathbb k$ is rather good for $G$, i.e., $ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautners characteristic-$ell$ `cleanness conjecture; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.
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In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the naive definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the correct one.
In type A we find equivalences of geometries arising in three settings: Nakajimas (``framed) quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are now all given by explicit formulas. As an application we provide a geometric version of symmetric and skew $(GL(m), GL(n))$ dualities.
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. In order to study the nilpotent bicone, we introduce another subvariety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed in http://arxiv.org/abs/math/0008002v5 .
Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.
Let $G=Sp_{2n}(mathbb{C})$, and $mathfrak{N}$ be Katos exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Lambda^+$, the dominant weights for a simple algebraic group $H$, and $textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $widetilde{mathfrak{N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(Coh^G(mathfrak{N}))$, and deduce that the resulting $t$-structure on $D^b(Coh^G(mathfrak{N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of this $t$-structure on $D^b(Coh^G(mathfrak{N}))$.
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