Semi-infinite flags. II. Local and global intersection cohomology of quasimaps’ spaces (original) (raw)
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Geometric properties of the Kazhdan-Lusztig Schubert basis
arXiv: Algebraic Geometry, 2020
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the KKK-theory and hyperbolic cohomology theory of flag varieties. We first show that, in KKK-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinsky's small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.
LECTURE 7: REALIZTING U−(g) USING LUSZTIG’S NILPOTENT VARIETY
2014
This will be the first in a series of lectures on a geometric way of realizing the algebra U−(g), the crystal B(∞), highest weight representations of g, and crystals of these highest weight modules. Note that, although we realize both the reprsentation of g and the crsytal of this representation, we do not realize it as a representation of Uq(]g). This can be done (see), but is much more difficult. The geometric spaces we use will be Lusztig’s varieties Λ(V) from [L] (sometimes called Lusztig’s nilpotent variety), and later on Nakajima’s varieties L(v, w) from [N]. Note that through-out this story we assume that g is a symmetric Kac-Moody algebra. Some constructions can be extended to the symmetrizable case by “folding ” arguments based on the observation that U(g) for symmetrizible g can be embedded into U(g) for a related symmetric g. Today we define Λ(V). Then we construct a product on ⊕vM(Λ(V)/GL(V)) (i.e. on the sum of the spaces of invariant constructible functions on Λ(V) as ...
Parabolic Kazhdan–Lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology
Journal of the Institute of Mathematics of Jussieu, 2019
We study the equivariant oriented cohomology ring mathtthT(G/P)\mathtt{h}_{T}(G/P)mathtthT(G/P) of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in mathtthT(G/P)\mathtt{h}_{T}(G/P)mathtthT(G/P) can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory mathfrakh\mathfrak{h}mathfrakh corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in mathfrakhT(G/P)\mathfrak{h}_{T}(G/P)mathfrakhT(G/P). We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.
On Schubert calculus in elliptic cohomology
Discrete Mathematics & Theoretical Computer Science, 2015
An important combinatorial result in equivariant cohomology and K-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We use these polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective K-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra. Résumé. Un résultat combinatoire important dans le calcul de Schubert pour la cohomologie et la K-théorie équivariante est représenté par les formules de Billey et Graham-Willems pour la localisation des classes de Schubert aux points fixes du tore. Ces formules sont uniformes pour tous les types de Lie, et sont basés sur le concept d'un polynôme de racines. Nous définissons les polynômes formels de racines associées à une loi arbitraire de groupe formel (et donc à une théorie de cohomologie généralisée). Nous utilisons ces polynômes pour simplifier les preuves de Billey et Graham-Willems, et aussi pour généraliser leurs résultats à la K-théorie connective et la cohomologie elliptique. Un autre résultat concerne la définition d'une base de Schubert dans cohomologie elliptique (c'est à dire, des classes indépendantes d'un mot réduit), en utilisant la base de Kazhdan-Lusztig de l'algèbre de Hecke correspondant.
On geometric constructions of the universal enveloping algebra U(slnÌ
1994
In the thesis, the universal enveloping algebra U(sl,,) and its modified version are constructed as subalgebras, of the algebra 6. The algebra U is constructed as a projective limit of finite dimensional subalgebras of convolution algebras of constructible functions on cotangent bundles of flag manifolds. The construction provides a canonical basis of &, which gives rise to distinguished bases of all irreducible finite dimensional representations of sl,. The basic steps follow those of Ginzburg's Lagrangian construction. We show how the latter is related to Lusztig's construction of the -part Uof U(sl"), which is done in terms of constructible functions on Lagrangian subvarieties; of spaces of representations of quivers. Using the geometric setting, we compute the canonical basis of for S12 and 12 series of monornials in the canonical basis for S13Thesis Supervisor: George Lusztig Title: Professor of Mathematics To my mother and the memory of my father
Some intersection properties of the fibres of Springer's resolution
Proc. Am. Math. Soc, 1984
Combinatorial results are used to calculate the dimension of the intersection of any two irreducible components of the set in the flag variety fixed by the action of a unipotent element of GL" whose Jordan decomposition has two blocks. This is then related to the "left cells" of Kazhdan and Lusztig, which are used to construct representations of S", the Weyl group of GL". 0. In this note, we study the fibres of Springer's resolution [St] of the singularities of the unipotent variety in G = GL"(/c), where k is an algebraically closed field. These fibres are fixed point sets for the action of G on the variety 38 parametrizing the complete flags in a vector space of dimension n. We use 38 u to fixed by a unipotent element u e SL". In general, S8U has several irreducible components. Suppose that the Jordan decomposition of u has block sizes X, > X2 ^ • • • > Xs, where a, + ■ ■ ■ + Xs = n. (We refer to this as the shape X, and, by abuse of language, we say that u has shape X.) Then each component of 38 u has dimension 2Zsi=l(i-1)a(, and there is one component for each standard Young tableau of shape X (see [S] and below). However, one does not know, in general, the codimension of the intersection of two components; only the "one-hook" case has been done [V]. Here, we calculate the codimension when the Jordan form of u has two blocks. This calculation depends on combinatorial techniques exposed in [LS]. The precise result is (2.1). This calculation has twofold significance. First, it enables one, in this case, to verify a conjecture of Kazhdan and Lusztig [KL, 6.3] concerning the configuration of components of 3 §u; see (4.3). Second, this casts new light on the combinatorial results in [LS] which are used to calculate Kazhdan-Lusztig polynomials, because there is no mention of the geometry of the Grassmannian here.
Filtering cohomology of ordinary and Lagrangian Grassmannians
arXiv: Combinatorics, 2020
This paper studies, for a positive integer mmm, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most mmm. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of kkk-conjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.
Representation theory of Geigle-Lenzing complete intersections
2014
Weighted projective lines, introduced by Geigle and Lenzing in 1987, are one of the basic objects in representation theory. One key property is that they have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative rings RRR graded by abelian groups LLL of rank 111, which we call Geigle-Lenzing complete intersections. We study their Cohen-Macaulay representations, and show that there always exists a tilting object in the stable category of mathsfCMLR{\mathsf CM}^LRmathsfCMLR. As an application we study when (R,L)(R,L)(R,L) is ddd-Cohen-Macaulay finite in the sense of higher dimensional Auslander-Reiten theory. Secondly, by applying the Serre construction to (R,L)(R,L)(R,L), we introduce the category mathsfcohX{\mathsf coh} XmathsfcohX of coherent sheaves on a Geigle-Lenzing projective space XXX. We show that there always exists a tilting bundle TTT on XXX, and study the endomorphism algebra rmEndX(T){\rm End}_X(T)rmEndX(T) which we call a ddd-canonical algebra. Further we study when mathsfcohX{\mathsf coh} XmathsfcohX is derived equivalent to a ddd-representation infinite algebra in the sense of higher dimensional Auslander-Reiten theory. Also we show that ddd-canonical algebras provide a rich source of ddd-Fano and ddd-anti-Fano algebras from non-commutative algebraic geometry. Moreover we observe Orlov-type semiorthogonal decompositions between the stable category of mathsfCMLR{\mathsf CM}^LRmathsfCMLR and the derived category Db(mathsfcohX)D^b({\mathsf coh} X)Db(mathsfcohX).
arXiv (Cornell University), 2020
We define a new geometric object-the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as ℓ-adic sheaves). We formulate a conjecture that makes precise the connection between the category of automorphic sheaves and the space of automorphic functions. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY 3.6. Associated pairs and semi-simple local systems 3.7. Analysis of connected/irreducible components 4. Comparison with the Betti and de Rham versions of LocSys dR G (X) 4.1. Relation to the Rham version 4.2. A digression: ind-closed embeddings 4.3. Uniformization and the proof of Theorem 4.1.10 4.4. Algebraic proof of Proposition 4.3.5 4.5. The Betti version of LocSys G (X) 4.6. The coarse moduli space of Betti local systems 4.7. Relationship of the restricted and Betti versions 4.8. Comparison of LocSys restr G (X) vs LocSys Betti G (X) via the coarse moduli space 5. Geometric properties of LocSys restr G (X) 5.1. "Mock-properness" of red LocSys restr G (X) 5.2. A digression: ind-algebraic stacks 5.3. Mock-affineness and coarse moduli spaces 5.4. Coarse moduli spaces for connected components of LocSys restr G (X) 6. The formal coarse moduli space 6.1. The coarse moduli space in the Betti setting 6.2. Property W 6.3. Property A 6.4. A digression: the case of algebraic groups 6.5. The case of pro-algebraic groups 6.6. Proof of Theorem 6.5.7 6.7. Proof of Theorem 5.4.2 6.8. Proof of Theorem 6.7.8 7. Quasi-coherent sheaves on a formal affine scheme 7.1. Formal affine schemes: basic properties 7.2. Proof of Proposition 7.1.12 7.3. Mapping affine schemes into a formal affine scheme 7.4. Semi-rigidity and semi-passable prestacks 7.5. Duality for semi-passable prestacks 7.6. The functor of !-global sections 7.7. The functor of !-global sections on a formal affine scheme 7.8. Applications of 1-affineness 7.9. Compact generation of QCoh(Maps(Rep(G), H)) 7.10. Enhanced categorical trace 8. The spectral decomposition theorem 8.1. Actions of Rep(G) ⊗X-lisse 8.2. The coHom symmetric monoidal category 8.3. Maps vs coHom 8.4. Spectral decomposition vs actions 8.5. A rigidified version 9. Categories adapted for spectral decomposition 9.1. The Betti case 9.2. The heriditary property of being adapted 9.3. Proof of Theorem 9.2.2: identifying the essential image 9.4. Proof of Theorem 8.3.7, Betti and de Rham contexts 9.5. Proof of Theorem 8.3.7,étale context over a field of characteristic 0 9.6. Proof of Theorem 8.3.7,étale context over a field of positive characteristic 9.7. A simple proof of Theorem 5.4.2 9.8. Complements: de Rham and Betti spectral actions 10. Other examples of categories adapted for spectral decomposition 10.1. The case of Lie algebras GEOMETRIC LANGLANDS WITH NILPOTENT SINGULAR SUPPORT 10.2. The space of maps of Lie algebras 10.3. Proof of Theorem 10.1.3 10.4. Back to the Betti case 11. Ran version of Rep(G) and Beilinson's spectral projector 11.1. The category Rep(G)Ran 11.2. Relation to the lisse version 11.3. Rigidity 11.4. Self-duality 11.5. The progenitor of the projector 11.6. The progenitor as a colimit 11.7. Explicit construction of the Hecke isomorphisms 12. The spectral projector and localization 12.1. The progenitor for coHom 12.2. Abstract version of factorization homology 12.3. Proofs of Propositions 12.1.2 and 12.1.7 12.4. Applications to C Betti Ran 12.5. Applications to CRan 12.6. Identification of the diagonal 12.7. Localization on LocSys G (X) 12.8. Tensor products over Rep(G) vs. QCoh(LocSys restr G (X)) 13. Spectral projector and Hecke eigen-objects 13.1. Beilinson's spectral projector-abstract form 13.2. A multiplicativity property of the projector 13.3. The spectral (sub)category 13.4. Beilinson's spectral projector-the universal case 13.5. Beilinson's spectral projector-the general case 13.6. A version with parameters 13.7.