Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group $G:=(\mathcal{F}\mathbf{G})(k)$ over $k$, such that $G\cong\mathbf{G}(A)$. We prove that if $\mathbf{G}$ is a reductive group scheme over $A$, and $\mathbf{T}$ is a maximal torus of $\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over $A$, and that the Greenberg functor preserves certain normaliser group schemes over $A$. Moreover, we prove that if $\mathbf{G}$ is reductive and $\mathbf{P}$ is a parabolic subgroup of $\mathbf{G}$, then $P$ is a self-normalising subgroup of $G$, and if $\mathbf{B}$ and $\mathbf{B}'$ are two Borel subgroups of $\mathbf{G}$, then the corresponding subgroups $B$ and $B'$ are conjugate in $G$.
The proof, but not the statement, of Proposition 18.2 contained an error which is repaired in this version. See Remark 18.3 in this version. No other changes. We extend Greenberg's original construction to arbitrary (in particular, non-reduced) schemes over (certain types of) local artinian rings. We then establish a number of basic properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor.
In this paper we show that it is possible to define a topology on the category of formal schemes over a ring of $p$-adic integers such that the left adjoint of the Greenberg Transform is a site cocontinuous functor when we equip the category of schemes over the residue field with the étale topology. We show furthermore that this topology allows us to give an isomorphism between the corresponding fundamental groups, and use this isomorphism to show that it is possible to geometrize the quasicharacters of a $p$-adic torus by a local system on a formal scheme over the ring of $p$-adic integers.
We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.
Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig's results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theory of group schemes over Artinian local rings.
In math.RT/0205144 we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of the ch…
We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and étale group schemes and we prove that, if $G$ is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of $G$-covers is reducible.
We study algebraicity and smoothness of fixed point stacks for flat group schemes which have a finite composition series whose factors are either reductive or proper, flat, finitely presented, acting on algebraic stacks with affine, finitely presented diagonal. For this, we extend some theorems of [SGA3.2] on functors of homomorphisms Hom(G, H) and functors of reductive subgroups Sub(H) for an affine, possibly non-flat group scheme H.
We present a generalized version of classical geometric invariant theory à la Mumford where we consider an affine algebraic group $G$ acting on a specific affine algebraic variety $X$. We define the notions of linearly reductive and of unipotent action in terms of the $G$ fixed point functor in the category of $(G,\Bbbk [X])$--modules. In the case that $X=\{\star\}$ we recuperate the concept of lineraly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant--Rosenlicht's…
This thesis is concerned with the mixed Tate property of reductive algebraic groups $G$, which in particular guarantees a Chow Kunneth property for the classifying space $BG$. Toward this goal, we first refine the construction of the compactly supported motive of a quotient stack. In the first section, we construct the compactly supported motive $M^c(X)$ of an algebraic space $X$ and demonstrate that it satisfies expected properties, following closely Voevodsky's work in the case of schemes. In the second section, we construct a functorial version of Totaro's definition of the compactly suppor…
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are t-exact on the category of tempered D-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for GL_n. We also show that Whittaker coefficient functors are t…
In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…
Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map H^1_{et}(R,G)\to H^1_{\et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop reductive, i.e. contains a maximal R-torus. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a red…
We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.
In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne--Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus generalised a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.
We prove that the perfect loop functor $LX$ of a quasi-projective scheme $X$ over a local non-archimedean field $k$ satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group $G$, the map $LG \rightarrow L(G/B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in étale topology) of Bouthier--Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne--Lusztig spaces $X_w(b)$ attached to unramified $p$-adic reductive …
A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an infinite field is given in [FP] recently. That proof is based significantly on Theorem 1.0.1 stated below in the Introduction and proven in the present preprint. Theorem 1.0.1 itself is a consequence of two purity theorems (Theorems A and 10.0.30) proven below in the present preprint. The geometric part of a new preprint [PSV] and the main result of an article [C-T-S] are used significantly in proofs of those two purity theorems. One of that purity result looks as follows. Let O be a sem…
We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $ω^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $ω^0_X$ and the computation of the motives E_X. In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the pro…
We generalize the Białynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Białynicki-Birula decomposition functorially: for a fixed ${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf G}$-action. The freedom of choice of $\ov…
The purpose of this paper is to introduce and study certain irreducible perverse l-adic sheaves on a reductive group G over a finite field (we call them gamma-sheaves). One can construct such a sheaf starting with (almost) every finite-dimensional representation of the Langlands dual group. We present conjecture connecting the above sheaves with generalized gamma-functions introduced in our previous paper. We also conjecture that the convolution functor with the above sheaves enjoys certain nice properties (in particular, we compute the convolution of a gamma-sheaf with a character sheaf). We …
We give geometric descriptions of the category C_k(n,d) of rational polynomial representations of GL_n over a field k of degree d for d less than or equal to n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category C_k(n,d) and the Schur functor.
Let X be a proper scheme over a field k which satisfies Serre's condition S2 and G a reductive group over k. We prove that the functor of principal G-bundles defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.
Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.
Let $G$ be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a $G$-variety $X$ admits a natural homomorphism with finite kernel to the Langlands dual group $G^\vee$ of $G$. Here, we prove that the dual group is functorial in the following sense: if there is a dominant $G$-morphism $X\to Y$ or an injective $G$-morphism $Y\to X$ then there is a canonical homomorphism $G^\vee_Y\to G^\vee_X$ which is compatible with the homomorphisms to $G^\vee$.
We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.
Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We show that K_1^G(k)=K_1^G(k[X_1,...,X_n]) for any n>= 1. This implies that K_1^G is A^1-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K_1^G(R)-> K_1^G(K) is injective for any regular local k-algebra R with the fraction field K.
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(\!(t)\!)$-equivariant and $U^-(\!(t)\!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and a…
Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constru…
We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal $\mathrm{Be}_{\check{G}}^{\dagger}$ on $\mathbb{G}_{m,\mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $\check{G}$-local system $\mathrm{Kl}_{\check{G}}$ constructed by Heinloth-Ngô-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $\mathrm{Be}_{\check{G}}^{\dagger}$ when $\check{G}$…
Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over $\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let $\mathcal L$ be an ample line bundle on $B$. Let $τ$ be a Harder-Narasimhan type of length 2, and let $δ\in\mathbb N$. We say a pure sheaf $\mathcal E$ on $B$ is $(τ,δ)$-stable if its Harder-Narasimhan filtration $0=\mathcal E_{\leq 0}\subsetneq\mathcal E_{\leq 1}\subsetneq\mathcal E_{\leq 2}=\mathcal E$ is non-splitting, of type $τ$, with stable subquotients, and $δ=\dim_\Bbbk\mat…
Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical equivalence. By refining a construction from a previous paper, we construct a local system of categories over a topological space V^0_C, where these categories are assigned to different points in V^0_C. Natural equivalence functors between these categories are parameterized by homotopy classes of paths between the corresponding points.
We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] wh…
Let k be an algebraically closed field of characteristic p>>0. Let $X\rightarrow Y$ be a symplectic resolution. There are two questions which motivates this work. One question is a construction of an action of a group on the category $\mathcal{C}:=D^b(Coh(X))$ - The bounded derived category of coherent sheaves of the symplectic resolution X. Second question is understanding equivalence functors between derived categories of coherent sheaves for different symplectic resolutions of Y. Let G/k be a reductive group. In this paper, we construct a local system on a topological space called $V^0_{\ma…
Let $G$ be a reductive group scheme over the $p$-adic integers, and let $μ$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,μ)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis étale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,μ)$-display, which extends the construction of the Dieudonné crystal ass…
Let W be a smooth complex quasiprojective variety with the action of a connected reductive group G. Adapting the stratification approach of Teleman to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections---i.e., of quantum Hamiltonian reduction---for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish an effective combinatorial criterion for exactness of the global sections functors of microlocalization theory. When combined with our earlier derived equivalence results, this gives precise criteria for "microlocalization of…
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 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 Langlands dual group is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in and some o…
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Deligne's classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of resi…
Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).
