We first introduce (parabolic) Hitchin systems on smooth projective curves and show the relation of singularities of generic spectral curves with Kazhdan-Lusztig maps on nilpotent orbits. Then we give an explicit description of generic fibers. As an application, we show the topological mirror symmetry for parabolic SL_n/PGL_n Hitchin systems. We will also discuss some generalizations to various SO_{2n+1}/Sp_{2n} Hitchin systems whose Higgs fields have Springer dual nilpotent residues at marked points. This talk is based on joint works with Xiaoyu Su and Xueqing Wen.
We review certain topics on algebraic cycles on hyperkahler manifolds. Our main goal is to present a proof (joint with Maulik) of the Beauville-Voisin conjecture for Hilbert schemes of points on K3 surfaces, and we will use representation theoretic tools to do so.
The good position elements for elliptic conjugacy classes in the Weyl group is defined by He and Nie in the study of minimal length elements. Later in a paper of He and Lusztig, they used these good elements to construct the transversal slices. In this talk, we will talk about the generalization of good position elements for non-elliptic conjugacy classes by considering braid monoids. In particular, for those conjugacy classes lying in the preimage of Lusztig’s map, we can construct the transversal slices. Moreover, we can define a partial order between these conjugacy classes based on these good position representatives such that Lusztig’s map is order-reversing. This is a generalization of the work of Adams, He and Nie.
I'll present a computation in Tanaka-Thomas's algebro-geometric approach to Vafa-Witten invariants of projective surfaces. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components; the physical notion of S-duality implies conjectural symmetries between these contributions.
One component, the "vertical" component, is a nested Hilbert scheme on a surface. I will explain work in progress with M. Kool and T. Laarakker, in which we express invariants of this component in terms of instanton moduli space of torsion-free framed sheaves on P^2. Applying a recent blow-up identity of Kuhn-Leigh-Tanaka, we obtain formulas conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for refined invariants of vertical component in rank 2.
Yakimov and Goodearl proved that the torus orbits of symplectic leaves of complex Grassmannian (more generally partial flag varieties of $G=GL_n(\CC)$) with respect to the standard Poisson structure are exactly Lusztig strata, therefore relates the study of Lusztig total positivity to Poisson geometry. I will provide an algebraic geometric perspective of Lusztig strata. It turns out that both the standard Poisson structure and the Lusztig strata admits a deformation whose parameter space is given by a family of elliptic curves degenerating to Kodaira E_n fiber. The key tool is our construction of modular Poisson structure on moduli space of complexes on Gorenstein Calabi-Yau curves. This is a joint work with Sasha Polishchuk.
June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence. But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group? The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points. I will present an equivariant log concavity conjecture for hyperplane arrangements (or more generally matroids) with symmetries, and show that one can use the theory of representation stability to prove infinitely many cases of this conjecture for configuration spaces.
We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibers. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.
Given a quiver, Nakajima introduced the quiver variety and the Hecke correspondence, which is a closed subvariety of Cartesian products of quiver varieties. We consider two nested quiver varieties as fiber products of Hecke correspondences along natural projections. After blowing up the diagonal, we prove that they are isomorphic to a quadruple moduli space which Neguţ observed for the Jordan quiver. Moreover, this blow up has a derived enhancement in the recent theory of Hekking.
Graded finite Hecke categories, also known as the homotopy categories of Soergel bimodules, play an important role in categorified link invariants as they are the main ingredient in one of the constructions of the Khovanov—Rozansky triply graded link homology. Their categorical traces, initially studied by Gorsky—Hogancamp—Wedrich, are the natural homes for the derived annular Khovanov—Rozansky link invariants and are closely related to the conjecture of Gorsky—Negut—Rasmussen. I will describe a geometric method to study their categorical traces and Drinfel’d centers, relating them to the categories of (graded) character sheaves, objects that are already extensively studied in geometric representation theory. One of the main ingredients is the new theory of graded sheaves introduced by Penghui Li and the speaker, which will be recalled in the talk. This is joint work with Penghui Li.
We will state a Koszul duality between filtered D-modules and circle equivariant sheaves on loop spaces for stacks after reviewing some of the requisite background. We will then compute some basic examples, give an indication of where one might find these mysterious sheaves on loop spaces in nature (i.e. around categorical traces), and discuss potential applications (still in progress) to the categorical and geometric local Langlands correspondences.
Nakajima quiver varieties are a class of combinatorially defined moduli spaces generalising the Hilbert scheme of points in the plane, defined with the aid of a quiver Q (directed graph) and a fixed framing dimension vector f. In the 90s Nakajima used the cohomology of these varieties (in fixed cohomological degrees, and for fixed f) to construct irreducible lowest weight representations of the Kac-Moody Lie algebras associated to the underlying graph of Q. Since the action is via geometric correspondences, the entire cohomology of these quiver varieties forms a module for the same Kac-Moody Lie algebras, suggesting the question: what is the decomposition of the entire cohomology into irreducible lowest weight representations?
In this talk I will explain that this question is somehow not the right one. I will introduce the BPS Lie algebra associated to Q, a generalised Kac-Moody Lie algebra associated to Q, which contains the usual one as its cohomological degree zero piece. The entire cohomology of the sum of Nakajima quiver varieties for fixed Q and f turns out to have an elegant decomposition into irreducible lowest weight modules for this Lie algebra, with lowest weight spaces isomorphic to the intersection cohomology of certain singular Nakajima quiver varieties. This is joint work with Lucien Hennecart and Sebastian Schlegel Mejia.
We will consider the recent developments in the study of the structure of the center (and higher Hochschild cohomologies) of the small quantum group associated to a simple Lie algebra g and a root of unity q. The results relating this structure with the geometry of the group G lead to a lower bound for the dimension of the center, which conjecturally provides an equality in type A_n.
By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirrorsymmetry of 3d N = 4 SUSY QFTs. Such a QFT is associated to a hyperkähler manifold X equipped with a hyperhamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as the 3d B-model or Rozansky–Witten theory, is a TQFT of algebro-geometric flavor and has been studied extensively by Kapustin, Rozansky and Saulina. The second twist, which is known as the 3d A-model or 3d Seiberg–Witten theory, is a more mysterious TQFT of symplecto-topological flavor. The 2-category of boundary conditions for each of these TQFTs is expected to categorify of category O for the hyperkähler quotient X///G and 3d mirror symmetry is expected to induce a categorification of the Koszul duality between categories O for mirror symplectic resolutions. In this talk I will describe an algebraic approach for abelian gauge theories due to myself, Ben Gammage, and Aaron Mazel-Gee. This generalizes works of Kapustin–Vyas–Setter and Teleman on pure gauge theory.
Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type (or more generally abelian type) constructed by Kisin and Kisin-Pappas. I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models.
A key ingredient in the parahoric version of the main theorem is a new CM lifting result on parahoric level integral models of Shimura varieties, which uses as input a new result on connected components of affine Deligne–Lusztig varieties (joint with Gleason and Lim). This resolves a long-standing conjecture on connected components of affine Deligne-Lusztig varieties.
If time permits, I will give a sketch on how to deduce the new CM lifting result, as well as various other applications to integral models of Shimura varieties.
Let C be a smooth projective curve. The non-abelian Hodge theory of Simpson is a homeomorphism between the character variety M_B of C and the moduli of (semi)stable Higgs bundles M_D on C. Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weightfiltration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on H^*(M_B) gets sent to the perverse filtration on H^*(M_D), associated to
the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on H^*(M_D). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.
Given a DG category acted by the loop group $LG$, one can define a factorization $D(Gr_G)$-module category, where $Gr_G$ is the Beilinson—Drinfeld affine Grassmannian, viewed as a factorization indscheme. We will show this construction loses no information: it gives a fully faithful 2-functor from $LG-mod(DGCat)$ to $Gr_G-mod_fact(DGCat)$. We will also explain the role of this result in the local Geometric Langlands program. This is a report of the incoming joint work of Dennis Gaitsgory, Yuchen Fu, David Yang and the speaker.
Symplectic duality predicts that symplectic singularities should come in pairs. For example, Nakajima quiver varieties are conjecturally dual to BFN Coulomb branches (of the corresponding quiver theories). Another family of potentially symplectically dual pairs was described recently in the works of Losev, Mason-Brown, and Matvieievskyi: they describe symplectically duals to Slodowy slices to nilpotent orbits. In this talk, we will discuss the Hikita-Nakajima conjecture that relates the geometry of symplectically dual varieties. We will restrict to the cases of certain quiver varieties and Slodowy slices and discuss the picture in these cases. Based on the joint work with Pavel Shlykov (arXiv:2202.09934) and the work in progress with Do Kien Hoang and Dmytro Matvieievskyi.
This talk is based on joint work with Reda Boumasmoud. We will discuss results that describe the Bernstein center of the Hecke algebra H(G(F), K) via the theory of types, where G is a connected, reductive group over a non-archimedean local field F (that satisfies some additional hypothesis), and K belongs to a nice family of compact open subgroups of G(F). Along the way, we will also describe the center of the Hecke algebra of a type attached to a Bernstein block.
In the first part of the talk, I will give a survey of the proof of the Dowling-Wilson conjecture using the Schubert variety of a hyperplane arrangement. The Schubert variety of a hyperplane arrangement is an equivariant compactification of the vector group with finitely many orbits. In the second part of the talk, we will discuss a recent work of Colin Crowley characterizing Schubert variety of hyperplane arrangements among all equivariant compactification of vector groups, and some on-going work on polymatroid Schubert varieties joint with Colin Crowley and Connor Simpson.
The Baker-Campbell-Hausdorff Theorem gives a formula for log(e^xe^y) where x and y don't commute. A solution to the Kashiwara-Vergne equations is a refinement of this, and the existence of such solutions has wide implications in harmonic analysis and Lie theory: in particular, it implies Duflo's Theorem. The symmetry groups of the set of Kashiwara-Vergne solutions, called the Kashiwara-Vergne groups, are closely related to the Grothendieck-Teichmuller groups. I will describe a topological (knot theoretic) approach to Kashiwara-Vergne theory, including a topological characterisation of the Kashiwara-Vergne groups, implications, related approaches, and open questions.
This talk is based on joint work with Marcy Robertson and Iva Halacheva, and earlier work with Dror Bar-Natan.
Nakajima's graded quiver varieties are complex algebraic varieties associated with quivers. They are introduced by Nakajima in
the study of representations of universal enveloping algebras of Kac-Moody Lie algebras, and can be used to study cluster algebras. In
the talk, I will explain how to precisely locate the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras by
applying the decomposition theorem to various morphisms related to quiver varieties, thus prove a conjecture proposed by
Lee-Li-Rupel-Zelevinsky in the skew-symmetric case.
In this talk I will discuss several connections between the small quantum group and a certain affine Springer fiber. I will mainly discuss some relation of the category of graded representations of the small quantum group and microlocal sheaves on the affine Springer fiber as part of ongoing work with R.Bezrukavnikov, M. McBreen and Z. Yun. I will also mention some connection of the center of the small quantum group and the cohomology of the affine Springer fiber, part of joint work with R. Bezrukavnikov, P. Shan and E. Vasserot.
The Langlands program posits that automorphic forms associated to a reductive group are parametrized by Galois representations valued in its dual group. About ten years ago, M. Weissman, W.-T. Gan, and F. Gao proposed an extension of the Langlands program which puts covering groups under the same framework. In this talk, I will introduce a formalism of covering groups based on étale cohomology and explain its applications to the Langlands program.
Classical theta series are generating functions for counting vectors in a lattice. They turn out to have a miraculous symmetry property called modularity, which is proved by some simple (by modern standards) Fourier analysis. Kudla discovered analogs of theta series in arithmetic geometry, called arithmetic theta series, which are generating functions composed of algebraic cycles in moduli spaces. These are also expected to enjoy modularity, but this is unknown in most cases and has been very difficult in the known cases. In joint work with Zhiwei Yun and Wei Zhang, we give a proof of a modularity property for arithmetic theta series in the function field context, which works in total generality. The argument is built upon a sheaf-cycle correspondence generalizing the classical sheaf-function correspondence, plus a theory of Fourier analysis on derived vector spaces.
Lusztig's braid group symmetries constitute an essential part in the theory of quantum groups. The i-quantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which are now viewed as natural generalizations of quantum groups. Recently, joint with Wang, we constructed relative braid group symmetries on i-quantum groups of arbitrary finite type, which are i-analogs of Lusztig's symmetries. In this talk, I will present this finite-type construction and its generalization to Kac-Moody types. I will define root vectors in i-quantum groups and show that relative braid group symmetries send root vectors to root vectors.
Let G be a real reductive Lie group. The set of irreducible unitary G-representations is one of the most mysterious objects in representation theory. Although this set has been computed in various special cases, the problem of providing a *uniform* description is unsolved. In this talk, I will sketch a conjectural description of this set in the case when G is complex. This talk is partially based on joint work with Ivan Losev.
Kudla-Rapoport conjecture is a precise identity between intersection number of special divisors on Rapoport-Zink space and derived local density, which is a key ingredient for arithmetic Siegel-Weil formula. The original Kudla-Rapoport conjecture is only formulated over unramified primes (the RZ space in this case has good reduction). In this talk, I will explain how to formulate a conjecture for Kramer models over ramified primes, and the strategy to prove it. On the geometric side, we can completely avoid the Tate conjecture for Deligne-Lusztig varieties. On the analytic side, we obtain a surprisingly simple formula for derived primitive local density. An induction and partial Fourier transform prove the conjecture. This is a joint work with Chao Li, Yousheng Shi and Tonghai Yang.
Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. The generalised Springer correspondence attaches to $(C,E)$ an irreducible representation $\rho$ of some Weyl group. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety. Waldspurger proved a 'maximality' result for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss and sketch the proof of an analogous result for $\text{SO}(N,\mathbb{C})$, which states: if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We will also present an algorithm to compute $\rho^{\text{max}}$. In particular, we focus on computing an example.
A basic problem in representation theory is to calculate the characters of irreducible highest weight modules. For semisimple Lie algebras this was resolved by Beilinson-Bernstein and Brylinski-Kashiwara in the early 1980s. For affine Lie algebras, this was resolved by Kashiwara-Tanisaki at non-critical central charges in the 1990s. At the remaining critical central charge certain cases are known by work of Arakawa-Fiebig, Feigin-Frenkel, Frenkel-Gaitsgory, Hayashi, and Malikov. In forthcoming joint work with David Yang, we calculate the irreducible characters for the regular block at critical level, confirming a conjecture of Feigin-Frenkel.
The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.
In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.
This is based on ongoing joint work with Adeel Khan and David Rydh.
We will discuss the basic geometry of certain v-sheaf analogues of formal schemes that we call kimberlites. We discuss the proof that a (nice enough) kimberlite is determined by its specialization triple. We discuss the proof that minuscule local models are representable by formal schemes. If time allows we discuss the normality of the local models and applications.
This talk is based on joint work with Anschütz-Gleason-Richarz. After motivating with p-adic shtukas, I'll define the p-adic local models in the sense of Scholze-Weinstein. Then we move to the study of their étale cohomology and of the associated nearby cycles functor. In particular, I'll explain why it is central and how this helps in determining the special fiber of the local model. Time permitting, I'll mention work in progress with Anschütz-Wu-Yu on Wakimoto filtrations and t-exactness of nearby cycles at Iwahori level.
Toric varieties are popular objects in algebraic geometry, modelled on convex polytopes. This is mainly because there is a dictionary between their geometric properties and the combinatorial features of their polytopes. This dictionary can be extended from toric varieties to arbitrary varieties through toric degenerations.
In this talk, I will first recall the notion of toric degenerations which generalizes the fruitful correspondence between toric varieties and polytopes to arbitrary varieties. Then I will show some prototypic examples of toric degenerations of Grassmannians. I will describe how to obtain such degenerations using the theory of Gröbner fans and tropical geometry and show the relations among their associated Newton-Okounkov bodies.
Rapoport and Zink introduce the p-adic period domain (also called the admissible locus) inside the rigid analytic p-adic flag varieties. The weakly admissible locus is an approximation of the admissible locus in the sense that these two spaces have the same classical points. On the flag variety, we have the Newton stratification parametrized by the isomorphism classes of the modifications of G-bundles on the Fargues-Fontaine curve. In this talk, we consider the condition that the weakly admissible locus is maximal (i.e. the weakly admissible locus is a union of Newton strata). This unifies the extreme cases when the weakly admissible locus equals to the admissible locus or the whole flag variety. We will give several equivalent criterions for the condition that the weakly admissible locus is maximal. We will also give a group theoretic classification for this condition when G is absolutely simple. Moreover, we give a criterion when a single Newton stratum is contained in the weakly admissible locus. This is a joint work in progress with Jilong Tong.
In the study of Shimura varieties, it is important to count the points reduction modulo p (Langlands-Rapoport conjecture) as it provides a way to compute the Hasse-Weil zeta function. The most interesting piece showing up in the point counting is affine Deligne-Lusztig variety (ADLV) and it has been studied in various level structures including the hyperspecial level and the Iwahori level. In this talk, we will see explicit examples of ADLV described as a set of certain lattices and flags. Moreover, we will discuss the nonemptiness criterion for ADLV along with the results already known and newly discovered. If time permits, the dimension formula will be discussed shortly.
(joint project with Aleksei Ilin, Vasily Krylov, and Inna Mashanova-Golikova)
Bethe subalgebras B(C) form a family of maximal commutative subalgebras in the Yangian Y(g) of a semisimple Lie algebra g depending on a group element C of the corresponding adjoint group G. The images of Bethe algebras in tensor products of fundamental representations can be regarded as the integrals of the quantum XXX Heisenberg magnet chain. On the other hand, according to Maulik and Okounkov, Bethe subalgebras arise as equivariant quantum cohomology rings of Nakajima quiver varieties. We give some reasonable sufficient conditions on a representation V of the Yangian of type A guaranteeing that B(C) acts on V without multiplicities. We show that this property holds for certain limits of Bethe subalgebras as well. This allows us to define a KR-crystal structure on the spectrum of a Bethe subalgebra on V. Conjecturally, the monodromy of the spectrum of B(C) along C is given by Schutzenberger involutions of this crystal.
This talk will be about affine Springer fibers and their generalizations that encode orbital integrals of functions in the spherical Hecke algebra of a p-adic reductive group. These objects can also be viewed as linear analogues of affine Deligne-Lusztig varieties. I will discuss current knowledge on basic geometric properties of these varieties, including non-emptiness, dimension, and irreducible components. Time permitting, I will also discuss local and global ingredients in the proofs.
In this talk, I will give an explicit description of affine Grassmannians for triality groups as functors classifying suitable lattices in a fixed space. These triality groups are of type $^3 D_4$ and can be constructed by certain twisted composition algebras. Further, I will briefly introduce global affine Grassmannians for triality groups. I combine this description with the Pappas-Zhu construction, to obtain corresponding local models; the singularities of these local models are supposed to model the singularities of certain orthogonal Shimura varieties.
In this talk, we introduce a toy model of shtukas and study moduli spaces of toy shtukas. In particular, we determine the space of principal horospherical divisors on the moduli spaces. As an application, we obtain analogues of modular units on the moduli stack of Drinfeld shtukas. This gives an explicit version of the Manin-Drinfeld theorem for function fields.
The affine flag variety of an algebraic group allows a decomposition into Iwahori double cosets, indexed by the affine Weyl group. This is the first step towards answering geometric questions via combinatorial methods.
The closure relations between these double cosets are described by the Bruhat order on the affine Weyl group, a classical order that has been fully understood only in special cases.
Related to this is the question of generic Newton points, which asks for the Newton point of the generic point of such an Iwahori double coset.
Both questions have partial answers in terms of the quantum Bruhat graph, given by Lam and Shimozono resp. Milićević. We extend these results and give a full answer to both problems in terms of this graph. As an application, we describe the cordial elements of certain groups.
Admissible representations of real reductive groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of a block of the LV module using Soergel bimodules.
I will explain a proof of the S=T conjecture proposed by Liang Xiao and Xinwen Zhu using Scholze’s theory of diamonds and v-stacks and Fargues-Scholze’s geometric Satake equivalence. I will explain the necessary background on Shtukas, and why the new geometric theory developed by Scholze is so natural to attack the conjecture.
I'll explain the motivation and construction of the action of the Braverman-Finkelberg-Nakajima (BFN) Coulomb branch algebra associated to a G representation N on the vertex algebra of chiral differential operators on the quotient stack N/G, whenever the latter is defined, following a conjecture of Costello-Gaiotto. Further, I'll state a generalization of the classical Atiyah-Bott localization theorem in the setting of equivariant chiral homology, and explain its relationship with the action constructed in the first part of the talk via the equivariant chiral-E_2 algebra of endomorphisms of the unit in the BFN variant of the coherent Satake category.
In this talk, I will introduce a virtual variant of the quantized Coulomb branch by Bravermann-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module.
In this talk, I will explain a connection between stable homotopy theory and representation theory. I will focus on one application of this idea to a problem arising from the modular representation theory. More explicitly, we study a family of new quantum groups labelled by a prime number and a positive integer constructed using the Morava E-theories. Those quantum groups are related to Lusztig's 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic. This talk is based on my joint work with Gufang Zhao.
We discuss a new definition of p-adic Deligne--Lusztig spaces, as arc-sheaves on perfect algebras over the residue field. Then we look then at some fundamental properties of these sheaves. In particular, we discuss cases in which these sheaves are ind-representable resp. even representable. Along the way we also discuss a general result saying that the (perfect) loop space of a quasi-projective scheme over Q_p is an arc-sheaf.
This talk will be about my paper of the same title with Gurbir Dhillon. It is well-known that the center of the enveloping algebra of an affine Kac-Moody algebra at noncritical level is trivial. Nonetheless, its representation theory shares many features with that of a finite-dimensional semisimple Lie algebra, including a block decomposition of category O. We propose an analogue, for any affine Weyl group orbit, of the category of Kac-Moody representations with the corresponding "generalized central character." Namely, we consider the subcategory generated by the relevant Verma modules under the categorical loop group action. We also construct equivalences relating various categories of affine Harish-Chandra bimodules, Whittaker modules, and Whittaker D-modules on the loop group, generalizing known equivalences in the finite-dimensional case proved by Bernstein-Gelfand, Beilinson-Bernstein, Milicic-Soergel, and others.
The talk is based on joint work with A.Smirnov. We obtain a factorization theorem about the limit of elliptic stable envelopes to a point on a wall in H^2(X,R), which generalizes the result of M.Aganagic and A.Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices, etc., to actions on the K-theory of the symplectic dual variety. In the case of the Hilbert scheme of points in the plane, our results imply the conjectures of E.Gorsky and A.Negut. As another application of this technique, we gain a better geometric understanding of the wall crossing operators and the quantum difference equations.
We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.
Also building upon work of He-Nie, our perspective furnishes a common generalisation and a simple geometric criterion for Weyl group elements to yield strictly transverse slices.
In the first half of the talk, I will give an overview of Tits cone intersections, which are structures that can be obtained from (possibly affine) ADE Dynkin diagrams, together with a choice of nodes. This is quite elementary, but visually very beautiful, and it has some really remarkable features and applications. In the second half of the talk I will highlight some of the applications to algebraic geometry in dimension two and three, including 3-fold flopping contractions, through mutation and stability conditions. This should be viewed as a categorification of the first half of my talk. Parts are joint work with Osamu Iyama, parts with Yuki Hirano.
The commuting scheme of a reductive Lie algebra has always been of great interest in Lie theory and Invariant theory but it was only recent that it appears as a primordial object in the study of the Hitchin fibration for higher dimensional varieties. I will explain how the invariant theory for the commuting scheme, in particular the Chevalley restriction theorem for the commuting scheme, is used in the study of Hitchin fibration and the proof of the Chevalley restriction theorem in the case of symplectic Lie algebras. The talk is based on joint work with Ngo Bao Chau.
Affine Hecke algebras are the key to many problems in the representation theory of reductive groups. Every affine Hecke algebra H(W,q) admits an explicit presentation as a deformation of the group algebra of an affine Weyl group W. That allows one to study them independently, without reductive groups.
In this talk we will compare the representations of affine Hecke algebras, of finite Weyl groups and of affine Weyl groups. Although H(W,q) is much more complicated than the group algebra of W (over C), we will show that there exists a canonical bijection between the sets of irreducible representations of these two algebras. We will also discuss the relation with Springer correspondences for affine Weyl groups.
Poles of finite-dimensional representations of Yangians.
The Yangian associated to a simple Lie algebra g, is a Hopf algebra which quantizes the Lie algebra of polynomials g[t]. Its finite-dimensional representation theory has remarkable connections with equivariant cohomology, combinatorics, integrable systems and mathematical physics. Concretely, a finite-dimensional representation of the Yangian is prescribed by a finite collection of operators whose coefficients are rational functions, satisfying a list of commutation relations.
In this talk I will give an explicit combinatorial description of the sets of poles of the rational currents of the Yangian, acting on an irreducible finite-dimensional representation. This result uses the generalization of Baxter's Q-operators obtained by Frenkel-Hernandez. Based on a joint work with Curtis Wendlandt (arxiv:2009.06427).
Unexpected fillings, singularities, and plane curve arrangements.
I will discuss joint work with Olga Plamenevskaya studying symplectic fillings of links of certain complex surface singularities, and comparing symplectic fillings with complex smoothings. We develop characterizations of the symplectic fillings using planar Lefschetz fibrations and singular braided surfaces. This provides an analogue of de Jong and van Straten's work which characterizes the complex smoothings in terms of decorated complex plane curves. We find differences between symplectic fillings and complex smoothings that had not previously been found in rational complex surface singularities.
In this talk, I will present a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. After motivating the problem and reviewing its history, I will explain the key idea, of geometric nature, behind the proof of its solution.
Semi-affineness of wrapped invariants on affine log Calabi-Yau varieties.
A general expectation in mirror symmetry is that the mirror partner to an affine log Calabi-Yau variety is "semi-affine" (meaning it is proper over its affinization). We will discuss how the semi-affineness of the mirror can be seen directly as certain finiteness properties of Floer theoretic invariants of X (the symplectic cohomology and wrapped Fukaya category). As an application of these finiteness results, we will show that for maximally degenerate log Calabi-Yau varieties equipped with a ``homological section," the wrapped Fukaya of X gives an (intrinsic) categorical crepant resolution of the affine variety Spec(SH^0(X)). This is based on https://arxiv.org/pdf/2103.01200.pdf.
Geometry of the Gopakumar-Vafa theory
Motivated by classical enumerative geometry and mathematical physics, counting curves in Calabi-Yau 3-folds has been studied intensively for decades, including Gromov-Witten theory and Donaldson-Thomas theory. In recent years, mathematical theory (Hosono-Takahashi-Saito, Kiem-Li, Maulik-Toda etc) has been developed to realize the idea of Gopakumar and Vafa to recover the curve-counting invariants using the geometry of 1-dimensional sheaves. These developments shed new light on both enumerative geometry and the classical geometry of the relevant moduli spaces. I will discuss 3 particular cases (1) Higgs bundles (2) K3 surfaces, and (3) P^2, where the Gopakumar-Vafa theory interacts with some other structures and conjectures in a surprising way.
Lagrangian Fillings of Legendrian links
In this talk, I will discuss the current state of the classification of Lagrangian fillings of Legendrian knots in the standard contact 3-sphere. This is a central problem in low-dimensional contact and symplectic topology. First, I will introduce the basic geometric objects and problems of interest. Second, I will explain the many developments that have taken place this last year, introducing the main new results and the different novel techniques that have become available. Finally, I will discuss work-in-progress complementing and expanding some of these results, and state some open questions and related conjectures.
Sheaf quantization and exact WKB analysis.
Sheaf quantization is a Betti-analogue of deformation quantization module, which has many applications: 1. A sheaf quantization contains symplectic-topological information of the underlying Lagrangian submanifold. 2. Sheaf quantization can be an appropriate notion to express "solution sheaf" for irregular D-modules. In this talk, I'd like to review sheaf quantization and then explain my work about 2, which is also very related to 1.
Let M be a (negatively) monotone symplectic manifold satisfying a strong non-degeneracy condition. The purpose of this talk is to construct an algebraic ``action'' of the algebraic torus G_m^{b_1(M)} on the Fukaya category of M and use this to study the behavior of Lagrangian Floer homology under symplectic isotopies. We use algebraicity of the action to show that the rank of Floer homology groups change in a tame way under symplectic isotopies. We also use this to describe ``stabilizer'' of a Lagrangian as an algebraic subtorus. If time remains, we will discuss applications to mirror symmetry.
The Rabinowitz Floer (co)homology of the boundary at infinity of a Liouville manifold is in some sense a symplectic-geometric analogue of the mapping cone from homology to cohomology. In this talk, I will discuss an A_{\infty}-categorical structure based on Rabinowitz Floer theory, which induces a non-trivial product on Rabinowitz Floer cohomology, extensively studied recently by Cieliebak-Oancea. We shall see that this A_{\infty}-structure can be algebraically computable from the wrapped Fukaya category, via the construction of the categorical formal punctured neighborhood, proving a conjecture due to Abouzaid.
This is joint work with Sheel Ganatra and Sara Venkatesh.
I will discuss the construction of a nil Hecke algebra in the category of bimodules over an algebra (full generality: bimodules over a small DG category) in the context of categorical representations of the Lie algebra sl_2.
We'll introduce a construction that turns an n-strand braid into a variety - which we call its cycloform - that lives (equivariantly) over the unipotent locus of SL_n. When the braid is the identity, the cycloform is the Springer resolution. In general, we can use the cycloform to build a "Steinberg-like variety" whose equivariant Borel-Moore homology admits an S_n-action, and indeed, the action of a larger algebra. The Steinberg-like variety is interesting in two ways: (1) The Khovanov-Rozansky homology of the link closure of the braid can be recovered from the S_n-action. We expect the algebra action to recover a conjugacy invariant of the braid called the horizontal trace. (2) We conjecture that if the braid maps to an n-cycle, then the variety deformation-retracts onto an (Iwahori) affine Springer fiber for SL_n, in a way compatible with the Springer action on both sides and reminiscent of nonabelian Hodge theory. This, in turn, implies unexpected geometric properties of the cycloforms. If time permits, we'll explain the proof of (1) and the numerical evidence for (2). Both extend beyond SL_n to other semisimple groups.
The motivic Chern class in K-theory is a natural generalization of the MacPherson class in homology. In this talk, we will talk about several applications of the motivic Chern classes of the Schubert cells. These classes can be used to give a smoothness criterion for the Schubert varieties, which is used to prove several conjectures of Bump- Nakasuji-Naruse about representations of p-adic dual groups and also conjectures of Lenart-Zainoulline-Zhong about Schubert classes in hyperbolic cohomology of flag varieties. The Euler characteristics of these classes are also related to the Iwahori-Whittaker functions of the dual groups. Based on several joint works with P. Aluffi, C. Lenart, L. Mihalcea, J. Schürmann, K. Zainoulline, and C. Zhong.
Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\fh$ of $H$, we define an analogous compactification $\bar \fh$ of $\fh$, to be referred to as the wonderful compactification of $\fh$. The wonderful compactification of $\fh$ is an example of an "additive toric variety". We establish a bijection between the set of irreducible components of the boundary $\bar \fh - \fh$ of $\fh$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1$, where $r$ is the dimension of $\fh$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems will be given. We prove that each irreducible component of $\bar \fh - \fh$ is isomorphic to the wonderful compactification of a Lie subalgebra of $\fh$ and is of dimension $r - 1$. In particular, the boundary $\bar \fh - \fh$ is equidimensional. We describe a large subset of the regular locus of $\bar \fh$. As a consequence, we prove that $\bar \fh$ is a normal variety.
After recalling the most important results about Kazhdan-Lusztig cells for symmetric groups, I will introduce the p-Kazhdan-Lusztig basis and give a complete description of p-cells for symmetric groups. After that I will mention important consequences of the Perron-Frobenius theorem for p-cells which provide one of the last missing ingredients for the proof of the cellularity of the p-Kazhdan-Lusztig basis in finite type A.
We will present recent developments in the theory of overconvergent F-isocrystals, the p-adic analogue of ell-adic lisse sheaves. For the most part of the talk, we will explain the proof of the parabolicity conjecture, a conjecture proposed by Crew in '92 on the algebraic monodromy groups of overconvergent F-isocrystals. In the end, we will explore some consequences.
In this talk I will sketch the construction of a determinant map for Tate objects and show two of its possible applications. The first is to construct central extensions of iterated loop groups and the second is to recover the notion of "determinantal" theory on reasonable ind-schemes. For the second application we will need a detour in the theory of Tate-coherent sheaves.
Hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. In my joint work with Masoud Kamgarpour, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric sheaves, thus confirm the geometric Langlands conjecture for hypergeometrics. In this talk, I will introduce the key tool we use called rigid automorphic data, due to Z. Yun. The definition of automorphic data for tame hypergeometrics involves the mirabolic subgroup, while in the wild case, Moy-Prasad subgroups of principal parahorics intervene.
Geometric approaches in constructing the Jacquet-Langlands transfers were pioneered by Ribet and Serre. Their works motivate many of the later progress in this direction. In this talk, I will discuss a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights. The essential ingredient of the construction is the integral coefficient geometric Satake equivalence in mixed characteristic obtained by the speaker and Scholze independently.
Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of principal block representations of a p-adic group with certain unipotent Langlands parameters (a q-commuting semisimple-nilpotent pair) in the Langlands dual group. We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category of coherent sheaves equivalent to modules for the affine Hecke algebra (or specializing at q, smooth principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category. This is a joint work with David Ben-Zvi, David Helm and David Nadler.