Related papers: Perverse $\mathbb{F}_p$-sheaves on the affine Gras…
This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We…
We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group $\check G$ is naturally equivalent to a…
The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived…
We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible…
We give a complete quiver description of the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the…
It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional…
We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg-Mirkovi\'c conjecture) of the Steinberg tensor product…
Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahori affine flag…
On an abelian variety $A$, sheaf convolution gives a Tannakian formalism for perverse sheaves. Let $X$ be an irreducible algebraic variety with generic point $\eta$. Let $K$ be a family of perverse sheaves (more precisely, a relative…
Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group $G$ over a field of positive characteristic, and following an idea of…
We study the constant term functor for $\mathbb{F}_p$-sheaves on the affine Grassmannian in characteristic $p$ with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of…
In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of…
The aim of this paper is to identify a certain tensor category of perverse sheaves on the real loop Grassmannian of a real form $G_{\mathbb R}$ of a connected reductive complex algebraic group $G$ with the category of finite-dimensional…
Let $X$ be a topologically stratified space, $p$ be any perversity on $X$, and $k$ be a field. We show that the category of $p$-perverse sheaves on $X$, constructible with respect to the stratification and with coefficients in $k$, is…
We study a category of Iwahori-equivariant modular perverse sheaves on some avatar of the semi-infinite flag variety, by adapting the work of Arkhipov-Bezrukavnikov-Braverman-Gaitsgory-Mirkovi\'c. We then construct a functor between the…
Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the…
The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\check G_\bZ$ over the integers. Given a field $k$, we give…
In this paper we provide a "combinatorial" description of the category of tilting perverse sheaves on the affine flag variety of a reductive algebraic group, and its free-monodromic variant, with coefficients in a field of positive…
We analyze irreducible perverse sheaves on abelian varieties, defined over the complex numbers or the algebraic closure of a finite field, whose Euler characteristic is zero. We give a description of such perverse sheaves under assumptions…
We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of G. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible…