Springer Resolution Research Articles

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Let $G$ be a reductive group, and let $\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags ${\operatorname {Fl}}_G$ is equivalent to the category of $\check {G}$ -equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on ${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category $\mathsf {O}$ is equivalent to the twisted Whittaker category on ${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on ${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the $\ell$ -adic setting.

Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules

We construct a co- t t -structure on the derived category of coherent sheaves on the nilpotent cone N \mathcal {N} of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co- t t -structure on N \mathcal {N} . We also demonstrate how the various parabolic co- t t -structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type A A for p > h p>h .

A new approach to the generalized Springer correspondence

The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig’s generalized Springer correspondence.

Representations of the elliptic affine Hecke algebras

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-A. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d N=2 gauge theory.

Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

AbstractWe compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

On subregular slices of the elliptic Grothendieck–Springer resolution

In \cite{davis19}, the author constructed an elliptic version of the Grothendieck-Springer resolution for the stack $\mathrm{Bun}_G$ of principal bundles under a simply connected simple group $G$ on an elliptic curve $E$. This is a simultaneous log resolution of a map from $\mathrm{Bun}_G$ to the union of the coarse moduli space of semistable $G$-bundles and a single stacky point. In this paper, we study singularities, resolutions and deformations coming from subregular slices of this elliptic Grothendieck-Springer resolution. More precisely, we construct explicit slices of $\mathrm{Bun}_G$ through all subregular unstable bundles, for every $G$. For $G \neq SL_2$, we describe the pullbacks of the elliptic Grothendieck-Springer resolution to these slices as concrete varieties, extending and refining earlier work of I. Grojnowski and N. Shepherd-Barron, who related these varieties to del Pezzo surfaces in type $E$. We use the resolutions to identify the singularities of the unstable locus of the subregular slices, and prove that that the extended coarse moduli space map gives deformations that are miniversal among torus-equivariant deformations with appropriate weights.

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A.

Graham's variety and perverse sheaves on the nilpotent cone

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 An, a key tool is a known description of an affine paving of Springer fibers.

On the K-theory stable bases of the Springer resolution

Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.

The center of small quantum groups II: singular blocks

We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, we present a linear algebro-geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an $l$th root of unity is formulated in type A.

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Let \mathbf G be a connected reductive group over an algebraically closed field \mathbb F of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov’s exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of \mathbf G , and Iwahori-constructible \mathbb F -parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirković–Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov–Bezrukavnikov–Ginzburg, and an extension of this equivalence.

Higher Cohomology Vanishing of Line Bundles on Generalized Springer Resolution

A conjecture of Michael Finkelberg and Andrei Ionov is proved on the basis of a generalization of the Springer resolution and the Grauert–Riemenschneider vanishing theorem. As a corollary, it is proved that the coefficients of the multivariable version of Kostka functions introduced by Finkelberg and Ionov are nonnegative.

The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck–Springer resolution

Let G = G L n ( C ) , the general linear group over the complex numbers, and let B be the set of invertible upper triangular matrices in G . Let b = Lie ( B ) . For μ : T ∗ ( b × C n ) → b ∗ , where b ∗ ≅ g ∕ u and u being strictly upper triangular matrices in g = Lie ( G ) , we prove that the Hamiltonian reduction μ − 1 ( 0 ) r s s ∕ ∕ B of the extended regular semisimple locus b r s s of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of C 2 n . We also show that the B -invariant functions on the regular semisimple locus of the Hamiltonian reduction of b × C n arise as the trace of a certain product of matrices.

Reductive groups, the loop Grassmannian, and the Springer resolution

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a graded version of a conjecture of Finkelberg-Mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic.

A lower bound for composition multiplicities for the Springer resolution

ABSTRACTLet be the Springer resolution of the nilpotent cone for a semisimple connected algebraic group G over ℂ and k be an arbitrary field. What happens to if the decomposition theorem fails for it? We show that in this case, some additional (with respect to the case char k=0) composition factors of this direct image in the (abelian) category of perverse sheaves may emerge. These factors emerge from the -composition factors of the radicals of certain intersection forms and from that of the top cohomologies of Springer fibres (in the non-semisimple case).

Quantum groups, quantum tori, and the Grothendieck–Springer resolution

We construct an algebra embedding of the quantum group Uq(g) into a central extension of the quantum coordinate ring Oq[Gw0,w0/H] of the reduced big double Bruhat cell in G. This embedding factors through the Heisenberg double Hq of the quantum Borel subalgebra U≥0, which we relate to Oq[G] via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck–Springer resolution studied in [10], and the quantum Beilinson–Bernstein theorem investigated in [2] and [36].

Dual pairs of quantum moment maps and doubles of Hopf algebras

Given a dual pair of topological Hopf algebras A,A⁎, under mild conditions there exists a natural associative algebra homomorphism D(A)→H(A) between the corresponding Drinfeld double D(A) and Heisenberg double H(A). We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck–Springer resolution arises in this context.

Comparing the commutative and non-commutative resolutions for determinantal varieties of skew symmetric and symmetric matrices

Let Y be the variety of (skew) symmetric n×n-matrices of rank ≤r. In paper we construct a full faithful embedding between the derived category of a non-commutative resolution of Y, constructed earlier by the authors, and the derived category of the classical Springer resolution of Y.

The affine Grassmannian and the Springer resolution in positive characteristic

An important result of Arkhipov–Bezrukavnikov–Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of ‘mixed modular sheaves’ recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov’s ‘exotic t-structure’ on the Springer resolution.

Restriction formula for stable basis of the Springer resolution

We give restriction formula for stable basis of the Springer resolution, and generalize it to cotangent bundles of homogeneous spaces. By a limiting process, we get the restriction formula of Schubert varieties.