Affine Springer fiber – sheaf correspondence
Eugene Gorsky
University of California, Davis
joint with Oscar Kivinen and Alexei Oblomkov

Gauge Theory, Moduli Spaces and Representation Theory
Kyoto 2023

1 / 20Affine Springer fibers

Let G be a connected reductive group with Lie algebra g. Our main
character is the affine Springer fiber associated to an element γ ∈ g((t)):
Spγ = {g ∈ G ((t)) ∶ g −1 γg ∈ G [[t]]} /G [[t]]which is a subvariety of affine Grassmannian GrG = G ((t))/G [[t]].
If γ is compact, regular and semisimple then Spγ is finite-dimensional, but
usually singular and can have infinitely many irreducible components.

2 / 20Affine Springer fibers in type A

In type A, the geometry of affine Springer fiber Spγ is controlled by thecharacteristic polynomial f (t, λ) = det(λI − γ(t)) and the spectral curve
Cγ = {f (t, λ) = 0} ⊂ C2t,λ .
Spγ is closely related to the compactified Jacobian of Cγ and the Hilbert
scheme of points on Cγ .
The spectral curve Cγ defines an n-strand braid β which describe thebehavior of eigenvalues of γ(t) as t goes around the origin. The
remarkable conjectures of Oblomkov, Rasmussen and Shende relate thehomology of Spγ to triply graded Khovanov-Rozansky homology of β.

3 / 20

Examples
Example
0 t2
The matrix γ = () has characteristic polynomial λ2 − t 3 andt 0
corresponds to a cuspidal curve in C2 . The corresponding two-strand braidis β = σ 3 which closes to the trefoil knot. The affine Springer fiber Spγ isisomorphic to CP1 .

Examplet 0The matrix γ = () has characteristic polynomial λ2 − t 2 and0 −tcorresponds to a pair of lines in C2 . The corresponding two-strand braid isβ = σ 2 which closes to the Hopf link. The affine Springer fiber is theinfinite chain of CP1 .
4 / 20Homology of Spγ
The centralizer of γ naturally acts on Spγ . If γ is quasihomogeneous, thenSpγ admits as additional C∗ action.
The equivariant Borel-Moore homology of Spγ was studied by
Bezrukavnikov, Hikita, Lusztig, Varagnolo, Vasserot, Yun and others. In
particular:H∗ (Spγ ) is a finitely generated module over C[T ∗ T ∨ ]W where T ∨ isthe Langlands dual torus. For G = GLn , we get a module overC[x1± , . . . , xn± , y1 , . . . , yn ]Sn .
Hence, it defines a coherent sheaf on T ∗ T ∨ /W which for G = GLn issimply (C∗ × C)n /Sn . It is supported on a certain Lagrangiansubvariety.
∗If γ is quasihomogeneous, then H∗C (Spγ ) is a representation of thespherical trigonometric Cherednik algebra associated to G .

5 / 20Infinite family
We will be interested in a family of affine Springer fibers
Spγ , Sptγ , Spt 2 γ . . .
In type A, going from γ to tγ corresponds to a blowdown of the spectral
curve. For example, if the characteristic polynomial of γ is fγ = λn − t mthen the characteristic polynomial of t k γ isft k γ = λn − t m+kn .
Topologically, going from γ to tγ adds a full twist to the corresponding
braid, so we are interested in a family of braids
β, FT β, FT2 β . . .

6 / 20Main theorem
Theorem (G.,Kivinen,Oblomkov)
There is a graded algebra AG = A0 ⊕ A1 ⊕ A2 ⊕ . . . (depending only on G )with the following properties:
For any γ the direct sum of homologies Mγ = ⊕k H∗ (Spt k γ ) is a
graded module over AG .
A0 ≃ C[T ∗ T ∨ ]W .

A1 ≃ C[T ∗ T ∨ ]ϵ where ϵ denotes the sign representation of W .
For all d ≥ 0, we have
Ad ≃ ed ⋂(1 − α∨ , yα )d ⊂ C[T ∗ T ∨ ]αwhere α runs over the roots of g and ed is the projector to theϵd -isotypic component.
7 / 20Main theorem

As a corollary, γ defines a quasicoherent sheaf Fγ on the variety
XG = Proj AG . Changing γ to tγ corresponds to twisting Fγ with O(1).
The variety XG is normal and admits a natural projection to T ∗ T ∨ /W , so
one can think of it as a partial resolution of the commuting variety for G ∨ .
However, for general G we expect that XG is not smooth and AG is not
generated by A0 and A1 . This is not a problem for G = GLn :Theorem (BFN; G.,Kivinen,Oblomkov)
For G = GLn , the algebra AG is generated in degrees 0 and 1, and
XGLn = Hilb(C∗ × C).
To sum up, to any γ ∈ gln ((t)) we associate a sheaf Fγ on Hilb(C∗ × C).

8 / 20

Examples: GLn
ExampleIf γ is a diagonal scalar matrix with distinct eigenvalues, then Fγ is theProcesi bundle P on Hilb(C∗ × C) restricted to a certain Lagrangiansubvariety L.

ExampleMore generally, if γ = diag(s1 t k , . . . , sn t k ) with si distinct and nonzero
then Fγ = P ⊗ O(k) restricted to L.

ExampleSuppose that det(λI − γ) = λn − t kn+1 . Then Fγ is isomorphic to therestriction of O(k) to the punctual Hilbert scheme at (1, 0) ∈ C∗ × C.

9 / 20

Examples: beyond GLn
Let γ be an equivalued element of valuation k ∈ Z≥0 , and G arbitrary.
Then the results of Kivinen imply thatMγ = Jγ /yJγ

where Jγ = ⊕ ⋂ (1 − α∨ , yα )jas a module overα

AG = ⊕ ed ⋂ (1 − α∨ , yα )jk+jdαThe module structure is clear, and therefore Mγ yields a sheaf Fγ on
XG = Proj AG which generalizes P ⊗ O(k) to arbitrary type.
It would be very interesting to understand the variety XG and thisProcesi-like sheaf on it geometrically, and relate it to the work of Losev.

10 / 20

Main theorem
The above theorems admit a quantization:

Theorem (G.,Kivinen,Oblomkov)
̵

̵

̵

̵

̵

There is a Z-algebra AhG = ⊕ i Ahj with multiplication i Ahj ⊗ j Ahk → i Ahk
such that:̵∗

For any quasihomogeneous γ the direct sum Mγh = ⊕k H∗C (Spt k γ ) is̵
a graded module over AhGh̵i Ai is a spherical trigonometric Cherednik algebra for G withparameter depending on ih̵h̵h̵
i Ai+1 is the shift bimodule between i Ai and i+1 Ai+1
̵At h̵ = 0, we get i Ah=0
≃ Aj−i and the Z-algebra degenerates into gradedjalgebra.
Similar Z-algebras built from shift bimodules for rational Cherednik
algebras were studied by Gordon and Stafford. We define the shift
bimodule in the trigonometric case (since studied by Wille Liu).
11 / 20

Action of antisymmetric polynomials

Since A1 is isomorphic to the space of W -antisymmetric polynomials, anyantisymmetric polynomial defines an operator from H∗ (Spγ ) to H∗ (Sptγ ).
Such an operator can be constructed explicitly as follows:̃ in affine flag variety FlG
Consider the affine Springer fiber Spγ̃ ) has an action of C[T ∗ T ∨ ] ⋊ W
The homology H∗ (Spγ̃ )]W = H∗ (Sp ), [H∗ (Sp̃ )]ϵ = H∗ (Sp )[N]
One has [H∗ (Spγγγtγ
An antisymmeric polynomial defines an operator̃ )W → H∗ (Sp̃ )ϵ .
H∗ (SpγγHowever, constructing Ak for k > 1 and verifying relations for A1 ⋅ A1 ⊂ A2
seems to be out of reach with this approach.

12 / 20

Coulomb branch algebras
̵Instead, we construct both A and Ah as graded Coulomb branch
algebras following Braverman, Finkelberg and Nakajima (BFN).
Consider the space
G [[t]] it g[[t]]i Rj = {[g , s] ∈ G ((t)) ×̵∶ gs ∈ t j g[[t]]} ,G [[t]]⋊C×then we define i Ahj = H∗(i Rj ). The multiplicationh̵h̵h̵
i Aj ⊗ j Ak → i Ak is defined by the BFN convolution product.

Theorem (Braverman, Finkelberg, Nakajima; Webster)̵The Z-algebra Ah is associative, and commutative for h̵ = 0. Furthermore,∗̵
for all i and j the component i Ahj is a free module over H∗G ×C (pt).

Theorem (Kodera, Nakajima)
̵For all i the algebra i Ahi is isomorphic to the spherical trigonometric
Cherednik algebra with parameter depending on i.
13 / 20

BFN Springer theory

̵The algebra A acts on Mγ = ⊕k H∗ (Spt k γ ) (or Ah acts in C∗ -equivarianthomology in quasihomogeneous case) by virtue of the BFN Springer
theory developed by Hilburn-Kamnitzer-Weekes and Garner-Kivinen.
The action is defined using a certain convolution between i Rj and Spt i γ ,which is compatible with BFN convolution on i Rj . Thus, themultiplication in the algebra is compatible with the action on the module.
In other words, the action of A on Mγ is similar in spirit to the BFNSpringer theory, but generalizes it to the Z-algebra level. One still needs,
though, to identify the algebra A explicitly.

14 / 20

Localization
The components i Aj can be embedded into difference operators on
C[treg ] using localization techniques. Furthermore, the homology of i Rj
has a basis [i R≤λj ] corresponding to Schubert cells in the affine
Grassmannian. Thus, we get an explicit basis of i Aj .

Theorem (G.,Kivinen, Oblomkov)
For arbitrary G the localization maps [i R≤λj ] to the difference operatori−α(λ′ )−j−1∏α(λ′ )+i<j ∏ℓ=0∑λ′ ∈W λ(yα + (α(λ′ ) + j + ℓ)h̵ + c)

max(0,α(λ′ ))−1∏α∈Φ ∏ℓ=0̵(yα + ℓh)uλ′ + . . .

′̵ ′ and . . . are lower order terms with
where u λ is the translation by hλrespect to some filtration.

15 / 20

Localization for GLn
For G = GLn and h̵ = 0 the localization formula simplifies to⎛⎞[i Rλj ] ↦ ±Sym ∆j−i
(yr − ys )∣j−i∣−∣λr −λs ∣ u λ∏⎠⎝r <s,∣λr −λs ∣<∣j−i∣where ∆ is the Vandermonde determinant in the yi .

ExampleAgain for G = GLn , assume that j = i + 1, then at h̵ = 0 we get[i Rλi+1 ] ↦ ±∆Alt⎛λ⎞∏ (yr − ys )u⎝r <s,λr =λs⎠

Up to ±∆, this is antisymmetrization of a certain monomial in ys and us ,which agrees with the description of A1 as the space of antisymmetricpolynomials.
16 / 20

Generation for GLn
We have an easy combinatorial lemma:

Lemma
Suppose that λ is an arbitrary integral coweight for GLn and d > 0. Then
there exist d coweights µ(0) , . . . , µ(d−1) such that µ(0) + . . . + µ(d−1) = λand for all i and j the following holds:1) If ∣λi − λj ∣ < d thend − ∣λi − λj ∣ =(k)2) If ∣λi − λj ∣ > d then µi(k)

≠ µj

∑

1.

(k)(k)
k,µi =µjfor all k.

By using the above formulas for the basis in i Aj , one can use this lemma̵
to prove that the graded algebra A (resp. Z-algebra Ah ) is generated indegrees 0 and 1 for G = GLn .
17 / 20

Further directions

Conjecture
The module Mγ = ⊕k H∗ (Spt k γ ) is finitely generated over AG and thesheaf Fγ on XG is coherent.
The conjecture is known in several cases but open in general.

Problem
BFN define Coulomb branch algebras AG ,N for arbitrary representations Nof the group G . One can also define generalized affine Springer fiber forγ ∈ N((t)). What could one say about graded algebras and modules inthis generality?

18 / 20

Further directions
The work of G.-Negut, -Rasmussen, G.-Hogancamp andOblomkov-Rozansky relates Khovanov-Rozansky homology to sheaves on
Hilbn (C2 ). In particular, for any braids α, β there is a naturalmultiplication
HHH(α) ⊗ HHH(β) → HHH(αβ).
Given a braid β, one can form a graded module ⊕k HHH(β FTk ) over thekk∞
graded algebra ⊕∞k=0 HHH(FT ), and a sheaf on Proj ⊕k=0 HHH(FT ).

Theorem (G.,Hogancamp)kThe variety Proj ⊕∞k=0 HHH(FT ) is isomorphic to the isospectral Hilbert
2scheme Xn (C ) of n points on C2 .

Problem

Given γ ∈ gln ((t)), we can construct a sheaf Fγ on Hilbn (C∗ × C) using
the methods in this talk, and another sheaf on Xn (C2 ) using link
homology. How are they related?
19 / 20

Happy birthday, Professor Nakajima!

20 / 20