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New York Journal of Mathematics
New York J. Math. 17 (2011) 251–267.

K-theory of weight varieties
Ho-Hon Leung
Abstract. Let T be a compact torus and (M, ω) a Hamiltonian T space. We give a new proof of the K-theoretic analogue of the Kirwan
surjectivity theorem in symplectic geometry (see Harada–Landweber,
2007) by using the equivariant version of the Kirwan map introduced in
Goldin, 2002. We compute the kernel of this equivariant Kirwan map,
and hence give a computation of the kernel of the Kirwan map. As an
application, we find the presentation of the kernel of the Kirwan map for
the T -equivariant K-theory of flag varieties G/T where G is a compact,
connected and simply-connected Lie group. In the last section, we find
explicit formulae for the K-theory of weight varieties.

Contents
1. Introduction
2. Equivariant Kirwan map in K-theory
3. K-theory of weight varieties
3.1. Weight varieties
3.2. Divided difference operators and double Grothendieck

polynomials
3.3. T -equivariant K-theory of flag varieties
3.4. Restriction of T -equivariant K-theory of flag varieties to
the fixed-point sets
3.5. Relations between double Grothendieck polynomials and
the Bruhat ordering
3.6. Main theorem
Acknowledgements
References

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266

1. Introduction
For M a compact Hamiltonian T -space, where T is a compact torus, we
have a moment map φ : M → t∗ . For any regular value µ of φ, φ−1 (µ) is
Received July 30, 2009.
2000 Mathematics Subject Classification. 53D20; 14M15, 19L47.
Key words and phrases. Kirwan surjectivity, flag variety, weight variety, equivariant
K-theory, symplectic quotient.
ISSN 1076-9803/2011

251

252

HO-HON LEUNG

a submanifold of M and has a locally free T -action by the invariance of φ.
The symplectic reduction of M at µ is defined as M//T (µ) := φ−1 (µ)/T .
The parameter µ is surpressed when µ = 0. Kirwan [K] proved that the

natural map, which is now called the Kirwan map,
κ : H ∗ (M ; Q) → H ∗ (φ−1 (0); Q) ∼
= H ∗ (M//T ; Q)
T

T
−1
φ (0)

induced from the inclusion
⊂ M is a surjection when 0 ∈ t∗ is a
regular value of φ. This result was done in the context of rational Borel
equivariant cohomology. In the context of complex K-theory, a theorem of
Harada and Landweber [HL1] showed that
κ : KT∗ (M ) → KT∗ (φ−1 (0))
is a surjection. This result was done over Z.
In Section 2, we give another proof of this theorem by using equivariant
Kirwan map, which was first introduced by Goldin [G1] in the context of
rational cohomology. It can also be seen as an equivariant version of the
Kirwan map.

Theorem 1.1. Let T be a compact torus and M be a compact Hamiltonian
T -space with moment map φ : M → t∗ . Let S be a circle in T , and φ|S :=
M → R be the corresponding component of the moment map. For a regular
value 0 ∈ t∗ of φ|S , the equivariant Kirwan map
κS : KT∗ (M ) → KT∗ (φ|−1
S (0))
is a surjection.
As an immediate corollary of a result in [HL1], we also find the kernel of
this equivariant Kirwan map.
In Section 3, for the special case G = SU(n), we find an explicit formula
for the K-theory of weight varieties, the symplectic reduction of flag varieties
SU(n)/T . The main result is Theorem 3.5. The results in this section are
the K-theoretic analogues of [G2].

2. Equivariant Kirwan map in K-theory
First we recall the basic settings of the subject. Let G be a compact connected Lie group. A compact Hamiltonian G-space is a compact symplectic
manifold (M, ω) on which G acts by symplectomorphisms, together with a
G-equivariant moment map φ : M → g∗ satisfying Hamilton’s equation:
hdφ, Xi = ιX ′ ω, ∀X ∈ g
where G acts on g∗ by the coadjoint action and X ′ denotes the vector field

on M generated by X ∈ g. In this paper, we only deal with a compact torus
action, so we will use the T -action on M as our notation instead. Let T ′
be a subtorus in T , φ|T ′ : M → t′ ∗ is the restriction of the T -action to the
T ′ -action. We call φ|T ′ the component of the moment map corresponding
to T ′ in T .

K-THEORY OF WEIGHT VARIETIES

253

We fix the notations about Morse theory. Let f : M → R be a Morse
function on a compact Riemannian manifold M . Consider its negative gradient flow on M , let {Ci } be the connected components of the critical sets
of f . Define the stratum Si to be the set of points of M which flow down
to Ci by their paths of steepest descent. There is an ordering on I: i ≤ j if
f (Ci ) ≤ f (Cj ). Hence we obtain a smooth stratification of M = ∪Si . For
all i, j ∈ I, denote
[
[
Sj , Mi− =
Sj

Mi+ =
j λ2 > · · · > λn , and λ1 +
· · · + λn = 0. Since M = Oλ is compact, M T has only a finite number of
points. The kernel of the Kirwan map κ is generated by a finite number of
components, see Theorem 2.5 and [HL2]. More specifically, let Mξµ ⊂ M, ξ ∈
t be the set of points where the image under the moment map φ lies to one
side of the hyperplane ξ ⊥ through µ = (µ1 , . . . , µn ) ∈ t∗ , i.e.,
Mξµ = {m ∈ M | hξ, φ(m)i ≤ hξ, µi}.
Then the kernel of κ is generated by
Kξ = {α ∈ KT∗ (M ) | Supp(α) ⊂ Mξµ }.
That is,
ker(κ) =

X

Kξ .

ξ∈t

Now, we are going to compute the kernel explicitly. Our proof is similar to the results in [G2]. In [G2], Goldin proved a very similar result

in rational cohomology by using the permuted double Schubert polynomials as a linear basis of HT∗ (M ) over HT∗ (pt). In K-theory, the permuted
double Grothendieck polynomials are used as a linear basis of KT∗ (M ) over

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HO-HON LEUNG

KT∗ (pt) ∼
= R(T ). The following lemma will be used in our proof of Theorem 3.5.
Lemma 3.6. Let Oλ be a generic coadjoint orbit of SU(n) through λ ∈ t∗ .
Let α ∈ KT∗ (Oλ ) be a class with Supp(α) ⊂ (Oλ )µξ . Then there exists some
γ ∈ W such that if α is decomposed in the R(T )-basis {Gγω }ω∈W as
X
aγω Gγω
α=
ω∈W

where
∈ R(T ), then
6= 0 implies Supp(Gγω ) ⊂ (Oλ )µξ . Indeed, γ can be

chosen such that ξ attains its minimum at φ(λγ ), where
aγω

aγω

λγ = (λγ −1 (1) , . . . , λγ −1 (n) ) ∈ t∗ .
Proof. The proof is essentially the same as Theorem 3.1 in [G2].



Proof of Theorem 3.5. Let ei be the coordinate functions on t∗ . That is,
for λ = (λ1 , λ2 , . . . , λn ) ∈ t∗ , ei (λ) = λi . For γ ∈ Sn , define ηkγ by
ηkγ =

n
X

eγ(i) .

i=k+1

We compute Mηµγ explicitly:
k

Mηµγ = {m ∈ M | hηkγ , φ(m)i ≤ hηkγ , µi}
k

= {m ∈ M | ηkγ (φ(m)) ≤ ηkγ (µ)}

Xn
= m ∈ M | ηkγ (φ(m)) ≤

i=k+1


µγ(i) .

For any ω ∈ W ,
ηkγ (λω )

=
=

n
X

i=k+1
n
X

eγ(i) (λω ) =

n
X

eγ(i) (λω−1 (1) , . . . , λω−1 (n) )

i=k+1

λω−1 γ(i) .

i=k+1

Notice that ηkγ
λ2 ≥ · · · ≥ λn )

attains minimum at λγ (due to our assumption that λ1 ≥
and respects the permuted Bruhat ordering, i.e.,
ηkγ (λv ) ≤ ηkγ (λω )

if v ≤γ ω. By restriction to the domain
Supp(Gγω ) = (Xωγ )T = {v ∈ W | v ≤γ w} = {v ∈ W | γ −1 v ≤ γ −1 ω},
ηkγ attains its maximum at λω and minimum at λγ . If
Xn
Xn
µγ(i) ,
λω−1 γ(i) <
ηkγ (λω ) =
i=k+1

i=k+1

K-THEORY OF WEIGHT VARIETIES

265

then for v ∈ (Xωγ )T ,
ηkγ (λv )

=

n
X

λv−1 γ(i) ≤

i=k+1

n
X

λω−1 γ(i) <

i=k+1

n
X

µγ(i)

i=k+1

and hence
Supp(Gγω ) = (Xωγ )T = {v ∈ W | γ −1 v ≤ γ −1 ω} ⊂ Mηµγ .
k

Since Gγω (x, y) = πω−1 γ G(x, yγ ), we have
πv G(x, yγ ) ∈ ker(κ)

if

Xn

i=k+1

λv(i) <

Xn

i=k+1

µγ(i) .

For the other direction, we need toP
show that the classes
πv G(x, yγ ) with
P
v, γ ∈ W having the property that ni=k+1 λv(i) < ni=k+1 µγ(i) for some
k ∈ {1, . . . , n − 1} actually generate the whole kernel. Let α ∈ KT∗ (M ) be a
class in ker(κ), so Supp(α) ⊂ Mξµ for some ξ ∈ t. We take γ ∈ W such that
ξ(λγ ) attains its minimum. Decompose α over the R(T )-basis {Gγω }ω∈W ,
α=

X

aγω Gγω

ω∈W

where aγω ∈ R(T ). By Lemma 3.6, we need to show that Supp(Gγω ) ⊂ Mηµγ
k

for some k. Since ηkγ is preserved by the permuted Bruhat ordering and
attains its maximum at λω in the domain Supp(Gγω ), we just need to show
that
(6)

ηkγ (λω ) < ηkγ (µ)

for some k. It is actually purely computational: Suppose (6) does not hold
for all k. We have
λω−1 γ(n) ≥ µγ(n)
..
.
λω−1 γ(2) + · · · + λω−1 γ(n) ≥ µγ(2) + · · · + µγ(n) .
P
For ξ = ni=1 bi ei , b1 , . . . , bn ∈ R (recall that ξ attains its minmum at λγ by
our choice of γ ∈ W ), we have ξ(λγ ) ≤ ξ(λsi γ ) where si is a transposition of
i and i + 1. And hence
bi λγ −1 (i) + bi+1 λγ −1 (i+1) ≤ bi λγ −1 (i+1) + bi+1 λγ −1 (i) .

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HO-HON LEUNG

By our assumption that λi > λi+1 , we get bγ(i) ≤ bγ(i+1) . And hence
bγ(1) ≤ bγ(2) ≤ · · · ≤ bγ(n) . Then,
(bγ(n) − bγ(n−1) )λω−1 γ(n) ≥ (bγ(n) − bγ(n−1) )µγ(n)
(bγ(n−1) − bγ(n−2) )(λω−1 γ(n−1) + λω−1 γ(n) ) ≥ (bγ(n−1) − bγ(n−2) )
· (µγ(n−1) + µγ(n) )
..
.
(bγ(2) − bγ(1) )(λω−1 γ(2) + · · · + λω−1 γ(n) ) ≥ (bγ(2) − bγ(1) )
· (µγ(2) + · · · + µγ(n) ).
Using
get

Pn

i=1 λi

=0=

Pn

i=1 µi

and summing up all the above inequalities to

n
X

bγ(i) λω−1 γ(i) ≥

n
X

bi µi

⇔

n
X

n
X

bi µi

i=1

i=1

bi λω−1 (i) ≥

i=1

i=1

⇔ ξ(λω ) ≥ ξ(µ).
The last inequality contradicts Supp(α) ⊂ Mξµ since λω has the property
that ω ∈ Supp(α). So (6) is true.
So the kernel
ker(κ) is P
generated by the set πv G(x, yγ ) for v, γ ∈ W
P
satisfying ni=k+1 λv(i) < ni=k+1 µγ(i) for some k = 1, . . . , n − 1. By (2)
and the surjectivity of the Kirwan map κ,
∼ K ∗ (Oλ //T (µ)).
κ : K ∗ (SU(n)/T ) = K ∗ (Oλ ) → K ∗ (φ−1 (µ)) =
T

T

T

This implies that
K ∗ (Oλ //T (µ)) = KT∗ (Oλ )/ ker(κ).
With ker(κ) explicitly computed and by (2), Theorem 3.5 is proved.



Acknowledgements
The author would like to thank Professor Sjamaar for all his encouragement, advice, teaching and finding calculational mistakes in the first edition
of this paper. The author is also grateful to the referee for critical comments.

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Department of Mathematics, Malott Hall, Cornell University, Ithaca, New
York 14853-4201 USA
hohonleung@math.cornell.edu
This paper is available via http://nyjm.albany.edu/j/2011/17-12.html.