STABLE ENVELOPES FOR SLICES OF THE AFFINE GRASSMANNIAN IVAN DANILENKO

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STABLE ENVELOPES FOR SLICES

OF THE AFFINE GRASSMANNIAN

IVAN DANILENKO

Abstract. The aﬃne Grassmannian associated to a reductive group Gis

an aﬃne analogue of the usual ﬂag varieties. It is a rich source of Poisson va-

rieties and their symplectic resolutions. These spaces are examples of conical

symplectic resolutions dual to the Nakajima quiver varieties. We study the

cohomological stable envelopes of D. Maulik and A. Okounkov[MO] in this

family. We construct an explicit recursive relation for the stable envelopes in

the G=PSL2case and compute the ﬁrst-order correction in the general

case. This allows us to write an exact formula for multiplication by a divisor.

1. Introduction

1.1. Overview. In their inﬂuential paper [MO] D. Maulik and A. Okounkov stud-

ied equivariant cohomology H∗

T(X∨) of the Nakajima quiver varieties X∨[N2, N,

N3]. They showed that H∗

T(X∨) are modules over a quantum group called Yan-

gian. The key tools in their constructions were the R-matrices, which essentially

came from a family of special bases, called stable envelopes. Their construction

of stable envelopes works quite generally for a symplectic resolution with a torus

action [see assumptions in Chapter 3 in [MO] or in Section 3].

From the perspective of gauge theories, Nakajima quiver varieties are Higgs

branches of moduli spaces of vacua in a 3d supersymmetric ﬁeld theory. The 3d

mirror symmetry is known to exchange the Coulomb (X) and the Higgs branch

(X∨) of vacua [IS, SW]. To understand the relations between the enumerative

geometry of Xand X∨, we study the Coulomb side directly.

The key object to get Coulomb branches is the moduli space called the aﬃne

Grassmannian Gr. It is associated with a complex connected reductive group G,

and one can think of it as an analogue of ﬂag varieties for the corresponding Kac-

Moody group. It is known to have deep connections with representation theory and

Langlands duality [G,MV,MV2]. A big family of Coulomb branches [BFN,BFN2]

is given by the transversal slices in aﬃne Grassmannians. To give an idea of what

these slices are, recall that the aﬃne Grassmannian Gr has a cell structure similar

to Schubert cells in the ordinary ﬂag variety. A transversal slice Grλ

µdescribes how

one orbit is attached to another. Similarly to Nakajima quiver varieties, they are

naturally algebraic Poisson varieties and, in some cases, admit a smooth symplectic

2020 Mathematics Subject Classiﬁcation. 55N25, 14J42, 14D21.

arXiv:2210.09967v2 [math.AG] 27 Jul 2024

2 IVAN DANILENKO

resolution Grλ

µ→Grλ

µ, which is a notion of independent mathematical interest

[K,K2,BK]. The goal of this work is to study the equivariant cohomology of these

resolutions.

We study the stable basis Stabϵ

C(p) introduced in [MO]. It is also given by

the images of 1p∈H∗

T(p) under a map

Stabϵ

C: H∗

TXA→H∗

T(X).

Note that these are going in the ”wrong” way, compared to the natural restriction

maps. We are working in ordinary equivariant cohomology, see our convetions is

Section 3.

The main choice to make is the choice of attracting directions in a torus,

which is here denoted by Cas a Weyl chamber. Informally, the classes Stabϵ

C(p) are

”corrected” versions of the (Poincar´e dual) fundamental classes of the attracting

varieties to p. The notion of attracting variety clearly depends on the choice of C.

This basis is a rich source of enumero-geometric data; see [MO]. Even for

purely computational convenience, they are useful because they are non-localized

classes of low degree, and this allows one to use the degree bounds eﬀectively.

However, this comes with a price. As opposed to the ﬁxed-point basis, the mul-

tiplication by a divisor is no longer diagonal. Fortunately, it is not that far from

it.

For a T-equivariant line bundle Twe have

1(L)∪Stabϵ

C(p) = ι∗

pcT

1(L)·Stabϵ

C(p) + ℏX

q<Cp

p,q Stabϵ

C(q)

for some cL

p,q ∈Q. Furthermore, these numbers cL

p,q can be easily recovered if one

knows ι∗

qStabϵ

C(p) modulo ℏ2.

We ﬁnd the restrictions ι∗

qStabϵ

C(p) in two steps.

•Use the wall-crossing behavior of ι∗

qStabϵ

C(p) modulo ℏ2to reduce the

computation to a similar computation on a wall in Lie A, which turns

out to be a slice for G=PSL2, the case of A1type. This is done in

Theorem 3.22

•Find the restrictions for A1-case via the action of Steinberg correspon-

dences. This computation ends with Theorem 3.21 and has a ﬂavor similar

to that of C. Su’s thesis [S].

For special line bundles Eispanning Pic(X)⊗ZQwe get the following main

formula

1(Ei)∪Stabϵ

C(p) = Stabϵ

C

Hi(p) + ℏX

j<i

Ωji

−C,ϵ (p)−ℏX

i<j

Ωij

−C,ϵ (p)



in Theorem 3.29.

Since multiplications by cT

1(Ei) act with a simple spectrum, this formula

uniquely determines the stable envelopes.

STABLE ENVELOPES FOR SLICES OF THE AFFINE GRASSMANNIAN 3

1.2. Structure. The paper is organized as follows. In Section 2we recall the main

facts about the slices in the aﬃne Grassmannian, and in Section 3we prove the

main results about the stable envelopes. For the reader’s convenience, we keep the

proofs of the technical facts until the Appendix A.

Acknowledgments. The author is thankful to Mina Aganagic, Joel Kamnitzer,

Henry Liu, Davesh Maulik, Andrei Negut, Andrei Okounkov, Andrey Smirnov,

Changjian Su, and Shuai Wang for discussions related to this paper.

2. Slices of the affine Grassmannian

In this section, we recall some facts about slices of the aﬃne Grassmannians.

2.1. Representation-theoretic notation. Let us present the notations we use

for common representation-theoretic objects. One major diﬀerence from standard

notation is that we use non-checked notation for coobjects (coweights, coroots, etc.)

and checked for ordinary ones (weighs, roots, etc.). This is common in the literature

on the aﬃne Grassmannian since it simpliﬁes notation. And the unexpected side

eﬀect is that weights in equivariant cohomology will have checks. We hope this

will not cause confusion.

•Gis a connected simple complex group unless stated otherwise,

•A⊂Gis a maximal torus,

•B⊃Ais a Borel subgroup,

•gis the Lie algebra of G

•X∗(A) = Hom(C×,A) is the cocharacter lattice, it’s equal to the coweight

lattice if Gis of adjoint type,

•X∗(A)+⊂X∗(A) the submonoid of dominant cocharacters.

•W=N(A)/Ais the Weyl group of G,

•g=h⊕L

α∨

gα∨is the root decomposition of g.

•ρ∨is the half-sum of positive roots.

•K(•,•) Weyl-invariant scalar product on cocharacter space X∗(A)⊗ZR,

normalized in such a way that the length squared of the shortest coroot is

2 (equivalently, the length squared of the longest root is 2).

We identify all weights with Lie algebra weights (in particular, we use additive

notation for the weight of a tensor product of weight subspaces).

2.2. Classical constructions.

2.2.1. Aﬃne Grassmannian. Let O=C[[t]] and K=C((t)). We will refer to

D= Spec Oand D×= Spec Kas the formal disk and the formal punctured disk,

respectively. The main references for this section are [BD, MV, KWWY].

Let Gbe a connected reductive algebraic group over C. The aﬃne Grass-

mannian GrGis the moduli space of

4 IVAN DANILENKO

(1) 



(P, ϕ)

Pis a G-principle bundle over D,

ϕ:P0|D×∼

−→ P|D×is a trivialization of P

over the punctured disk D×





where P0is the trivial principal G-bundle over D. It is representable by an ind-

scheme (see [BD] 4.5.1).

One can give a less geometric deﬁnition of the C-points of the aﬃne Grass-

mannian

GrG(C) = G(K)/G(O)

where we use notation G(R) for R-points of the scheme Ggiven a C-algebra R.

We will use this point of view when we talk about the points in GrG.

The see that these are exactly the C-points of GrG(i.e. the pairs (P, ϕ)

as in (1)), note that over Dany G-principle bundle is trivializable, so take one

trivialization ψ:P0

∼

−→ P. This gives a composition of isomorphisms

(2) P0|D×

ψ|D×

−−−→ P|D×

φ−1

−−→ P0|D×

which is a section of P0|D×, i.e. an element of G(K). The change of trivialization

ψpre-composes with a section of P0, in other words, pre-composes with an element

of G(O). This gives a quotient by G(O) from the left.

We restrict ourselves to the case of connected simple (possibly not simply-

connected) Gin what follows without losing much generality. Restating the results

to allow for an arbitrary connected complex reductive group is straightforward.

From now on, we ﬁx a connected simple complex group G. Since it cannot

cause confusion, we later omit Gin the notation GrGand just write Gr.

The group G(K)⋊ C×naturally acts on Gr. In the coset formulation, the

G(K)-action is given by left multiplication and C×-acts by scaling variable tin K

with weight 1. In the moduli space description G(K) acts by changes of section

g·(P, ϕ) = P, ϕg−1(these two actions are the same action if one takes into

account identiﬁcation (2)), C×scales Dsuch that the coordinate thas weight 1.

We will later denote this C×as C×

ℏwhen we want to emphasize that we use this

algebraic group and its natural action.

We will need actions by subgroups of this group, namely A⊂G⊂G(O)⊂

G(K). It will also be useful to consider extended torus T=A×C×

ℏ, where the

C×

ℏ-part comes from the second term in the product G(K)⋊ C×

ℏ.

The canonical projection T→C×

ℏgives a character of Twhich we call ℏ

(this explains the subscript ℏin the notation C×

ℏ). Then the weight of coordinate

ton Dis ℏby the construction of the C×

ℏ-action.

Remark. We call the maximal torus of Gby Ato have notation similar to [MO],

i.e. Tis the maximal torus acting on the variety and Ais the subtorus of T

preserving the symplectic form.

STABLE ENVELOPES FOR SLICES OF THE AFFINE GRASSMANNIAN 5

The partial ﬂag variety G/P(where Pis parabolic) has a well-known decom-

position by orbits of the B-action called Schubert cells. The aﬃne Grassmannian

has a similar feature.

One can explicitly construct the ﬁxed points of the A-action. Given a cochar-

acter λ:C×→A, one can construct a map using natural inclusions

D×→C×λ

−→ A→G

i.e. an element of G(K) which we denote by tλ. Projecting it naturally to Gr we

get an element [λ]∈Gr.

Using these elements we deﬁne

Grλ=G(O)·[λ]

as their orbits.

Let us present the main properties of these points and subspaces.

Proposition 2.1.

(1) GrA=F

λ∈X∗(A)

{[λ]}.

(2) Moreover, GrT=GrA.

(3) For any w∈Wone has Grwλ =Grλ.

(4) Gr =F

λ∈X∗(A)+

Grλ.

(5) Grλ=F

µ≤λ

µ∈X∗(A)+

Grµ.

(6) As a G-variety, Grλis isomorphic to a the total space of a G-equivariant

vector bundle over the G-orbit G·[λ].

(7) As a G-variety, G·[λ]is a partial ﬂag variety G/P−

λwith parabolic P−

whose Lie algebra contains all roots α∨such that ⟨α∨, λ⟩ ≤ 0.

(8) D-scaling C×

ℏin Tacts on Grλby scaling the ﬁbers of the vector bundle

(6) with no zero weights.

(9) Grλis the smooth part of Grλ. In particular, Grλis smooth iﬀ λis a

minuscule coweight or zero.

Proof. The main reference for most of the statements here is [BD]. Part (4) is 4.5.8

(see also 3.1.7 in [CG] for a ﬁnite analog). Part (5) is 4.5.12. Part (6) is Lemma

9.1.5(iii). Part (7) is 9.1.3. Part (8) is Remark 9.1.6. Part (9) follows from parts

(6) and (5).

For part (1) let us ﬁrst show that for any A-cocharacter λthe point [λ] is

A-ﬁxed. Ais commutative and A⊂G(O), so for any a∈Awe have

a[λ] = atλG(O) = tλaG(O) = tλG(O)=[λ].

This gives one inclusion in (1). To prove the other inclusion, let p∈GrA. By (4)

there is a dominant coweight λ, such that p∈Grλ. Since pis A-ﬁxed, it must be

over an A-ﬁxed point in G/P−

λ. The A-ﬁxed points in G·[λ] = G/P−

λare of the

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STABLEENVELOPESFORSLICESOFTHEAFFINEGRASSMANNIANIVANDANILENKOAbstract.TheaffineGrassmannianassociatedtoareductivegroupGisanaffineanalogueoftheusualflagvarieties.ItisarichsourceofPoissonva-rietiesandtheirsymplecticresolutions.ThesespacesareexamplesofconicalsymplecticresolutionsdualtotheNakajimaquiverv...

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