Equivariant Cohomology of Infinite-Dimensional Grassmannian and Shifted Schur Functions

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by  Jia-Ming Liou

EQUIVARIANT COHOMOLOGY OF INFINITE-DIMENSIONAL
GRASSMANNIAN AND SHIFTED SCHUR FUNCTIONS
JIA-MING(FRANK)LIOU,ALBERTSCHWARZ
2 Abstract. We study the multiplication and comultiplication in equivariant
1
cohomology of Sato Grassmannian .
0
2

n
a
J 1. Introduction

2 LetusconsidertheHilbertspaceH = L2(S1)andits subspacesH ,H definedasclosed
1 + −
subspacesof H spannedby {zi :i ≥ 0} and{zj :j < 0} respectively . Usingthe decomposi-

] tion H = H ⊕H one can define the (Segal-Wilson version of) Sato Grassmannian Gr(H)
+ −
G
as a space of all closed linear subspaces W ⊂ H such that the projection π : W → H is
− −
A
a Fredholm operator and the projection π : W → H is a compact operator (see [11] for
+ +
h. more detail).
t The group S1 acts naturally on H: to every α obeying |α| = 1 we assign a map
a
m f(z) → f(αz). This action (we will call it standard action) generates an action of S1
on Grassmannian . Representing a function on a circle as a Fourier series f(z) = a zn,
[ n
1 we see that the standard action sends ak → αkak. The standard action is relatedPto the
S1-action on various moduli spaces embedded into Grassmannian by means of Krichever
v
4 construction, see [12] and [6] for more detail. Motivated by the desire to construct nonper-
5 turbative string theory we considered in [6] equvariant cohomology of Sato Grassmannian
5
and homomorphism of this cohomology induced by the Krichever map. The results of
2
. present paper will be used in [7] to study this homomorphism in more detail.
1
Onecanconsidermoregeneralactions ofS1 onHsendinga → αnka wheren ∈ Zisan
0 k k k
2 arbitrary doubly infinite sequence of integers. This action also generates an action of S1 on
1 Grassmannian. The Grassmannian Gr(H) is a disjoint union of connected components
:
v Gr (H) labeled by the index of the projection π : W → H . All components are
d − −
i homeomorphic. It was proven in [11] that every component is homotopy equivalent to a
X
subspace having a cell decomposition K = ∪σ consisting of even-dimensional cells (finite-
r λ
a dimensional Schubert cells). The cells are labeled by partitions. This decomposition is
S1-invariant with respect to any action of S1 from the class of actions we are interested in.
This allows us to say that the equivariant cohomology HS1(Grd(H)) has a free system of
generators ΩTλ = [Σλ] as a module over HS1(pt). These generators can be interpreted also
as cohomology classes dual to Schubert cycles Σ having finite codimension, In this letter
λ
we calculate the multiplication table in the basis ΩT.
λ
Theorem 1.1. For the standard S1-action on Gr (H) the coefficients in the decomposition
d
(1.1) ΩTΩT = Cν (u)ΩT
λ µ X λµ ν
ν
can be expressed in terms of coefficients in the decomposition
(1.2) s∗s∗ = Cν s∗
λ µ X λµ ν
ν
1
2 JIA-MING(FRANK)LIOU,ALBERTSCHWARZ
by the formula Cν (u) = Cν u|λ|+|µ|−|ν|. Here s∗ stands for the shifted Schur function in
λµ λµ λ
the sense of [10] and |λ| = λ is the weight of the partition λ = (λ ).
Pi i i
Combining this theorem with the results of [8] we obtain the following expression for the
coefficients.
Corollary 1.1.
h(ρ)
Cν (u) = u|λ|+|µ|−|ν| (−1)|ν|−|ρ| .
λµ X h(ν/ρ)h(ρ/λ)h(ρ/µ)
λ,µ⊂ρ,ν
Here the function h is defined to be h(ν/µ) = |ν/µ|!/dim(ν/µ) for any skew diagram ν/µ
and dim(ν/µ) is the number of standard ν/µ-tableaux.
TheTheorem1.1can beappliedtotheanalysis ofmultiplication intautological cohomol-
ogy ring of the modulispaces of complex curves (more precisely, to the studyof intersection
of cycles defined in terms of Weierstrass points), see [7]. To generalize the theorem to the
non-standard action of S1 we introduce the notion of shifted double Schur function (it is
closely related to double Schur functions of infinite number of arguments introduced in [9]).
Let x = (x1,··· ,xn) be an n-tuple of variables and y = (yi)i∈Z be a doubly infinite
sequence. Recall that, the double Schur function ns (x ,··· ,x |y) is 1 a symmetric poly-
λ 1 n
nomial in x = (x ,··· ,x ) with coefficients in C[y] defined by
1 n
ns (x ,··· ,x |y) = det (x |y)λj+n−j /det (x |y)n−j ,
λ 1 n h i i i
(cid:2) (cid:3)
where (x |y)p = p (x −y ). Let us introduce the shift operator on C[y] given by
i Qj=1 i j
(τy) = y , i ∈Z.
i i−1
Then the double Schur function satisfies the generalized Jacobi-Trudi formula [1]:
(1.3) ns (x ,··· ,x |y) = det h (x ,··· ,x |τj−1y) n ,
λ 1 n (cid:2) λi+j−i 1 n (cid:3)i,j=1
where
h (x ,··· ,x |y)= (x −y )···(x −y ), p ≥ 1.
p 1 n X i1 i1 ip ip+p−1
1≤i1≤···≤ip≤k
The Littlewood-Richardson coefficients ncν (y) of the double Schur functions are defined
λµ
by
(1.4) ns (x ,··· ,x |y) ns (x ,··· ,x |y)= ncν (y) ns (x ,··· ,x |y).
λ 1 n µ 1 n X λµ µ 1 n
ν
Thesecoefficients ncν (y)werecalculatedin[8]. WedefinetheshifteddoubleSchurfunction
λµ
by
(1.5) ns∗(x ,··· ,x |y) =n s (x +y ,x +y ,··· ,x +y |τn+1y).
λ 1 n λ 1 −1 2 −2 n −n
Under the change of variables x′ = x +y for 1≤ i ≤ n, the shifted double Schur function
i i −i
ns∗(x ,··· ,x |y) becomes the double Schur function ns (x′,··· ,x′ |τn+1y). Notice that
λ 1 n λ 1 n
the shifted double Schur function ns∗(x ,··· ,x |y) is the shifted Schur function defined in
λ 1 n
[10] if y = (y ) is the sequence defined by the relation y = constant+k for all k. The
k k k
shifted double Schur functions ns∗(x ,··· ,x |y) have the following stability property:
λ 1 n
1
Weaddtheindexntotheconventional notationsλ toemphasizethatthefunctiondependsonnvariablesxk.
EQUIVARIANT COHOMOLOGY OF INFINITE-DIMENSIONAL GRASSMANNIAN AND SHIFTED SCHUR FUNCTIONS3
Proposition 1.1. If l(λ) < n, then
n+1s∗(x ,··· ,x ,0|y) =n s∗(x ,··· ,x |y).
λ 1 n λ 1 n
Here l(λ) is the length of a partition λ.
Using this property, we can define the shifted double Schur function s∗(x|y) depending
λ
on infinite number of arguments x = (xi)i∈N and y = (yj)j∈Z (we assume that only finite
number of variables x does not vanish). Namely, we define
i
(1.6) s∗(x|y) = ns∗(x ,··· ,x |y)
λ λ 1 n
where n is chosen in such a way that n > l(λ) and x = 0 for i > n. The coefficients in the
i
decomposition
(1.7) s∗(x|y)s∗(x|y) = Cν (y)s∗(x|y).
λ µ X λµ ν
canbeobtainedfrom(1.4)andfromtheresultsof[8],[9]or[5]. Infact, therelationbetween
Cν and ncν is given by
λµ λµ
Cν (y) = ncν (τn+1y)
λµ λµ
for n > l(λ),l(µ),l(ν).
TheshifteddoubleSchurfunctionsbelongtotheringΛ∗(xky)thatconsistsofpolynomial
functions that depend on sequences x = (xn)n∈N and y = (yi)i∈Z and are symmetric with
respect to shifted variables x′ = x +y (we assume that x = 0 for n >> 0). Moreover,
i i −i n
theyformafreesystemofgenerators of Λ∗(xky)consideredas C[y]-module. Notice thatthe
ring Λ∗(xky) is obviously isomorphic to the ring Λ(xky) constructed in [9]. The following
theorem describes the multiplication in the equivariant cohomology HS1(Grd(H)) for non-
standard action of S1:
Theorem 1.2.
ΩTΩT = Cν (n)ΩT,
λ µ X λµ ν
ν
where Cν (n) are the coefficients in (1.7) calculated for y = n u and n denotes the
λµ k k+d k
sequence specifying the action of S1.
The Theorem 1.1 is a particular case of Theorem 1.2 for n = k.
k
Let us consider the infinite-dimensional torus T and its action on the Grassmannian.
Algebraically the infinite torus T is the infinite direct product S1. The action of T
on Grassmannian corresponds to the action on H transforming aQ→i∈Zα a , where (α ) ∈T
k k k k
and f = a zn ∈ H. This action specifies an embedding of T into the group of unitary
transformPatnionns of H; the topology of T is induced by this embedding.
One can prove that the equivariant cohomology HT(pt) (cohomology of the classifying
space BT) is isomorphic to the polynomial ring C[u] where u stands for the doubly infinite
sequence u . The proof is based on the consideration of homomorphisms of T onto finite-
k
dimensional tori and homomorphisms of S1 into T. The finite codimensional Schubert cells
areT-invariant andhencetheequivariant cohomology HT(Grd(H))is afreeHT(pt)-module
generated by cohomology classes labeled by partitions; we denote these classes by ΩT.
λ
Thesubmanifold Grl of Gr (H) consisting of points W so that the orthogonal projection
d d
π : W → z−lH is surjective. The intersection of Schubert cycle Σ and Grl is denoted
l − λ d
4 JIA-MING(FRANK)LIOU,ALBERTSCHWARZ
by Σ . The equivariant cohomology class in H∗(Grl) corresponding to Σ is denoted by
λ,l T d λ,l
ΩT . The Schubert cycle Σ and Grl are in general position if l(λ) < d+l; then we have
λ,l λ d
f∗ΩT = ΩT ,
l λ λ,l
wheref∗ : H∗(Gr (H)) → H∗(Grl)ismapinducedbytheinclusionmapf :Grl → Gr (H).
l T d T d l d d
The classes ΩT for l(λ) < d + l form an additive system of generators of equivariant
λ,l
cohomology. It follows from (1.4) that
Theorem 1.3. The multiplication table in H∗(Grl) is given by
T d
ΩT ΩT = d+lcν (y)ΩT ,
λ,l µ,l X λµ ν,l
ν
where y = (yi)i∈Z is the sequence defined by y = τl+1u.
Using the coefficients in (1.7), we can calculate the multiplication table in equivariant
cohomology of Grassmannian Gr (H) with respect to the infinite torus action. Namely,
d
Theorem 1.4. The multiplication table in H∗(Gr (H)) is given by:
T d
ΩTΩT = Cν (y)ΩT,
λ µ X λµ ν
ν
where (y ) is the sequence given by the relation y = τ−du. The coefficient in this equation
k
comes from the formula (1.7); it is considered as an element of HT(pt) .
Using homomorphisms of S1 into T one can derive this theorem from Theorem 1.2 (con-
versely, one can deduce Theorem 1.2 from Theorem 1.4). It follows from Theorem 1.4 that
the equivariant cohomology H∗(Gr (H)) is isomorphic to the ring Λ∗(xky).
T d
We can introduce a comultiplication in the ring H∗(Gr (H)) using the map
T d
(1.8) ρ: Gr (H′)×Gr (H′′)→ Gr (H′⊕H′′)
0 0 0
defined by (V,W) 7→ V ⊕W. Namely, we take H′ = H (the subspace spanned by z2k)
even
andH′′ = H (thespacespannedbyz2k+1). ThenH′⊕H′′ = H andthemapρdetermines
odd
a homomorphism of T- equivariant cohomology
HT(Gr0(H)) → HT(Gr0(Heven)×Gr0(Hodd)).
It is easy to prove that HT(Gr0(Heven)×Gr0(Hodd)) is isomorphic to tensor product of two
copies of H∗(Gr (H)); we obtain a comultiplication ∆ in H∗(Gr (H)).
T 0 T 0
Let us introduce the notion of the k-th shifted power sum function in Λ∗(xky):
∞
(1.9) p (x|y) = (x +y )k −yk .
k Xh i −i −ii
i=1
Here we assume that x = 0 for n ≫ 0; hence (1.9) is a finite sum. We will prove that
n
(1.10) ∆p = p ⊗1+1⊗p , k ≥ 1.
k k k
In other words, we have the following theorem:
Theorem 1.5. The comultiplication ∆ in the ring H∗(Gr (H)) defined above coincides
T d
with the comultiplication in the ring Λ∗(xky) = Λ(xky) constructed in [9].
EQUIVARIANT COHOMOLOGY OF INFINITE-DIMENSIONAL GRASSMANNIAN AND SHIFTED SCHUR FUNCTIONS5
2. Equivariant Schubert classes
The Grassmannian Gr(H) has a stratification in terms of Schubert cells having finite
codimension; it is a disjoint union of T-invariant submanifolds Σ labeled by S, where S is
S
a subset of Z such that the symmetric difference Z ∆S is a finite set. The Schubert cells
−
Σ are in one-to-one correspondence with the T-fixed points H , where H is the closed
S S S
subspace of H spanned by {zs : s∈ S} (the fixed point H is contained in the Schubert cell
S
Σ ). Instead of a subset S of Z, we can consider a decreasing sequence (s ) of integers.
S n n
It is easy to check that s = −n+d for n ≫ 0, where d is the index of H . The complex
n S
codimension of the Schubert cell Σ is given by the formula
S
∞
codimΣ = (s +i−d).
S X i
i=1
The closure Σ of Σ is called the Schubert cycle of characteristic sequence S. It defines a
S S
cohomology class in H∗(Gr(H)) having dimension equal to 2codimΣ . Since the Schubert
S
cycleΣ isT-invariant,itspecifiesalsoanelementΩT inH∗(Gr(H)). Denoteλ = s +n−d
S S T n n
for n ≥ 1. Then (λ ) form a partition. Instead of using the sequence S to label the
n
(equivariant) cohomology class ΩT, we use the notation ΩT. Then the dimension of ΩT is
S λ λ
equal to 2|λ|. Similarly, we denote Σ by Σ . For more details see [6], [11].
S λ
Let us consider the submanifold Grl consisting of points W in Gr (H) such that the
d d
orthogonal projection π :W → z−lH is surjective. There is an equivariant vector bundle
l −
ofrankn = d+loverGrl whosefiberoverW isthekerneloftheprojectionπ :W → z−1H .
d l −
The action of T (or any action of S1) on Gr (H) induces an action on Grl. The equivariant
d d
SchubertcycleΣ inGrl istheintersection of theequivariant Schubertcycle Σ inGr (H)
λ,l d λ d
and Grl. The dual equivariant cohomology class of Σ in H∗(Grl) is denoted by ΩT . If
d λ,l T d λ,l
l(λ) < d+l where l(λ) means the length of the partition λ then the Schubert cycle Σ and
λ
Grl are in general position; hence we have
d
f∗ΩT = ΩT ,
l λ λ,l
where f∗ : H∗(Gr (H)) → H∗(Grl) is the homomorphism induced by the inclusion map
l T d T d
f :Grl → Gr (H).
l d d
Proposition 2.1. The equivariant Schubert class ΩT in Gr (H) is given by
λ,l d
(2.1) ΩT = det cT (H −E ) n .
λ,l (cid:2) λi+j−i −l,λi−i+d−1 l (cid:3)i,j=1
Here H is the equivariant vector bundle H × Grl and the vector space H is the
i,j i,j d i,j
subspace of H spanned by {zk : i≤ k ≤ j}.
Proof. This statement follows from the Kempf-Laksov’s formula, see [1] and [4]. (cid:3)
3. Structure Constants of Schubert Classes with respect to the Standard
S1-Action
Let H be a fixed point of T–action on Gr (H). The inclusion map ι :{H } → Gr (H)
S d S S d
induces a homomorphism:
ι∗ : H∗ (Gr (H)) → H∗ ({H })
S S1 d S1 S
called the restriction map. Denote by δ = (δ ) the partition corresponding to S, i.e. δ =
i i
s +i−d. Assume that λ = (λ ) is a partition such that l(λ) < l(δ). Then
i i
6 JIA-MING(FRANK)LIOU,ALBERTSCHWARZ
Lemma 3.1.
ι∗ΩT = u|λ|s∗(δ ,··· ,δ ).
S λ,l λ 1 d+l
Here s∗ is the shifted Schur function defined in [10].
λ
Proof. Denoten= d+l. Assumethatx′,··· ,x′ areequivariantChernrootsofE∨ (thedual
1 n l
equivariant vector bundle of E ). Then cT(E ) = n (1−x′). We also have cT(H ) =
m (1−ju). This gives us:lfor each p ≥ 0l Qi=1 i −l,m
Qj=−l
cT (H −E ) = h (x′,··· ,x′ )(−1)be (y ,y ,··· ,y ),
p+j−i −l,λi−i+d−1 l X a 1 n b −n −n−1 λi−i−1
a+b=p+j−i
where y = (j +d)u for j ∈ Z. By the generalized Jacobi-Trudi formula (1.3), we find
j
ΩT = ns (x′,··· ,x′ |τn+1y).
λ,l λ 1 n
Applying the restriction homomorphism ι∗, we find
S
ι∗ΩT = ns (δ −1+d,··· ,δ −n+d|τn+1a)u|λ| = ns∗(δ ,··· ,δ |a)u|λ|,
S λ,l λ 1 n λ 1 n
where a = (a ) with a = j + d for all j ∈ Z. The shifted double Schur function
j j
ns∗(δ ,··· ,δ |a) coincides with the shifted Schur function defined in [10]. This completes
λ 1 n
the proof. (cid:3)
Now we are ready to prove the theorem 1.1. Since Cν (u) is of the form Cν u|λ|+|µ|−|ν|,
λµ λµ
(1.1) can be rewritten as
ΩTΩT = Cν u|λ|+|µ|−|ν|ΩT.
λ µ X λµ ν
ν
Applying ι∗ to the above equation2 and using lemma 3.1, we find that the constants (Cν )
S λµ
satisfy (1.2).
In order to compute the structure constants for the equivariant Schubert classes of
Gr (H), we have to introduce the notion of shifted symmetric functions.
d
4. Algebra of Shifted Symmetric Functions
Let x = (xn)n∈N bea sequence of variables obeyingxn = 0 for n >>0 and y = (yi)i∈Z be
a doubly infinite sequence of variables. Denote the set of pairs (x,y) by R. Let us consider
a function f(x|y) such that its restriction f to the subset R specified by the condition
n k
x = x = ··· = 0 is a polynomial for every k ∈ N. We say that f is shifted symmetric
k+1 k+2
if f symmetric with respect to the variables x′ = x +y for 1≤ i ≤ n. In other words,
n i i −i
f′(x′,...,x′ |y) = f (x′ −y ,...,x′ −y |y)
n 1 n n 1 −1 n −n
is symmetric with respect to x′ = (x′,...,x′ ).
1 n
This definition is motivated by the definition in [10]. (If we replace y by constant+j,
j
we obtain the definition of the shifted symmetric functions given in [10].) An essentially
equivalent notion was introduced in [9]. Instead of R one can consider a set R˜ of pairs (x,y)
where x = (xn)n∈N and y = (yi)i∈Z are sequences obeying xn = y−n for n >> 0. Shifted
symmetricfunctionsonR correspondtosymmetricfunctionsonR˜; this correspondencecan
be used to relate our approach to the approach of [9]. It is obvious that shifted symmetric
functions on R constitute a ring; we denote this ring by Λ∗(xky). It is clear that this ring
is isomorphic to the ring Λ(xky) of symmetric functions on R˜ considered in [9]. It follows
2Herewemayassumethatl(λ),l(µ),l(ν)<l(δ). Ifl(λ)>l(δ),ι∗ΩT =0
S λ
EQUIVARIANT COHOMOLOGY OF INFINITE-DIMENSIONAL GRASSMANNIAN AND SHIFTED SCHUR FUNCTIONS7
immediately fromwellknownresults thattheshifteddoubleSchurfunctions{s∗(x|y)}form
λ
a linear basis for Λ∗(xky) considered as C[y]-module..
Now we are ready to finish the proof of theorem 1.4. Let {x′,··· ,x′ } be the equivariant
1 n
Chern roots of of the equivariant vector bundle El∨. Denote u = (ui)i∈Z the sequence of
weights of the action of T on H and y = τ−du. Define x = x′ −y for 1 ≤ i ≤ n.
i i −i
Theorem 4.1. The equivariant Schubert class ΩT is given by
λ,l
ΩT = d+ls∗(x ,··· ,x |y).
λ,l λ 1 d+l
Proof. The proof is the same as that in lemma 3.1. (cid:3)
This proposition allows us to define the algebra isomorphism between the equivariant
cohomology ring H∗(Gr ) and the algebra of shifted symmetric functions Λ∗(xky). The
T d
isomorphism between them is given by
ΩT 7→ s∗(x|y).
λ λ
This implies that the multiplication table in H∗(Gr (H)) with respect to the basis {ΩT} is
T d λ
the same as the multiplication table of {s∗(x|y)} in Λ∗(xky). This completes the proof of
λ
the theorem 1.4.
5. Comultiplication
Our main goal in this section is to prove theorem 1.5. Since the algebra of shifted
symmetric functions Λ∗(xky) is isomorphic to the algebra of symmetric functions Λ(xky)
with an isomorphism given by the change of variables, we will use Λ(xky) for convenience.
Without loss of generality, we may assume that all indices d are zero, i.e. we consider the
product map
ρ :Gr (H )×Gr (H )→ Gr (H).
0 even 0 odd 0
The infinite torus acts on H and thus acts on H and H in a natural way; the map ρ
even odd
is equivariant. Since the Grassmannian is equivariantly formal (see [2],[1]) we obtain that
H∗(Gr (H)) = H∗(pt)⊗H∗(Gr (H))
T 0 T 0
as a H∗(pt)-module. Using the Ku¨nneth theorem and relation C[u] = C[u ]⊗C[u ],
T even odd
we find that
H∗(Gr(H )×Gr(H )) = H∗(Gr(H ))⊗H∗(Gr(H ))
T even odd T even T odd
as HT∗(pt)-modules, where ueven = (u2k)k∈Z and uodd = (u2k+1)k∈Z.
For each l ≥ 1, let Grl(H ) and Grl(H ) be respectively the submanifolds of
0 even 0 odd
Gr (H ) and Gr (H ) consisting of points W so that theorthogonal projection π :
0 even 0 odd even,l
W → z−2[l/2]H is surjective if W ∈ Gr (H ) and π : W → z−2[l/2]−2H is
even,− 0 even odd,l odd,−
surjective if W ∈ Gr (H ). We can restrict the equivariant map (1.8) to the equivariant
0 odd
map
ρ: Grl(H )×Grl(H ) → Grl .
0 even 0 odd 0
Then the pull back bundle ρ∗E over Grl(H )×Grl(H ) splits into the direct sum of
l 0 even 0 odd
T-equivariant vector bundlesE and E , wherethe vector bundleE and E are
l,even l,odd l,even l,odd
respectively theT-equivariant bundleover Grl(H )andover Grl(H ))whosefiberover
0 even 0 odd
W isthekerneloftheorthogonalprojectionπ : W → z−2[l/2]H ifW ∈ Grl(H )
even,l even,− 0 even
and is the kernel of π : W → z−2[l/2]−2H if W ∈ Grl(H ). Similarly, the T-
odd,l odd,− 0 odd
equivariantvectorbundleρ∗H overGrl(H )×Grl(H )splitsintoadirectsumofT-
−l,−1 0 even 0 odd
equivariantbundlesF andF ,whereF andF aretheproductbundlesofthe
l,even l,odd l,even l,odd
8 JIA-MING(FRANK)LIOU,ALBERTSCHWARZ
formF ×(Grl(H )×Grl(H ))andF ×(Grl(H )×Grl(H ));F andF
even 0 even 0 odd odd 0 even 0 odd l,even l,odd
are the linear spaces spanned by {z2s : −l ≤ 2s ≤ −1} and by {z2s+1 : −l ≤ 2s+1 ≤ −1}
respectively. In other words, we have
ρ∗E = E ⊕E and ρ∗H = F ⊕F
l l,even l,odd −l,−1 l,even l,odd
as T-equivariant vector bundles over Grl(H )×Grl(H ).
0 even 0 odd
Assume that x ,··· ,x are the equivariant Chern roots of E and define the power sum
1 l l
functions p (x ,··· ,x |u), k ≥ 1 by
k 1 l
l
(5.1) p (x ,··· ,x |u) = xk −uk .
k 1 l Xh i −ii
i=1
Then the power sum function p equals to the k-dimensional component of the equivariant
k
CherncharacterofthedifferencebundleE −H ,i.e. p (x ,··· ,x |u) = chT(E −H ).
l −l,−1 k 1 l k l −l,−1
Since chT(E −H )= chT(E −F )+chT(E −F ), we find that
l −l,−1 l,even l,even l,odd l,even
(5.2) ρ∗p (x ,··· ,x |u) = p (x ,x ··· ,x |u )+p (x ,x ,··· ,x |u ).
k 1 l k 2 4 2[l/2] even k 1 3 2[(l−1)/2]+1 odd
Notice that the function (5.1) can be extended to an element of the ring Λ(xku) ( to a
symmetric function on R˜). More precisely, the formula
∞
(5.3) p (x|u) = xk −uk
k Xh i −ii
i=1
is a finite sum on R˜ and therefore specifies an element of Λ(xku). This formula is also
equivalent to (1.9) under a change of variables. By (5.2) and (5.3), we obtain the following
formula:
(5.4) ρ∗p (x|u) = p (x |u )+p (x |u )
k k even even k odd odd
where (x ) = x and (x ) = x for k ∈ Z.
even k 2k odd k 2k+1
Letψ andψ bethealgebraisomorphismsΛ(x ku ) → Λ(xku)andΛ(x ku )→
even odd even even odd odd
Λ(xku)definedbyψ (x ) = x ,ψ (u )= u andbyψ (x )= x ,ψ (u ) =
even 2k k even 2k k odd 2k+1 k even 2k+1
u fork ∈ Zrespectively. Thenψ ⊗ψ determinesanisomorphismfromH∗(Gr (H ))⊗
k even odd T 0 even
H∗(Gr (H )) to H∗(Gr (H))⊗H∗(Gr (H)). Combining the maps ρ∗and ψ ⊗ψ we
T 0 odd T 0 T 0 even odd
obtain a homomorphism
∆ :H∗(Gr (H)) → H∗(Gr (H))⊗H∗(Gr (H))
T 0 T 0 T 0
so that (1.10) holds by (5.4). This completes the proof of the theorem 1.5.
Acknowledments Both authors thank Max-Planck Institute fu¨r Mathematik in Bonn
for generous support and wonderful environment. The second author was partially sup-
ported by the NSF grant DMS-0805989.
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