We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers.
The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.
Let $Π$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $λ$ of the field of algebraic numbers which is prime to p, consider the $λ$-adic pro-semisimple completion of $Π$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $λ$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its c…
In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conje…
In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig's character formula, and the $G_{r}$T-radical series for baby Verma modules.
For those deformations that satisfy a certain non-degeneracy condition, we describe the structure of certain simple modules of the deformations of the subcharacter algebra of a finite group. For finite abelian groups, we prove that the deformation given by the inclusion of the natural numbers, which corresponds to the algebra generated by the fibred bisets over a field of characteristic zero, is not semisimple. In the cyclic group of prime order case, we provide a complete description of the semisimple deformations.
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve G$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$. Denote by $\Bun_G$ the moduli stack of $G$-bundles on $C$ and $ \Loc_{\breve G}$ the moduli stack of $\breve G$-local systems on $C$. Let $D_{\Bun_G}$ be the sheaf of crystalline differential operators on $\Bun_G$. In this paper we construct an equivalence between the bounded derived category $D^b(\on{QCoh}(\Loc_{\breve G}^0)…
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of $\mathfrak{g}$-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of $\mathfrak{g}$. In this expository paper, we give four detailed examples of this theorem when $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$. Specifically, we describe the $\mathcal{D}$-modules associated to finite-dimensional irreducible $\mathfrak{g}$-modules, Verma modules, Whittaker modules, discre…
Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let $V$ be an irreducible ${\mathbb F}H$-module and $U$ a quotient of the induced ${\mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$, which is semisimple when ${\rm char}({\mathbb F})\ne p$ and uniserial if ${\rm char}({\mathbb F})=p$. Furthermore, we describe the division rings arising as endomorphism algebras of the simple components of $U$. We use techniques from noncommutative ring theory to study ${\rm End}_{{\mathbb F}G}(V\kern-3pt\uparrow)$ and relate the right ideal structure of ${\rm End}…
In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.
