An introduction to perverse sheaves

A class of perverse sheaves on a partial flag manifold

We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G defined over a finite field which are equivariant under the action of the group of rational points of G. The definition of this class is similar to the definition of parabolic character sheaves.

On parabolic restriction of perverse sheaves

We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.

Z/m-graded Lie algebras and perverse sheaves, I

We give a block decomposition of the equivariant derived category arising from a cyclically graded Lie algebra. This generalizes certain aspects of the generalized Springer correspondence to the graded setting.

Stacks similar to the stack of perverse sheaves

We introduce, on a topological space X, a class of stacks of abelian categories we call "stacks of type P." This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification), and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimension…

Mixed modular perverse sheaves on moment graphs

We study an analogue of the Achar-Riche "mixed modular derived category" for moment graphs. In particular, given a Coxeter group $W$ and a reflection faithful representation $\mathfrak{h}$, we introduce a category that plays the role of Schubert-stratified mixed modular perverse sheaves on "the flag variety associated to $(W, \mathfrak{h})$." We show that this Soergel-theoretic generalization of graded category $\mathcal{O}$ is graded highest weight.

Modular perverse sheaves on flag varieties III: positivity conditions

We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.

Modular perverse sheaves on flag varieties II: Koszul duality and formality

Building on the theory of parity sheaves due to Juteau-Mautner-Williamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche-Soergel-Williamson).

Addendum to "Quivers with loops and perverse sheaves"

We prove a character formula for the Hopf algebra defined in arXiv:1401.5302 that generalizes quantum groups, as well as for the simple modules associated to dominant integral weights defined in arXiv:1403.0846.

仿射旗流形上的混合模 perverse sheaves 与 Koszul 对偶

Mixed modular perverse sheaves on affine flag varieties and Koszul duality

本文基于技术假设，证明了仿射旗流形上不可分解 tilting perverse sheaves 的重数公式，并解释了连接混合模 perverse sheaves 与普通 perverse sheaves 的“degrading functor”的构造。

Nonsplit Hecke algebras and perverse sheaves

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over F_q induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is of type B_4 and that of L is of type B_2.

Mixed perverse sheaves on flag varieties of Coxeter groups

In this paper we construct an abelian category of "mixed perverse sheaves" attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

Graham's Variety and Perverse Sheaves on the Nilpotent Cone

In recent work, Graham has constructed a variety with a map to the nilpotent cone which is similar in some ways to the Springer resolution. One aspect in which Graham's map differs is that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone, and we discuss here the problem of describing the summands of this perverse sheaf. For type $A_n$, a key tool is a known combinatorial description of an affine paving of Springer fibers.

Calculations with graded perverse-coherent sheaves

In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the coho…

半无限旗 manifold 上的 perverse sheaves 与 Frobenius 核的表示

Perverse sheaves on the semi-infinite flag variety and representations of the Frobenius kernel

本文研究了半无限旗 manifold 上的 Iwahori-等变模算子 perverse sheaves，通过借鉴 Arkhipov 等人的工作，构建了与 Frobenius 核的 graded 表示之间的 functor，并猜想其为等价。

Lecture notes on sheaves and perverse sheaves

Informal lecture notes with examples on sheaf theory and the derived category of sheaves; sheaves and Morse theory; perverse sheaves, and some applications to representation theory. Added Oct 2021: cellular perverse sheaves. Proofs are outlined with references for further details.

Non-perverse parity sheaves on the flag variety

We give examples of non-perverse parity sheaves on Schubert varieties for all primes.

球面反常层的Levi等变限制

Levi-Equivariant Restriction of Spherical Perverse Sheaves

本文研究仿射Grassmannian上球面反常层在Levi子群支集下的等变上同调，推广Ginzburg-Riche工作，并用Langlands对偶群将其等同于超球面Hamiltonian簇上的函数代数。

Quivers with loops and perverse sheaves

In this article we consider the general definition of Lustig sheaves for arbitrary quivers, possibly carrying loops. We answer a conjecture raised by Lusztig asking if the more "simple" Lusztig perverse sheaves are enough to span the whole Grothendieck group usually built with these sheaves. Our proof is based on induction, with the help of restriction and induction functors, but with non trivial first steps, consisting in the study of quivers with one vertex but possible loops. We also need to consider regularity conditions on the support of our perverse sheaves to perform efficient restricti…

Green functions via hyperbolic localization

Let G be a reductive algebraic group, with nilpotent cone N and flag variety G/B. We construct an exact functor from perverse sheaves on N to locally constant sheaves on G/B, and we use it to study Ext-groups of simple perverse sheaves on N in terms of the cohomology of G/B. As an application, we give new proofs of some known results on stalks of perverse sheaves on N.

Perverse coherent sheaves on the nilpotent cone in good characteristic

In characteristic zero, Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a simply connected semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent to the category of (ordinary) coherent sheaves. We prove that graded versions of these results also hold in good positive characteristic.

Z/m-graded Lie algebras and perverse sheaves, IV

Let G be a reductive group over C. Assume that the Lie algebra g of G has a given grading (g_j) indexed by a cyclic group Z/m such that g_0 contains a Cartan subalgebra of g. The subgroup G_0 of G corresponding to g_0 acts on the variety of nilpotent elements in g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of the closures of these orbits with coefficients in irreducible G-equivariant local systems which are in the "principal block". We show that these can be computed by a purely combinatorial algorithm.

Tensor product varieties, perverse sheaves and stability conditions

We show that the space spanned by the class of simple perverse sheaves in [Zh08] without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. We also show that the two stability conditions coincide in the localization process in [Zh08], by using supports and singular supports of complexes of sheaves, respectively.

$\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \ci…

Perverse sheaves and finite-dimensional algebras

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 equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if $X$ has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

辛奇点上的反常凝聚层

Perverse coherent sheaves on symplectic singularities

本文提出辛奇点上反常凝聚层的概念，证明其给出泊松层Grothendieck群的简单对象基，并涵盖幂零锥与仿射格拉斯曼流形的特例。

Theory of homotopes in application to mutually unbiased bases, harmonic analysis on graphs and perverse sheaves

The paper is the survey of the modern results and applications of the theory of homotopes. The notion of a well-tempered element in an associative algebra is introduced and it is proven that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued t-structure. Hochschild and global dimensions of the homotopes are calculated. The homotopes constructed by generalized Lapalce operators in Poincare groupoids of graphs are studied. It is shown that they are quotients of Temperley-Lieb algebras of general graphs. The perverse sheaves on …

Parabolic induction and perverse sheaves on h/W

For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.

通过反常拉回构造周期层

Period sheaves via perverse pullbacks

本文利用反常拉回函子为 Hamiltonian 空间构造周期层，通过维数约化同构揭示其对普通拉回的细化及与微茎函子的联系，并恢复余切与 Whittaker 情形的已知构造。

Perverse sheaves on affine flags and Langlands dual group

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 categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.

Perverse coherent sheaves

We describe an analogue of the notion of a perverse sheaf in the setting of the derived category of coherent sheaves on an algebraic stack. Under strong additional assumptions the construction of coherent "intersection cohomology" complexes works. This paper supersedes the preprint arXiv:math/0005152 by the second author.

关于Lusztig的perverse sheaves的奇异支撑

On singular supports of Lusztig's perverse sheaves

本文证明了Lusztig关于其perverse sheaves的微局部刻画猜想：无环有限quiver的等变简单perverse sheaf是Lusztig的，当且仅当其奇异支撑包含于Lusztig的Lagrangian簇（即preprojective代数的幂零表示簇）。

Perverse sheaves and modular representation theory

This paper is an introduction to the use of perverse sheaves with positive characteristic coefficients in modular representation theory. In the first part, we survey results relating singularities in finite and affine Schubert varieties and nilpotent cones to modular representations of reductive groups and their Weyl groups. The second part is a brief introduction to the theory of perverse sheaves with an emphasis on the case of positive characteristic and integral coefficients. In the final part, we provide some explicit examples of stalks of intersection cohomology complexes with integral or…

Decomposition numbers for perverse sheaves

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.

Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group

In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.

Recollement for perverse sheaves on real hyperplane arrangements

We consider a hyperplane arrangement in $\mathbb{C}^n$ defined over $\mathbb{R}$, and the associated natural stratification of $\mathbb{C}^n$. The category of perverse sheaves smooth with respect to this stratification was described by Kapranov and Schechtman in terms of quiver representations. Using work of Weissman, we reinterpret this category as the category of finite-dimensional modules over an explicit algebra. We also describe recollement (open-closed decomposition) of perverse sheaves in terms of this module category. As an application, we identify the modules associated to all interse…

Perverse coherent sheaves (after Deligne)

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic group; though proof of the general statement in this case does not require new ideas, it provides examples (such as sheaves on the nilpotent cone of a semi-simple group equivariant under the adjoint action) where construction of coherent "intersection cohomology" sheaves works.

Cluster algebras and singular supports of perverse sheaves

We propose an approach to Geiss-Leclerc-Schroer's conjecture on the cluster algebra structure on the coordinate ring of a unipotent subgroup and the dual canonical base. It is based on singular supports of perverse sheaves on the space of representations of a quiver, which give the canonical base.

Coxeter 排列与建筑上的等变 perverse 层

Equivariant perverse sheaves on Coxeter arrangements and buildings

本文描述了有限 Coxeter 群作用下反射超平面排列上的等变 perverse 层范畴，并利用 Kapranov–Schechtman 分析得到其与显式代数模的等价；同时推广到仿射建筑，建立与深度零表示的联系。

Ringel duality for perverse sheaves on hypertoric varieties

Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field $k$ on any affine unimodular hypertoric variety. Our main result is that this is a highest weight category whose Ringel dual is the corresponding category for the Gale dual hypertoric variety. On the way to proving our main result, we confirm a conjecture of Finkelberg-Kubrak in the case of hypertoric varieties. We also show that our category is equivalent to representations of a combinatorially-defined al…

Perverse sheaves on infinite-dimensional stacks, and affine Springer theory

The goal of this work is to construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the loop Lie algebra Lg and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of ``compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because LG and Lg are infinite-dimensional ind-schemes.

Serre 函子与 $\mathbb{P}^n$ 上反常层的 $\mathbb{P}$-对象

Serre functor and $\mathbb{P}$-objects for perverse sheaves on $\mathbb{P}^n$

本文证明了 $\mathbb{P}^n$ 上可构造导出范畴的逆 Serre 函子可由开层简单反常层的 $\mathbb{P}$-扭转实现，并证明所有不可分解反常层均为 $\mathbb{P}$-like 对象且显式构造了其自同态空间的生成态射。

Perverse Sheaves on Real Loop Grassmannians

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 representations of a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The root system of $\check H$ is closely related to the restricted root system of the real form $G_{\mathbb R}$. The fact that $\check H$ is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition…

扭曲仿射旗簇上的反常层与Langlands对偶

Perverse sheaves on twisted affine flag varieties and Langlands duality

本文用Langlands对偶数据描述温和分歧约化群仿射旗簇上的Iwahori-Whittaker等变反常层，推广了Arkhipov-Bezrukavnikov的工作，并通过扩展Wakimoto层理论及建立倾斜性质解决了非分裂群的中心函子缺失问题。

Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)

We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D_k constructible with respect to the Schubert stratification. The connection is given by an explicit isomorphism to the endomorphism algebra of a projective generator described in by Braden. The algebra is obtained by a "folding" procedure from the generalized Khovanov arc algebras. We relate this algebra to the category of finite dimensional representations of the …

Holomorphic Lagrangian branes correspond to perverse sheaves

Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotangent bundle of $X$. We prove that under the quasi-equivalence between $D^b_c(X)$ and $DFuk(T^*X)$ established in [NaZa09] and [Nad09], holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.

Macdonald polynomials, Laumon spaces and perverse coherent sheaves

Let $G$ be an almost simple simply connected complex Lie group, and let $G/U_-$ be its base affine space. In this paper we formulate a conjecture, which provides a new geometric interpretation of the Macdonald polynomials associated to $G$ via perverse coherent sheaves on the scheme of formal arcs in the affinization of $G/U_-$. We prove our conjecture for $G=SL(N)$ using the so called Laumon resolution of the space of quasi-maps (using this resolution one can reformulate the statement so that only "usual" (not perverse) coherent sheaves are used). In the course of the proof we also give a $K$…

Character sheaves on certain spherical varieties

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construct…

双轨道层与单位根处的仿射赫克代数

Bi-orbital sheaves and affine Hecke algebras at roots of unity

本文在 $\mathbb{Z}/m$-分次李代数幂零锥上引入超尖层概念，证明双轨道层由超尖层经抛物诱导产生，并据此证明了 Grojnowski 关于单位根处仿射赫克代数单模参数化的定理。

Perverse sheaves, Koszul IC-modules, and the quiver for the category O

For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category O for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.

Arthur packets for $G_2$ and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classifica…

Perverse Sheaves on affine Grassmannians and Langlands Duality

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…

Parabolic character sheaves, I

We study a class of perverse sheaves on the variety of pairs (P,gU_P) where P runs through a conjugacy class of parabolics in a connected reductive group G and gU_P runs through G/U_P. This is a generalization of the theory of character sheaves.