Springer Fibers Research Articles

Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

Abstract We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture

Abstract We prove that $\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta $-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $\Delta $-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^{2}$ coefficients of $\omega \Delta ^{\prime}_{e_{k}}e_{n}$.

The companion section for classical groups

We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text].

A type C study of Braverman–Gaitsgory–Ginzburg’s construction of sln representations

In [Z. Fan, H. Ma and H. Xiao, Equivariant K-theory approach to [Formula: see text]-quantum groups, Publ. Res. Inst. Math. Sci. 58 (2022), no.3, 635–668], it is proved that the convolution algebra of the top Borel–Moore homology on the Steinberg variety of type [Formula: see text] realizes [Formula: see text], where [Formula: see text] is the fixed point subalgebra of involution on [Formula: see text]. So, the top Borel–Moore homology of the partial Springer fibers gives the representations of [Formula: see text]. In this paper, we study these representations using Schur–Weyl duality and Springer theory.

Parabolic induction for Springer fibres

Let G G be a reductive group satisfying the standard hypotheses, with Lie algebra g \mathfrak {g} . For each nilpotent orbit O 0 \mathcal {O}_0 in a Levi subalgebra g 0 \mathfrak {g}_0 we can consider the induced orbit O \mathcal {O} defined by Lusztig and Spaltenstein. We observe that there is a natural closed morphism of relative dimension zero from the Springer fibre over a point of O 0 \mathcal {O}_0 to the Springer fibre over O \mathcal {O} , which induces an injection on the level of irreducible components. When G = G L N G = GL_N the components of Springer fibres were classified by Spaltenstein using standard tableaux. Our main result explains how the Lusztig–Spaltenstein map of Springer fibres can be described combinatorially, using a new associative composition rule for standard tableaux which we call stacking.

Springer Fibers of Hook Type and Schubert Points

Springer Fibers of Hook Type and Schubert Points

Littlewood-Richardson coefficient, Springer fibers and the annihilator varieties of induced representations

For G=GL(n,C) and a parabolic subgroup P=LN with a two-block Levi subgroup L=GL(n1)×GL(n2), the space G⋅(O+n), where O is a nilpotent orbit of l, is a union of nilpotent orbits of g. In the first part of our main theorem, we use the geometric Sakate equivalence to prove that O′⊂G⋅(O+n) if and only if some Littlewood-Richardson coefficients do not vanish. The second part of our main theorem describes the geometry of the space O∩p, which is an important space to study for the Whittaker supports and annihilator varieties of representations of G.

Algebra and geometry of link homology: Lecture notes from the IHES 2021 Summer School

AbstractThese notes cover the lectures of the first named author at 2021 IHES Summer School on “Enumerative Geometry, Physics and Representation Theory” with additional details and references. They cover the definition of Khovanov‐Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro‐geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

Exotic t-structures for two-block Springer fibres

We study the category of representations of s l m + 2 n \mathfrak {sl}_{m+2n} in positive characteristic, with p p -character given by a nilpotent with Jordan type ( m + n , n ) (m+n,n) . Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to D m , n 0 \mathcal {D}_{m,n}^0 , the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories D m , n \mathcal {D}_{m,n} (i.e. for different values of n n ). This allows us to describe the irreducible objects in D m , n 0 \mathcal {D}_{m,n}^0 and enumerate them by crossingless ( m , m + 2 n ) (m,m+2n) matchings. We compute the E x t \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov’s arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category O \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al.

Paving Springer fibers for E7

Paving Springer fibers for E7

Twisting operators and centralisers of Lie type groups over local rings

We extend the classical result asserting that the twisting operator preserves certain Deligne–Lusztig character values for truncated formal power series; along the way we discuss some properties of centralisers. Then, combining with the orbit construction, we apply it to analyse SL2 over finite dual numbers. In the appendix we discuss an action of GLn(Fq[[π]]/πr) on a Springer fibre intersected by Deligne–Lusztig varieties.

Irreducible components of two-row Springer fibers for all classical types

We give an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. We use these diagrams to explicitly write down relations between the vector spaces of the flags contained in a given irreducible component. This generalizes results by Stroppel–Webster and Fung for type A to all classical types.

Generalized Affine Springer Theory and Hilbert Schemes on Planar Curves

AbstractWe show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky–Kottwitz–MacPherson. Using a generalization of affine Springer theory for Braverman–Finkelberg–Nakajima’s Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn–Kamnitzer–Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp–Mellit, we also show the rational Cherednik algebra acts on the HOMFLY-PT homologies of torus knots. This work was inspired in part by a construction in 3D ${\mathcal {N}}=4$ gauge theory.

On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

Let W be the Weyl group of type BCn. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of W which is of type BCn− 1. Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.

Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic p, and with coefficients in a field of odd characteristic ℓ≠p): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.

TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

AbstractWe explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

A study of irreducible components of Springer fibers using quiver varieties

It is a remarkable theorem by Maffei–Nakajima that the Slodowy variety, which is a subvariety of the resolution of the nilpotent cone, can be realized as a Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which the variables are linear maps. In this paper, we construct solutions to this system under certain assumptions. This establishes an explicit and efficient way to compute the image of a complete flag contained in the Slodowy variety under the Maffei–Nakajima isomorphism and describe these flags in terms of quiver representations. As Slodowy varieties contain Springer fibers naturally, we can use these results to provide an explicit description of the irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations.

Hessenberg varieties associated to ad-nilpotent ideals

We consider Hessenberg varieties in the flag variety of with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties of this kind are connected.

Differential operators on quantized flag manifolds at roots of unity III

We describe the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. It follows from this and our previous results that for the De Concini-Kac type quantized enveloping algebra, where the parameter q is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This gives a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.