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THE PARTIAL COMPACTIFICATION OF THE UNIVERSAL CENTRALIZER

ANA BALIBANU

Abstract. The universal centralizer of a semisimple algebraic group G is the family of centralizers

of regular elements in Lie(G), parametrized by their conjugacy classes. It has a natural symplec-

tic structure inherited via Hamiltonian reduction from the cotangent bundle T∗

G. We construct

a smooth log-symplectic partial compactification of the universal centralizer where each fiber is

compactified inside the wondeful compactification of G.

Contents

1. Introduction 1

2. The universal centralizer 2

3. The partial compactification of Z 4

4. The universal family of Hessenberg varieties 11

5. Relation to Coulomb branches 18

References 20

1. Introduction

The universal centralizer Z of a semisimple algebraic group G of adjoint type is the family

of centralizers of regular elements in g = Lie G, indexed by their conjugacy classes. It can be

obtained by a two-sided Whittaker Hamiltonian reduction from the cotangent bundle T ∗G of G,

and this equips it with a natural symplectic structure. This variety has appeared as an important

technical tool in the geometric Langlands program, for instance in [BFM], [BF], and [Ngo1, Ngo2].

The wonderful compactification of G is a distinguished equivariant embeddingG whose boundary

is a divisor with normal crossings that encodes the asymptotic behavior of the group “at infinity.”

It was first introduced by DeConcini and Procesi [DP] in the context of symmetric spaces, and since

then its study has been generalized to the setting of spherical varieties, for instance in [BP], [Bri1],

and [Kno2]. The logarithmic cotangent bundle of G has a concrete description as a subbundle of

G × g × g, and it is equipped with a canonical log-symplectic structure. By taking a Whittaker

reduction, we construct a log-symplectic smooth partial compactification Z of Z in which every

centralizer fiber is compactified inside G.

Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138

E-mail address: [email protected].

1

http://arxiv.org/abs/1710.06327v2

THE PARTIAL COMPACTIFICATION OF THE UNIVERSAL CENTRALIZER 2

These compactified centralizers have been studied in [Bal], where they are identified with certain

subvarieties of the flag variety known as Hessenberg varieties, which were first introduced by De

Mari, Procesi, and Shayman [MPS]. Hessenberg varieties have many applications in combinatorics

and representation theory, notably in the recent proof of the Shareshian-Wachs conjecture [BC], in

the description of the quantum cohomology of flag varieties [Kos4, Rie1, Rie2], and in the study

of affine Springer fibers [GKM]. The partial compactification Z is an example of a more general

type of universal flat family of regular Hessenberg varieties, parametrized by the regular conjugacy

classes of g and naturally equipped with a contracting C∗-action. We use this connection to compute

the cohomology of Z, and also to produce a log-symplectic partial compactification of the twisted

cotangent bundle of the base affine space.

In [BFM] it is shown that the coordinate ring C[Z] of the universal centralizer is isomorphic to

the equivariant homology ring of the affine Grassmannian of the Langlands dual group G. This is an

example of a Coulomb branch in the sense of Nakajima [Nak]. The partial compactification Z can

be obtained directly from this equivariant homology ring through a Rees construction, analogous

to the way the wonderful compactification G is obtained from the coordinate ring C[G] via the

Vinberg monoid.

In Section 2 we review the classic construction, due to Kostant, of the universal centralizer Z as

a Whittaker reduction of the cotangent bundle T ∗G. In Section 3 we recall the logarithmic cotangent

bundle T ∗G,D

of G. We obtain the partial compactification Z as a Whittaker reduction of T ∗G,D

, we

show that its induced Poisson structure is log-symplectic, and we describe its symplectic leaves.

In Section 4 we construct universal families of regular Hessenberg varieties parametrized by the

regular conjugacy classes in g and equipped with a contracting C∗-action. We identify the partial

compactification Z with one such family, and we use this identification to compute its singular

cohomology. Lastly, in Section 5, we explain how the partial compactification Z is related to the

realization of the universal centralizer as a Coulomb branch.

Acknowledgements: The author would like to thank Victor Ginzburg, Sam Evens, Ioan

Marcut, Sergei Sagatov, and Travis Schedler for many interesting discussions. Part of this work

was completed while the author was supported by a National Science Foundation MSPRF under

award DMS–1902921.

2. The universal centralizer

Let G be a connected complex semisimple algebraic group of adjoint type, and let g = Lie(G)

be its Lie algebra. Let

e, h, f ⊂ g

be a principal sl2-triple—a triple of regular elements that generates a copy of sl2 inside g. Write

ge = Lie(Ge) for the annihilator of e in g, and consider the principal slice

S = f + ge.


It intersects each regular orbit of G on g exactly once and trasversally [Kos1], giving a section of

the adjoint quotient g −→ g // G.

Definition 2.1. The universal centralizer of G is the variety

Z = (g, x) ∈ G× S | g ∈ Gx .

It is the family of centralizers of regular elements of g, parametrized by representatives of their

conjugacy classes.

An outline of the following construction, which is due to Kostant [Kos3], can also be found in

[Tel] and in [Gin]. The action of G on itself by left- and right-multiplication induces a G × G-

action on T ∗G. Trivializing the cotagent bundle T ∗

G∼= G× g∗ with respect to left multiplication, the

G×G-action becomes

(a, b) · (g, x∗) = (agb−1, b · x∗) for a, b ∈ G, (g, x∗) ∈ G× g∗.

The bundle T ∗G has a canonical symplectic structure, and the action of G × G is Hamiltonian.

Identifying g ∼= g∗ via the Killing form, the corresponding moment map is

µ : T ∗G∼= G× g −→ g× g(2.2)

(g, x) 7−→ (g · x, x).

The image of µ is the set of conjugate pairs inside g× g, and the fiber above the point (g · x, x) is

µ−1(g · x, x) ∼= g ·Gx,

where Gx is the centralizer of x in G.

Let B ⊂ G be the unique Borel subgroup whose Lie algebra b contains e, let N ⊂ B be

its unipotent radical, and denote by n the Lie algebra of N . The maximal unipotent subgroup

N × N ⊂ G × G acts on T ∗G. The killing form gives an identification n∗ ∼= g/b, and the moment

map

µN×N : T ∗G −→ n∗ × n∗

factors through the moment map for the G×G-action:

(2.3)

T ∗G∼= G× g g× g

g/b× g/b.

µ

µN×N

The coset (f, f) ∈ g/b × g/b, which corresponds to a regular character in n∗ × n∗, is fixed by the

action of N ×N .

We will repeatedly use the following essential fact, due to Kostant:

Proposition 2.4 ([Kos2], Theorem 1.2). The action map N×(f+ge) −→ f+b is an isomorphism

of affine spaces.


The fiber of the moment map µN×N above the coset (f, f) is

µ−1N×N (f, f) = (g, x) ∈ G× g | x ∈ f + b, g · x ∈ f + b

By Proposition 2.4, N × N acts freely on this fiber, which implies that (f, f) ∈ g/b × g/b is a

regular value of µN×N and that µ−1N×N (f, f) is smooth.

Again by Proposition 2.4, there is a natural isomorphism

N ×N ×Z −→ µ−1N×N (f, f)

(n1, n2, (g, x)) 7−→ (n1gn−12 , n2 · x).

It follows that the universal centralizer

Z ∼= µ−1N×N (f, f)/(N ×N)

is a smooth variety with a natural symplectic structure.

3. The partial compactification of Z

3.1. The logarithmic cotangent bundle of G. Let G be the wonderful compactification of G,

and let D = G\G be its boundary divisor. A detailed survey of its structure can be found in [EJ],

and here we collect only the facts that will be used in what follows.

The wonderful compactification G is a smooth variety with a natural G×G-action that extends

the left- and right-multiplication inside the group G. It contains a distinguished open affine space

X ∼= CdimG called the big cell. There is an open cover of G consisting of finitely many G × G-

translates of X.

Let l = rk(g) be the rank of g. The boundary D of G is a normal crossing divisor with exactly l

irreducible components

D = D1 ∪ . . . ∪Dl,

and it is a union of finitely manyG×G-orbits OI . These orbits are indexed by subsets I ⊂ 1, . . . , l,

in the sense that the closure of the orbit OI is the corresponding intersection of irreducible divisor

components:

OI =⋂

i∈I

Di.

Let T = Gh be a fixed maximal torus in the Borel B chosen above, and let

∆ = α1, . . . , αl

be the set of simple roots. Let PI be the standard parabolic subgroup generated by the simple

roots αi | i 6∈ I, and let P−I be the opposite parabolic with respect to T . Let UI and U−

I be their

respective unipotent radicals, LI = PI ∩ P−I their shared Levi component, and Z(LI) its center.

The orbit OI is a fibration over the product of partial flag varieties G/PI × G/P−I with fiber

isomorphic to LI/Z(LI). In each orbit OI there is a distinguished basepoint

zI ∈ T ∩X


that lies in the intersection of the big cell with the closure of the torus. The stabilizer of this

basepoint in G×G is

StabG×G(zI) =(us, vt) ∈ PI × P−

I | u ∈ UI , v ∈ U−I , s, t ∈ LI , st

−1 ∈ Z(LI).

We will work with the logarithmic cotangent bundle T ∗G,D

of G. This is the vector bundle

associated to the locally-free sheaf of logarithmic differential forms on G—differential forms with

at most logarithmic poles along the boundary divisor D. The pullback of T ∗G,D

to G ⊂ G is the

usual cotangent bundle T ∗G.

The logarithmic cotangent bundle T ∗G,D

can be viewed as a G×G-stable subbundle of the trivial

bundle

G× (g× g),

called the bundle of isotropy Lie subalgebras. (See [Bri3], Proposition 2.1.2 and [Bri2], Example

2.5.) Its fibers are identified with subalgebras of g× g as follows.

Fix a point a ∈ G and let O = (G × G) · a be its G × G-orbit. The stabilizer StabG×G(a) acts

on the normal space

NO,a = TG,a/TO,a.

Denote by g× g(a) the Lie algebra of the kernel of this action, known as the isotropy Lie subalgebra

at a. The fiber of the logarithmic cotangent bundle at a is

T ∗G,D,a

∼= g× g(a).

Remark 3.1. In particular,

g× g(e) = Lie(StabG×G(e)) = g∆,

where g∆ is the diagonal copy of g inside g× g. For any g ∈ G,

g× g(g) = Lie(StabG×G(g)) = g · g∆.

There is a realization of G inside the Grassmannian Gr(dim g, g×g) obtained by taking the closure

of the image of the embedding

ι : G −→ Gr(dim g, g× g)

g 7−→ g · g∆.

(See [EJ], Section 3.2.) The bundle of isotropy Lie subalgebras, which we have identified with T ∗G,D

,

is precisely the restriction of the tautological bundle on Gr(dim g, g× g) to ι(G).

Let pI , p−I , uI , u

−I , and lI denote the Lie algebras of PI , P

−I , UI , U

−I , and LI respectively. Recall

that zI ∈ T ∩X is the basepoint of the G×G-orbit OI .

Proposition 3.2. The fiber g× g(zI) of T∗G,D

at zI is

pI ×lI p−I ,

the subalgebra of pairs of elements in pI × p−I that have the same component in lI .


Proof. The stabilizer StabG×G(zI) acts on the normal space

NOI ,zI = TG,zI/TOI ,zI .

Since the orbit OI is open and dense in the intersection of the hypersurfaces Di | i ∈ I, the

normal space NOI ,zI decomposes naturally as a sum of one-dimensional subspaces

NOI ,zI =⊕

i∈I

ℓi,

where each ℓi corresponds to the normal direction to Di at zI .

The stabilizer

StabG×G(zI) = (us, vt) ∈ PI × P−I | u ∈ UI , v ∈ U−

I , s, t ∈ LI , st−1 ∈ Z(LI)

preserves each ℓi. The torus T acts on ℓi by −αi (see [EJ], Lemma 2.7), so the kernel of this action

is (us, vt) ∈ StabG×G(zI) | αi(st

−1) = 1 for all i ∈ I,

and its Lie algebra is(m+ x, n+ x) ∈ pI × p−I | m ∈ uI , n ∈ u−I , x ∈ lI

= pI ×lI p

−I .

Consider the G×G-equivariant projection

µ :T ∗G,D

−→ g× g(3.3)

(a, x1, x2) 7−→ (x1, x2).

This map is studied in [Bri4] and [Kno1], where it is called the compactified moment map because

its restriction to the cotangent bundle T ∗G ⊂ T ∗

G,Dcoincides with the moment map µ defined in

(2.2).

Proposition 3.4. The image of µ is the variety g×g//G g of pairs of elements in g× g that lie in

the closure of the same G-orbit.

Proof. The map µ is proper because it factors through the projection

T ∗G,D

G× g× g

g× g.

µ

Because it is proper, its image is closed, so it is the closure of the image of µ. As described in

Section 2, this is the set of conjugate pairs in g× g.

Proposition 3.5. The variety g×g//G g is normal.

Proof. Let f1, . . . , fl ∈ C[g]G be a minimal set of generators for the algebra of G-invariant polyno-

mials on g. The variety g×g//G g has dimension

2 dim(g)− dim(g // G) = 2dim(g)− l.


It is a complete intersection because it is the zero-locus in g × g of l algebraically-independent

polynomials:

g×g//G g = (x, y) ∈ g× g | fi(x)− fi(y) = 0 for all i = 1, . . . , l.

The subset

greg ×g//G greg = (x, y) ∈ greg × greg | x ∈ G · y ⊂ greg × greg

is a smooth open subset of g ×g//G g, because the differentials df1, . . . dfl are linearly independent

at every point of greg. (See, for instance, Claim 6.7.10 in [CG].) Its complement has codimension

at least two, because adjoint orbits are even-dimensional. It follows that g ×g//G g is a complete

intersection with no codimension-one singularities, so it is normal.

3.2. Whittaker reduction of T ∗G,D

. Suppose that X is a smooth connected algebraic variety

containing a normal crossing divisor Z = ∪Zi. A log-symplectic Poisson structure on (X,Z) is a

generically non-degenerate Poisson bivector whose top wedge power vanishes transversally along the

nonsingular locus of Z, and with minimal multiplicity along the nonsingular locus of each partial

intersection ∩i∈IZi.

Equivalently, a log-symplectic Poisson bivector induces an isomorphism

T ∗X,Z

∼−−→ TX,Z

between the logarithmic cotangent bundle and the logarithmic tangent bundle of (X,Z). Under

this isomorphism, the bivector corresponds to a closed non-degenerate logarithmic two-form, which

restricts to a symplectic form on the open dense symplectic leaf X\Z.

For completeness, we include the following log-symplectic analogue of the Marsden-Weinstein-

Meyer Theorem. Although it appears to be known to experts, we were unable to find a reference

in the existing literature. A similar statement is discussed in Section 7.7 of [GLPR].

Proposition 3.6. Let (X,Z) be a log-symplectic Poisson variety with log-symplectic form ω as

above. Suppose that H is a connected algebraic group with a Hamiltonian action on X, let

ρ : X −→ h∗

be the induced moment map, and let x ∈ h∗ be a point fixed by the coadjoint action of H. Suppose

that H acts on the fiber X = ρ−1(x) freely, with quotient variety X0 = X/H containing the divisor

Z0 = (X ∩ Z)/H. Then

(1) x is a regular value of ρ, X0 is smooth, and Z0 has normal crossings;

(2) the pullback of ω to X induces a log-symplectic structure ω0 on (X0, Z0) such that

(3.7) ι∗ω = q∗ω0,

where ι : X −→ X is the inclusion and q : X −→ X0 is the quotient map.


Proof. Let L ⊂ X be any symplectic leaf that intersects the fiber X = ρ−1(x)—sinceH is connected,

L is H-stable, and we can consider the restricted moment map

ρ|L : L −→ h∗.

The group G acts freely on ρ−1|L (x) and L is symplectic, so x is a regular value of ρ|L. Being a

regular value of ρ|L for every symplectic leaf L, x is also a regular value of ρ. This proves (1), and

implies that X is smooth.

Let S ⊂ Z be a stratum of the divisor Z. Since S is a union of symplectic leaves, x is a regular

value of ρ|S. It follows that X∩S is smooth, and that Z = X∩Z is a normal crossing divisor. Since

the quotients X0 and Z0 are are varieties, they are also smooth. The restriction of ω to (X, Z) is

a degenerate logarithmic two-form whose kernel is generated by the Hamiltonian vector fields:

〈ϑx | x ∈ g〉 ⊂ TX,Z .

The quotient morphism q induces a surjection of logarithmic tangent bundles

q∗ : TX,Z −→ TX0,Z0

whose kernel is exactly 〈ϑx | x ∈ g〉. Therefore, the two-form ω|X descends to a non-degenerate

logarithmic two-form ω0 on (X0, Z0), which satisfies (3.7) by definition.

Define a partial compactification of the universal centralizer by

Z =(a, x) ∈ G× S | a ∈ Gx

.

We will use a Whittaker log-symplectic reduction of T ∗G,D

to prove the following theorem:

Theorem 3.8. The partial compactification Z is a smooth variety with a log-symplectic structure

that extends the symplectic structure on Z.

Consider the bundle map

β : T ∗G,D

−→ G.

The preimage D′ = β−1(D) is a normal crossing divisor in T ∗G,D

, and the pair (T ∗G,D

,D′) has a

canonical log-symplectic structure

ω ∈ Ω2(T ∗G,D

,D′)

—an exact logarithmic two-form that restricts to the canonical symplectic structure on T ∗G. Ex-

plicitly, if a1, . . . , an are local coordinates on G such that

D = a1 · . . . · ak = 0,

and if x1, . . . , xn are induced coordinates on the fiber, the log-symplectic form is given locally by

(3.9) ω =k∑

i=1

daiai

∧ dxi +n∑

i=k+1

dai ∧ dxi.

The action of G ×G is Hamiltonian with respect to this Poisson structure, and the moment map

is precisely the compactified moment map µ defined in (3.3).


Restricting to the action of the subgroupN×N ⊂ G×G, and making once again the identification

n∗ ∼= g/b, we obtain a moment map

µN×N : T ∗G,D

−→ g/b× g/b

and a commutative diagram

(3.10)

T ∗G,D

g× g

g/b× g/b

µ

µN×N

Proposition 3.11. The fibers of µN×N are connected.

Proof. The moment map µN×N factors through the moment map µ, as in diagram (3.10), and the

fibers of the vertical projection are connected, so it is sufficient to show that the fibers of µ are

connected.

It is shown in Example 2.5 of [Bri4] that the fiber of µ above a regular semisimple pair is a

connected toric variety. Moreover, by Propositions 3.4 and 3.5, the image of µ is normal. Since

µ is proper and the fiber above a generic point is connected, it follows by Stein factorization that

every fiber of µ is connected.

Proof of Theorem 3.8. Consider once again the coset (f, f) ∈ g/b × g/b. The points in the fiber

µ−1N×N (f, f) are of the form

(a, x1, x2) ∈ G× g× g

with x1, x2 ∈ f + b. In view of Proposition 2.4, N × N acts freely on µ−1N×N (f, f). Then, by

Proposition 3.6, the coset (f, f) is a regular value of µN×N , so µ−1N×N (f, f) is a smooth variety.

By Proposition 3.11, µ−1N×N (f, f) is connected. Being connected and smooth, it is irreducible. It

follows that the intersection

µ−1N×N (f, f) ∩ T ∗

G = µ−1N×N (f, f)

is dense in µ−1N×N (f, f). But this intersection is precisely

µ−1N×N (f, f) =

(g, n1x, n2x) ∈ T ∗

G,D| n1, n2 ∈ N, x ∈ f + ge, g ∈ n1G

xn−12

,

and therefore

µ−1N×N (f, f) =

(a, n1x, n2x) ∈ T ∗

G,D| n1, n2 ∈ N, x ∈ f + ge, a ∈ n1Gxn−1

2

,

This gives an isomorphism

µ−1N×N (f, f) ∼= N ×N ×Z.

By Proposition 3.6, the quotient

Z ∼= µ−1N×N (f, f)/N ×N

is smooth with normal crossing boundary divisor Z\Z, and it inherits a natural log-symplectic

structure.


Corollary 3.12. Let x ∈ g be a regular element. The fiber of the moment map µ over the diagonal

point (x, x) ∈ g× g is the closure Gx of the centralizer of x in the wonderful compactification.

Remark 3.13. The compactified centralizer fibers of Z have been studied in [Bal], where it is

shown that for any x ∈ S, there is a Gx-equivariant isomorphism between Gx and the closure of

the orbit Gx · b inside the flag variety, which is a Hessenberg variety. This connection is discussed

in more detail in Section 4.

The generic fiber, above a regular semisimple element, is the projective toric variety whose fan

is the fan of Weyl chambers. The fiber above the principal nilpotent f is the well-studied Peterson

variety [Kos4, Rie1, Rie2]. In particular, away from the semisimple locus, the fibers of Z are

singular and generally not normal.

3.3. The symplectic leaves of Z. Log-symplectic varieties have an open dense symplectic leaf,

which is the non-degeneracy locus of the associated Poisson bivector. The open dense symplectic

leaf of Z is the universal centralizer Z, the open dense symplectic leaf of the logarithmic cotangent

bundle T ∗G,D

is the ordinary cotangent bundle T ∗G.

The symplectic leaves of Z are Hamiltonian reductions of the symplectic leaves of T ∗G,D

, so we first

give a general description of the latter. In view of (3.9), the restriction of the logarithmic cotangent

bundle to each G×G-orbit OI is a union of symplectic leaves, so we fix a subset I ⊆ 1, . . . , l.

The big cell X—the distinguished open affine space inside G whose translates form a finite open

cover—decomposes as a product

(3.14) X = N− × V ×N,

where V ∼= Cl is the closure of the maximal torus T inside X. The torus T embeds into V via

t 7−→ (α1(t)−1, . . . , αl(t)

−1).

Let aj and a−j be coordinates on N and N− respectively, and let zi | i = 1, . . . , l be

coordinates on V at zI as above, so that a±j , zi is a coordinate system on X centered at zI .

Then the divisor D at zI is cut out by the equation∏

i∈I

zi = 0.

Let x±j , ζi be the induced coordinates on the fiber of T ∗G,D

at zI . In these coordinates, the

log-symplectic form ω is given by

ω =∑

j

da±j ∧ dx±j +∑

i 6∈I

dzi ∧ dζi +∑

i∈I

dzizi

∧ dζi.

The form ω is nondegenerate along the subspace ζi = 0 | i ∈ I.

By Proposition 3.2, the fiber of T ∗G,D

at zI is pI ×lI p−I , and the set ζi | i ∈ I gives a set of

coordinates on the center Z(lI). It then follows that two points

(x1, x2), (y1, y2) ∈ pI ×lI p−I


are in the same symplectic leaf of ω if and only if they have the same component in the center Z(lI)

of the Levi. In other words, if and only if x1 and y1 have the same image in the “universal Cartan”

aI = pI/[pI , pI ].

We have obtained the following proposition.

Proposition 3.15. For every subset I ⊂ 1, . . . , l and every point c ∈ aI , define

AIc =

(x1, x2) ∈ pI ×lI p

−I | x1 ≡ c mod [pI , pI ]

.

The symplectic leaves of ω are exactly the G×G-saturations

(G×G) · AIc .

Now, the symplectic leaves of the Hamiltonian reduction Z are the Hamiltonian reductions of

the symplectic leaves of T ∗G,D

. To each point a = (g, h) · zI ∈ OI of the wonderful compactification,

we associate the pair of parabolic subalgebras

pa = g · pI and qa = h · p−I .

This assignment is well-defined because StabG×G(zI) ⊆ PI × P−I . By Proposition 3.2 the fiber of

the logarithmic cotangent bundle at a is contained in pa × qa.

Corollary 3.16. Two points (a, x), (b, y) ∈ Z are contained in the same symplectic leaf if and only

if the following two conditions are satisfied:

(1) a and b are in the same G×G-orbit OI ;

(2) there is an equality

x mod [pa, pa] ≡ y mod [pb, pb]

in the universal Cartan aI .

4. The universal family of Hessenberg varieties

Definition 4.1. A Hessenberg subspace of g is a B-submodule H of g that contains b.

Let H be such a space, and consider the associated vector bundle

G×B H.

It has a canonical Poisson structure, observed also in a special case in [AC], which comes from a

Hamiltonian reduction as follows. The right action of G on the cotangent bundle T ∗G given by

a · (g, x) = (ga, a−1x), for a ∈ G, (g, x) ∈ G× g

is Hamiltonian. We restrict to the action of the Borel B to get a moment map

T ∗G −→ b∗ ∼= g/n,


and we consider the preimage of the B-stable subset H/n. Reducing, we obtain the smooth Poisson

variety

µ−1R (H)/B ∼= G×B H.

Its symplectic leaves are in bijective correspondence with the B-orbits on the quotient H/n.

The variety G×B H inherits the Hamiltonian left action of G with moment map

µH : G×B H −→ g

[g : x] 7−→ g · x.

Definition 4.2. The Hessenberg variety associated to the space H and a point x ∈ g is the fiber

Hess(x,H) = µ−1H (x) =

gB ∈ G/B | g−1 · x ∈ H

.

Remark 4.3. When H = b, the varieties Hess(x, b) are exactly the Grothendieck-Springer fibers.

We will give more examples in the next subsection.

Now consider the family of regular Hessenberg varieties

H = (gB, x) ∈ G/B × S | gB ∈ Hess(x,H) −→ S,

parametrized by the principal slice.

Proposition 4.4. H is a smooth Poisson variety and a flat family over S of relative dimension

dim(H/b).

Proof. We will take a Whittaker reduction of the vector bundle G ×B H. Consider the moment

map

G×B H −→ n∗ ∼= g/b

for the action of N , and the commutative diagram

G×B H g

g/b.

µH

As before, take the fiber above the coset of f in g/b:

µ−1H (f + b) = (gB, x) ∈ G/B × (f + b) | g−1 · x ∈ H.

By Proposition 2.4, the action of N on this fiber is free, so µ−1H (f + b) is smooth. The action map

N ×H −→ µ−1H (f + b)

(n, (gB, x)) 7−→ (ngB, nx)

is an isomorphism, and this realizes H as the smooth Poisson variety:

H ∼= µ−1H (f + b)/N.


The fibers of the family H −→ S are regular Hessenberg varieties, which all have dimension

dim(H/b) ([Pre], Corollary 2.7). Since H and S are both smooth, the morphism H −→ S is

flat.

Recall that e, h, f is our fixed principal sl2-triple, and choose a one-parameter subgroup

γ : C∗ −→ G(4.5)

whose Lie algebra is spanned by the regular semisimple element h. There is a natural action of C∗

on H given by

t · (b, x) = (γ(t)b′, t2γ(t) · x).

As t → 0, this action contracts the principal slice S to the principal nilpotent f ∈ S, and it

contracts the variety Z to the C∗-fixed points of the fiber Hess(f,H). Because H is a proper flat

family, we obtain the following description of its cohomology:

Proposition 4.6. There is an isomorphism of singular cohomology rings

H∗(H,C) ∼= H∗(Hess(f,H),C).

4.1. The standard Hessenberg space. For the rest of this section we consider the standard

Hessenberg space

H =

(∑

α∈∆

g−α

)⊕ b

consisting of the sum of the positive Borel subalgebra and the negative simple root spaces, and we

write H for the corresponding universal family of regular Hessenberg varieties.

Remark 4.7. When x = s is regular and semisimple, the variety Hess(s,H) is the toric variety

whose fan is the fan of Weyl chambers [MPS]. When x = f is a principal nilpotent, the variety

Hess(f,H) is the Peterson variety.

It was proved in [Bal] that there is a Gx-equivariant isomorphism of varieties

Gx −→ Hess(x,H)

between the compactification of Gx inside G and the standard Hessenberg variety associated to x.

We give a new proof of this result in the framework of universal families.

Proposition 4.8. There is an isomorphism Z ∼= H compatible with the structure maps over S.

Lemma 4.9. Let x ∈ S. Then Gx ∩B = 1.

Proof. Write x = f + v ∈ f + ge and suppose that g ∈ Gx ∩ B. Let g = tu with t ∈ T and u ∈ N .

The Lie algebra g is graded by eigenvalues for the adjoint action of the regular semisimple h. The

regular nilpotent f sits in degree −2, and the Borel b consists of the non-negative eigenspaces.

Then

f + v = g · (f + v) = tu · f + g · v = t · f + (higher degree terms).

It follows that t = 1 and g ∈ N . But by Proposition 2.4, N acts freely on f + b, so g = 1.


Proof of Proposition 4.8. Define a morphism

α :Z −→ H

(g, x) 7−→ (gB, x).

By Lemma 4.9, α is injective. Denote its image by H, and write πZ : Z −→ S and πH : H −→ S

for the structure maps. Define

Zss

= Z ∪ π−1Z

(Sss)

Hss = H ∪ π−1H (Sss),

where Sss ⊂ S is the semisimple locus.

The fiber of Z above a semisimple element s ∈ S is the closure of the maximal torus Gs in

the wonderful compactification G. It is shown, for example, in Remark 4.5 of [EJ], that this toric

variety corresponds to the fan given by the Weyl chambers determined by Gs. In view of Remark

4.7, α extends to an isomorphism

αss : Zss

−→ Hss.

Both Z and H are smooth, and the complements of Zss

and Hss have codimension at least 2,

so αss extends to an isomorphism

α : Z −→ H.

Corollary 4.10. There is a Gx-equivariant isomorphism of varieties Gx −→ Hess(x,H).

As explained above, the variety Z ∼= H is equipped with the action of a one-parameter subgroup

of G whose Lie algebra is Ch. This C∗-action contracts H to the C

∗-fixed points of the Peterson

variety Hess(f,H). Because h is regular, these coincide with the T -fixed points of G/B that lie in

Hess(f,H), and these are known to be exactly the Borel subalgebras

wI · b, I ⊂ ∆,

where wI ∈ W is the longest word of the parabolic Weyl group indexed by the subset of simple

roots I. (See [HT], Proposition 5.8.)

For each I ⊂ ∆, consider the attracting set

XI =(b′, x) ∈ H | lim

t→0t · (b′, x) = (wI · b, f)

=(b′, x) ∈ H | b′ ∈ Hess(x,H) ∩BwIb

.

It is shown in [Bal], Proposition 6.3 that

dim (Hess(f,H) ∩BwIb) = |I|.

Since H is flat, the dimension of the attracting set XI is l+ |I|. Then the following theorem follows

from Byalinicki-Birula:

Theorem 4.11. The attracting sets XI form a stratification of H by affine spaces, and the classes

[XI ] | I ∈ ∆


form an additive basis for the singular cohomology H∗(H,C), where the degree of the class [XI ] is

2l − 2|I|.

4.2. The Poisson structure on G×BH. The symplectic leaves of G×BH correspond bijectively

to the B-orbits on the quotient space H/n. They are described in detail in [AC]. There is a unique

open dense orbit whose preimage in H is

H =

(∑

α∈∆

g−α \ 0

)⊕ b.

It follows that G×B H contains a unique open dense symplectic leaf given by

G×B H ∼= G×N (f + b).

This leaf is an affine bundle over G/N , called a twisted cotangent bundle. The family G ×B H is

its canonical partial compactification, and in this section we will show that the Poisson structure

defined on G×B H is log-symplectic.

The Whittaker reduction of G ×B H is the open dense symplectic leaf of H, which is a family

of quasiprojective varieties over S. The fiber of this family over each x ∈ S is

(4.12) gB ∈ G/B | g−1 · x ∈ H.

Lemma 4.13. The open dense symplectic leaf of H is precisely the image H of the morphism α

defined in the proof of Proposition 4.8.

Proof. Recall that alpha was defined by

α :Z −→ H

(g, x) 7−→ (gB, x).

It is enough to check that each fiber in (4.12) is a single Gx-orbit. Let g1B, g2B be elements of

the fiber over x ∈ S, so that g−11 · x, g−1

2 · x ∈ H. By Proposition 2.4, the action map gives an

isomorphism

B × S −→ H.

This implies that any two elements in H that are G-conjugate are in fact B-conjugate. So there

is an element b ∈ B with

g−11 · x = bg−1

2 · x.

It follows that g1bg−12 ∈ Gx, so g1B and g2B are Gx-conjugate.

Proposition 4.14. The isomorphism

α : Z −→ H

is an isomorphism of Poisson varieties. In particular, the Poisson structure on H is log-symplectic.

Proof. The isomorphism α is Poisson if and only if it maps the Poisson bivector of Z to the Poisson

bivector of H. It is sufficient to check this condition on the open dense symplectic leaf—that is, to


check that the isomorphism of varieties

α : Z −→ H

is a symplectomorphism.

Consider again the moment map

µR : T ∗G −→ g

for the right action of G on T ∗G. Since µR is G-equivariant, we have an isomorphism of coisotropic

varieties induced by the action map:

T × µ−1R (f + b) ∼= µ−1

R (H).

The induced isomorphism

(4.15) G×N (f + b) ∼= µ−1R (f + b)/N ∼= µ−1

R (H)/B ∼= G×B H

is a symplectomorphism.

The universal centralizer Z is the Whittaker reduction of the left-hand side with respect to the

left action of N , and the leaf H is the Whittaker reduction of the right-hand side with respect to

µH . Since the isomorphism α is induced from (4.15) by taking Whittaker reduction of both sides,

it is a symplectomorphism.

Theorem 4.16. The Poisson structure on the vector bundle G×B H is log-symplectic.

The proof will use the theory of Poisson transversals. We recall briefly the essential facts. Let

X be a smooth Poisson variety with a stratification by algebraic symplectic leaves.

Definition 4.17. A Poisson transversal in X (also called a cosymplectic subvariety) is a smooth

subvariety V whose intersection with every symplectic leaf is transverse and symplectic. In other

words, let π ∈∧2 TX be the Poisson bivector, and let

π# : T ∗X −→ TX

be the corresponding Poisson homomorphism. Then V is a Poisson transversal if it induces a

splitting

TX|V = TV ⊕NV ,

where NV = π#(N∗V ) is the image in TX of the conormal bundle to V .

A Poisson transversal V gives a decomposition

π|V = πV + ωV ∈∧2

TV ⊕∧2

NV,

where πV is an induced Poisson structure on V and ωV is nondegenerate.

Example. Consider X = g with the standard Kostant-Kirillov Poisson structure induced by the

identification g ∼= g∗. The symplectic leaves of this structure are exactly the G-orbits, and the

principal slice S = f + ge is a Poisson transversal.


The following lemma appears in [FM] in the real analytic setting. The proof goes through

identically in the complex algebraic context, so we do not reproduce it.

Lemma 4.18 ([FM], Lemma 2). Let ϕ : X1 −→ X2 be a Poisson morphism between smooth

Poisson varieties as above, and suppose that V ⊂ X2 is a Poisson transversal. Then the preimage

ϕ−1(V ) is a Poisson transversal in X1. In particular, ϕ−1(V ) is smooth.

Remark 4.19. The variety H can also be obtained by taking the preimage of the principal slice

S under the moment map µH : G×B H −→ g. Similarly, one obtains both Z and Z. Independent

of the previous discussion, Lemma 4.18 implies that these varieties are smooth and Poisson.

We now prove a result on log-symplectic Poisson transversals in the setting of complex manifolds.

Because all our varieties are smooth and all our Poisson structures are algebraic, we will be able to

apply it to the proof of Theorem 4.16.

Proposition 4.20. Let (M,π) be a complex Poisson manifold and suppose that V ⊂ M is a

Poisson transversal that intersects every symplectic leaf of M . If the induced Poisson structure on

V is log-symplectic, then the Poisson structure on M is also log-symplectic.

Proof. Choose a point m ∈ V . By the Weinstein splitting theorem [Wei], there is an open neigh-

borhood U ⊂ M containing m such that U is Poisson-diffeomorphic to the product of V ∩ U and

a symplectic manifold—that is,

(U, π|U ) ∼= (U ∩ V, πV )× (L,ωL).

If V is log-symplectic, then the Poisson structure π|U is also log-symplectic. It follows that there is

an open neighborhood of V in M where the Poisson bivector is log-symplectic at every point.

If V intersects every symplectic leaf, then any point in M is reached from a point of V by flowing

along Hamiltonian vector fields. Since the Poisson structure is invariant under this flow, it follows

that it is log-symplectic at every point of M .

Proof of Theorem 4.16. The moment map

µH : G×B H −→ g

is a Poisson morphism and S is a Poisson transversal in g. By Lemma 4.18, the preimage

µ−1H (S) ∼= H

is a Poisson transversal in G×B H, and by Proposition 4.14 is it log-symplectic. It remains to show

that it intersects every symplectic leaf in G×B H—then the theorem will follow from Proposition

4.20.

The symplectic leaves in G×BH are in bijection with the B-orbits on the quotient space H/n. If

O ⊂ H/n is such an orbit, we write O+n for its preimage in H. Then the corresponding symplectic

leaf is G×B (O + n).

For each simple root α ∈ ∆, let α be the corresponding coroot and let fα ∈ g−α be a fixed choice

of negative simple root vector. Then the B-orbits on H/n are indexed by the data


• a subset of simple roots I ⊂ ∆

• a collection of elements hα ∈ Cα | α 6∈ I,

in the sense that each orbit O ⊂ H/n contains a unique element of the form

(∑

α∈I

fα

)+

∑

α6∈I

hα

+ n.

Then O + n contains the element

y =

(∑

α∈I

fα

)+

∑

α6∈I

hα

+ e ⊂ b− + e ⊂ greg.

Since y is regular, there is some g ∈ G such that gy ∈ S. Then the point [g : y] ∈ G×B H lies both

in the symplectic leaf corresponding to O and in the Poisson transversal µ−1H (S).

5. Relation to Coulomb branches

Let G be the Langlands dual group of G, and denote by K = C((t)) the field of Laurent series

and by O = C[[t]] its ring of integers. The affine Grassmannian of G is the ind-scheme

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

The G(O)-orbits on GrG are finite-dimensional varieties indexed by Λ+, the set of dominant char-

acters of the maximal torus T of G. They form a stratification of GrG, ordered by the standard

partial order on the cocharacter lattice. (See, for example, [Zhu].)

The equivariant homology space

HG(O)• (GrG)

has a ring structure given by the convolution product (see [CG], Section 2.7). It is a Poisson algebra

whose Poisson structure [BFM] comes from the non-commutative one-parameter deformation

HG(O)⋊C∗

• (GrG).

It is an example of a Coulomb branch in the sense of Nakajima [Nak].

In [BFM] the authors construct an isomorphism of Poisson algebras

(5.1) HG(O)• (GrG)

∼= C[Z].

In this section we will explain, through the lens of this isomorphism, how to obtain the partial

compactification Z directly from the Coulomb branch HG(O)• (GrG).

First notice that both sides of (5.1) have natural filtrations indexed by the lattice Λ of characters

of T . The filtration on the equivariant homology ring HG(O)• (GrG) is induced by the support in

G(O)-orbit closures. The filtration on the coordinate ring C[Z] is inherited through the surjection

C[G]⊗ C[S] ∼= C[G× S] ։ C[Z]

from the Peter-Weyl filtration on C[G].


Proposition 5.2. The isomorphism (5.1) is an isomorphism of filtered algebras.

Proof. We recall an outline of the construction of (5.1). Let R be the set of roots of g and W the

Weyl group. In Proposition 2.8 of [BFM], the universal centralizer Z is identified with an invariant

affine blow-up of T × h:

C[Z] ∼= C

[T × h,

(α− 1)× 0

0× α

]W.

Here we abuse notation to write α ∈ R both for the function on T and for the function on h. The

right-hand side is filtered by dominant weights of T , and from the construction it is clear that this

isomorphism is compatible with the filtrations on both sides.

Let T be the maximal torus of G. Using the fixed-point localization theorem, it is shown in

[BFM] Section 6.3 that there is an isomorphism of localized C[h]W -modules

HG(O)(GrG)|hreg/W∼= H T (O)(GrT)

W|hreg/W

∼= C[T × h]W|hreg/W∼= C[Z]|hreg/W .(5.3)

The authors then prove that this restricts to the desired isomorphism (5.1). The filtrations we

are interested in are compatible with the localization and with the first and third isomorphisms

above. But it is clear that they also coincide under the second, which is induced by

H T (O)(GrT)∼= CΛ⊗H T (O)(pt) ∼= CΛ⊗C[h] ∼= C[T × h].

Now, the wonderful compactification G admits a Rees-type construction from the Peter-Weyl

filtration on C[G] as follows. One considers the Rees algebra

ReesΛC[G] =⊕

λ∈Λ

C[G]≤λtλ ⊂ C[G× T ],

where

C[G]≤λ =⊕

µ≤λ

V ∗µ ⊗ Vµ.

The resulting affine variety

VG = Spec (ReesΛ C[G])

carries a natural action of T . The wonderful compactification G is precisely the GIT quotient VG//T .

(For details, see for example [BK] Chapter 6.) In other words, G is obtained from ReesΛ C[G] by

a multi-proj construction.

An analogous construction produces the partial compactification Z. The affine S-scheme

VZ = Spec (ReesΛC[Z])

carries an action of T , and there is an isomorphism

Z ∼= VZ // T.

In view of Proposition 5.2, we have proved the following result:

Proposition 5.4. There is an isomorphism Z ∼= Spec(ReesΛH

G(O)• (GrG)

)// T.


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