The moment graph model

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 February 2013

The moment graph model

T-fixed points and the map Φ

Following Goresky-Kottwitz-MacPherson [GKM1489894, Theorem 1.2.2] a powerful way to think about this theory is via the moment graph model. This means that for a T-variety X where the imbeddings of the T-fixed points of X into X are

ιw: pt X * w considerι*= wWιw*: ΩT*(X) wWΩT (pt), (3.1)

where the sums are over an index set W for the T-fixed points in X. When X is a “GKM-space” (see [GKM1489894, Theorem 14] for several equivalent characterization of a GKM space for equivariant ordinary cohomology and [?, HHH0409305] or equivariant generalized cohomology theories) the ring homomorphism ι* is injective with image

imι*= { (gw) wW0 wW0 ΩT(pt) , gw-gw yαΩT (pt)if there is a 1-dimensionalT-orbit containingw andw } ,

where yα is the T-equivariant Chern class of the tangent along the 1-dimensional orbit connecting w and w.

Computations are facilitated by encoding the information of imι* with a moment graph, which has vertices corresponding to the T-fixed points of X and labeled edges wαw corresponding to 1-dimensional T-orbits in X. For example, for G/B for type GL3 the graph is

1 s1 s2 s1s2 s2s1 s1s2s1=s2s1s2 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 (3.2)

A moment graph section is a tuple (g2)wW of elements of ΩT(pt) which is an element of imι*.

A morphism of GKM-spaces is a morphism of T-spaces

f:XYwhich provides, by restriction, f:WV

from the set W of T-fixed points of X to the set V of T-fixed points of Y. Viewing elements of HT(X) and HT(Y) as moment graph sections the maps

f*:HT(Y) HT(X)and f!:HT(X) HT(Y)

are given by

(f*(c))w= cf(w),and (f!(γ))v =wf-1(v) γw 1e(f)wv, (3.3)

where the Euler class of f from v to w is

e(f)wv= ( edges ofW adjacent tow yβ ) ( edges ofV adjacent tov yβ ) -1 .

The second formula in (3.3) is a form of the familiar formula for push forwards by “localization at the T-fixed points” as found, for example, in [ABo0721448, (3.8)]. The Euler class of f from v to w is the contribution measured by the difference between the tangent space at the T-fixed point w in X to the tangent space to the T-fixed point v=f(w) in Y.

The Borel model and the moment graph model for G/B for equivariant algebraic cobordism ΩT(G/B) are summarized in the following Theorem, which is a combination of [KKr1104.1089, Theorem 4.7] and [HHH0409305, Theorem 3.1]. The ring S which takes the role of ΩT(pt) is as in [CPZ0905.1341, §2.4]. For comparison to the K-theory case see [KKu0895705, Theorem 3.13] and [LSS2660675, Theorem 3.1].

Theorem 3.1. ([HHH0409305, Theorem 3.1], [KKr1104.1089, Theorem 4.7] and [CPZ0905.1341, §2.4] combined) Let GBT be a reductive group as in (2.1) and let W0 and 𝔥* be the Weyl group and the weight lattice 𝔥* as in (2.2). Let 𝕃 be the Lazard ring generated by aij as in (2.13) and let S be the 𝕃-algebra

S=𝕃 [ [ yλ λ𝔥* ] ] ,with yλ+μ=yλ+ yμ+a11yλ yμ+a12yλ yμ2+a21 yλ2yμ+. (3.4)

The Weyl group

W0acts 𝕃-linaerly onS bywyλ= ywλ,

for wW0, λ𝔥*. Define a product on wW0S pointwise,

(fw) wW0 · (gw) wW0 = (fwgw) wW0 , (3.5)

and let SSW0S be the coinvariant ring as defined in (2.15). The S-algebra homomorphism

Φ: SSW0S ΩT (G/B)im Φ wW0S fg (f·(w-1g)) wW0 (3.6)

is well defined and injective with

imΦ= { (gw)wW0 wW0S gw- gwsα y-αSfor αR+ andwW0 } ,

where R+ is the set of positive roots corresponding to B and sαW0 denotes the reflection corresponding to α.

To provide a feel for the ring S of (3.4), let us provide some formulas which will be useful for computations later. To recapitulate and summarize previous definitions,

S=𝕃 [ [ yλ λ𝔥* ] ] withyλ+μ =yλ+yμ- p(yλ,yμ) yλyμ, (3.7)

where p(yλ,yμ)𝕃 [ [ yλ,yμ ] ] is a power series

p(yλ,yμ)= -a11-a12yμ- a21yλ-a31 yλ2-a22yλ yμ-a13yμ yλ-, (3.8)

with aij𝕃 satisfying relations such that

y-λ+λ=y0 =0,yλ+μ= yμ+λ, y(λ+μ)+ν= yλ+(μ+ν). (3.9)


yα= -y-α 1-p (yα,y-α) y-α ,1y-α+ 1yα=p (yα,y-α), (3.10)

and the formula

y-α y-α =-j=1-1 p(y-αy-jα) y-jα=1+ j=1-1 ( 1-p (y-αy-jα) y-jα ) ,for>0, (3.11)

is proved by induction on . Using (3.11) and the formula siλ=λ- λαi αi for the action of a simple reflection on 𝔥* produces

ysiλ-yλ y-αi = ( 1-p ( yλ, y -λ,αi αi ) yλ ) ( 1+ j=1 λ,α-1 ( 1-p ( y-αi, y-jαi ) y-jαi ) ) , (3.12)

for λ,αi 0. Formula (3.12) generalizes one of the favorite formulas for the action of a Demazure operator (see [Kum1923198, Lemma 8.2.8]). This cobordism case specializes to HT and KT by setting

p(yλ,yμ)= { 0, inHT, 1, inKT, andyλ= { yλ, inHT, 1-eλ, inKT. (3.13)

The nil affine Hecke algebra

Let S be as in (3.4) and (3.7). The point of view of [GRa0405333] is that the homomorphism Φ of (3.6) arises naturally from the nil affine Hecke algebra.

The nil affine Hecke algebra H is

H = (S𝕃S)𝕃 [W0] = S-span { gtw gS,wW0 } =𝕃-span { (fg)tw f,gS,w W0 }


tutv=tuv andtw (fg)= (f(wg))tw, (3.14)

for u,v,wW0 and f,gS. The nil affine Hecke algebra H acts on S𝕃S and on SSW0S by

tw(fg)=f wgand (hp) (fg)=hf pg, (3.15)

for h,p,f,gS and wW0. These actions arise from the realization of SSW0S as an induced up H-module in (3.16) below.

Let b1 be a symbol and let Sb1 be the S𝕃S module (a rank 1 free S-module with basis {b1}) corresponding to the ring homomorphism

ε: S𝕃S S fg fg so that theS𝕃S action onSb1is given by (fg)b1=fgb1,

for f,gS. The induced module

Hb1= IndS𝕃SH (Sb1)has S-basis {bwwW0} ,wherebw=tw b1.

Let 10= wW0tw. With the definition of the H action on S𝕃S as in (3.15), the sequence of maps (see [GRa0405333, Theorem 2.12])

S𝕃S H10 H Hb1 wW0S (fg) (fg)10 h hb1 (3.16)

is a homomorphism of H-modules (with kernel generated by { f1-1f fSW0 } ). The maps in (3.16) allow for the expansion of any element of S𝕃S in terms of the basis {bwwW0} of Hb1, giving

(fg)10b1 = (fg) ( wW0 tw ) b1 = wW0tw ( f(w-1g) ) b1 = wW0tw ( f·(w-1g) ) b1 = wW0 ( f· (w-1g) ) bw.

This formula illustrates that computing Φ(fg) in (3.6) is equivalent to expanding (fg)b1 in terms of the bw. Because of this we use (3.6) and (3.16) to

identifyΩT (G/B)=Hb1=S -span { bww W0 } wW0S

and write elements

fΩT (G/B)as f=wW0 fwbw. (3.17)

The product in ΩT(G/B) is then given by (3.5). To more easily keep track of the left and right factors in S𝕃S use the notation

xμ=1yμ and yμ=yμ1. (3.18)

Then the formulas

xλ·1=xλ wW0tw b1=wW0 twxw-1λ b1=wW0 yw-1λbw, and (3.19) tvwW0 fwbw= wW0fw tvbw= wW0fw bvw= zW0 fv-1zbz, (3.20)

provide the formulas for action of the nil affine Hecke algebra in terms of moment graph sections (see (3.15)). We often view the values fw as labels on the vertices of the moment graph so that, for exmaple, in type GL3 where the moment graph is as in (3.2), (3.19) can be written

yλ xλ = ys1λ ys2s1λ ys2λ ys1s2λ ys1s2s1λ

Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.

page history