# Quantum $K$-theory of $G/P$ and $K$-homology of affine Grassmannian

Chi Hong Chow  and  Naichung Conan Leung

###### Abstract.

This paper is the $K$-theoretic analogue of a recent new proof, given by the
first named author, of Peterson-Lam-Shimozono’s theorem via Savelyev’s
generalization of Seidel representations. The outcome is a new proof of Lam-
Li-Mihalcea-Shimozono’s conjecture, including its extension to the parabolic
case, which was first verified by Kato.

##  1\. introduction

Let $G$ be a simple and simply connected complex Lie group. In Schubert
calculus, an unpublished result of Peterson [35], first proved by Lam-
Shimozono [30], states that the Pontryagin product of the homology of the
affine Grassmannian $Gr_{G}$ determines completely, via an explicit ring
homomorphism defined in terms of the (affine) Schubert bases, the quantum cup
product of the quantum cohomology of any flag variety $G/P$. Recently, Chow
[13] has given a new proof of this result by computing Savelyev’s parametrized
version [38] of Seidel representations [39].

In $K$-theory, Lam-Li-Mihalcea-Shimozono [28] conjectured a similar
homomorphism for the case of $G/B$ where the bases are replaced by the
structure sheaves of the (affine) Schubert varieties. Their conjecture was
first proved by Kato [20, 22]. Kato also extended the result to the general
parabolic case [21]. In this paper, we give an alternative proof of Kato’s
result by following the approach in [13].

######  Theorem 1.1.

The map

$\begin{array}[]{ccccc}\Phi&:&K^{T}(Gr_{G})&\rightarrow&QK_{T}(G/P)[\Lambda^{-1}]\\\\[5.0pt]
&&\mathcal{O}_{wt_{\lambda}}&\mapsto&q^{\lambda+Q^{\vee}_{P}}\mathcal{O}_{\widetilde{w}}\end{array}$

is an $R(T)$-algebra homomorphism, where $\widetilde{w}$ is the minimal length
coset representative of $wW_{P}$.

As already pointed out by Lam-Li-Mihalcea-Shimozono, Theorem 1.1 implies
immediately the finiteness property of the quantum $K$-product $\star$.

######  Corollary 1.2.

For any $v_{1},v_{2}\in W^{P}$,

$\mathcal{O}_{v_{1}}\star\mathcal{O}_{v_{2}}\in
K_{T}(G/P)\otimes\mathbb{Z}[\Lambda].$

Corollary 1.2 is not as obvious as the case of quantum cohomology because the
moduli spaces of stable maps of arbitrary dimension contribute. What’s more,
as proved by Givental [18], in order for $\star$ to be associative, it is
necessary to introduce a deformation of the Poincaré pairing
$\chi_{G/P}(-\otimes-)$ by two-pointed $K$-theoretic GW invariants which is in
general an infinite sum. Thus, the finiteness must follow from a non-trivial
cancellation of terms. This issue has already been settled by Kato [20, 21,
22] and Anderson-Chen-Tseng [4]. See also the earlier work [3] of the authors
of [4] and the work of Buch-Chaput-Mihalcea-Perrin [10, 11].

## Outline of the proof

Our approach is to define a map by Gromov-Witten theory and show that it is an
$R(T)$-algebra homomorphism and has the desired form.

We first recall the proof of Peterson’s original result given in [13]. Take
$Gr_{G}$ to be Pressley-Segal’s model [36]. In loc. cit., they constructed,
for any holomorphic map $f:\Gamma\rightarrow Gr_{G}$, a holomorphic principal
$G$-bundle $P_{f}$ over $\mathbb{P}^{1}\times\Gamma$ with a trivialization
over $(\mathbb{P}^{1}\setminus\\{|z|\leqslant 1\\})\times\Gamma$. In
particular, we obtain a $G/P$-fibration $P_{f}(G/P)$ over
$\mathbb{P}^{1}\times\Gamma$ by reduction of fibers. One should think of
$P_{f}(G/P)$ as a holomorphic family of $G/P$-fibrations over $\mathbb{P}^{1}$
parametrized by $\Gamma$. For any section class $\beta$, define

$\overline{\mathcal{M}}(f,\beta):=~{}$moduli stack of holomorphic sections in
$P_{f}(G/P)$ representing $\beta$

and

$\operatorname{ev}:\overline{\mathcal{M}}(f,\beta)\rightarrow G/P$

to be the evaluation map at $\infty\in\mathbb{P}^{1}$. Thanks to the above
trivialization, $\operatorname{ev}$ is well-defined.

For our purpose, we consider two classes of $f$: the $T$-fixed points of
$Gr_{G}$ and Bott-Samelson resolutions of the affine Schubert varieties. They
give rise to the localization basis $\\{\eta_{\mu}\\}_{\mu\in Q^{\vee}}$ and
the affine Schubert basis $\\{\xi_{wt_{\lambda}}\\}_{wt_{\lambda}\in
W_{af}^{-}}$ of $H^{T}(Gr_{G})$ respectively. Denote by
$\overline{\mathcal{M}}(\mu,\beta)$ and
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ the corresponding moduli stacks
defined above. Define the Savelyev-Seidel homomorphism [38, 39]

$\Phi_{QH}:H^{T}(Gr_{G})\rightarrow QH_{T}(G/P)[\Lambda^{-1}]$

either by

$\Phi_{QH}(\eta_{\mu}):=\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\overline{\mathcal{M}}(\mu,\beta)]^{vir}$
(1.1)

or

$\Phi_{QH}(\xi_{wt_{\lambda}}):=\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\overline{\mathcal{M}}(wt_{\lambda},\beta)]^{vir}.$
(1.2)

By the virtual localization formula [19], these two definitions are
equivalent. (1.1) is used when we show that $\Phi_{QH}$ is a ring homomorphism
and (1.2) is used for the computation. The former follows from a degeneration
argument and the latter relies on the following key observation:

$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is smooth and of expected
dimension.

Since $\operatorname{ev}$ is $B$-equivariant,
$\operatorname{ev}_{*}[\overline{\mathcal{M}}(wt_{\lambda},\beta)]$ is equal
to a multiple of a Schubert class or zero depending on whether the generic
fiber of $\operatorname{ev}$ has zero or positive dimension. This reduces our
computation to a purely combinatorial problem which can be solved in a
straightforward way.

Back to the situation in the present paper, we will prove Theorem 1.1 by
adapting the above approach to the $K$-theoretic settings. By recalling the
definition of the quantum $K$-product, one expects that the $K$-theoretic
Savelyev-Seidel homomorphism should be defined by incorporating a new feature
that the contribution of each $\overline{\mathcal{M}}(\mu,\beta)$ or
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ be corrected by the two-pointed
$K$-theoretic GW invariants (see the paragraph following Corollary 1.2).
Define

$A_{G/P}:=\sum_{\beta\neq
0}q^{\beta}(\operatorname{ev}^{\beta}_{2})_{*}(\operatorname{ev}^{\beta}_{1})^{*}$

where $\operatorname{ev}^{\beta}_{1},\operatorname{ev}^{\beta}_{2}$ are the
evaluation maps on $\overline{\mathcal{M}}_{0,2}(G/P,\beta)$. The
aforementioned correction is given by $(\operatorname{id}+A_{G/P})^{-1}$.
Therefore, we define

$\Phi_{QK}:K^{T}(Gr_{G})\rightarrow QK_{T}(G/P)[\Lambda^{-1}]$

by

$\Phi_{QK}(\mathcal{O}_{wt_{\lambda}}):=(\operatorname{id}+A_{G/P})^{-1}\left(\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]\right).$

where $\mathcal{O}_{wt_{\lambda}}$ is the $K$-theoretic analogue of
$\xi_{wt_{\lambda}}$. That $\Phi_{QK}$ is a ring homomorphism follows from
similar localization and degeneration arguments as well as an argument used by
Givental [18] and Lee [31] in their proof of the $K$-theoretic WDVV equation.

The heart of the paper is the computation of
$\Phi_{QK}(\mathcal{O}_{wt_{\lambda}})$. Our strategy is to introduce a
$\mathbb{C}^{\times}$-action on $\overline{\mathcal{M}}(wt_{\lambda},\beta)$
by rescaling the domain of free loops in $G$, and apply Oprea’s stacky version
[34] of Białynicki-Birula’s theorem [8] to this action. We show that if
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$, there exists a
unique component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}}$ whose
Białynicki-Birula cell is open, and hence

$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is either empty or irreducible.

A more in-depth analysis of this component gives

$\\{\beta|~{}\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset\\}=[\lambda]+\Lambda$

and

$\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]=(\operatorname{ev}^{\beta-[\lambda]}_{2})_{*}(\operatorname{ev}^{\beta-[\lambda]}_{1})^{*}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}],\quad\beta\in[\lambda]+(\Lambda\setminus\\{0\\})$
(1.3)

where we put $[\lambda]:=\lambda+Q^{\vee}_{P}$ for simplicity, $\Lambda$ is
the semigroup of effective curve classes and
$\operatorname{ev}^{\beta-[\lambda]}_{1},\operatorname{ev}^{\beta-[\lambda]}_{2}:\overline{\mathcal{M}}_{0,2}(G/P,\beta-[\lambda])\rightarrow
G/P$ are the evaluation maps. Summing up (1.3) over all $\beta$, weighted by
$q^{\beta}$, we get

$\displaystyle\Phi_{QK}(\mathcal{O}_{wt_{\lambda}})$
$\displaystyle=(\operatorname{id}+A_{G/P})^{-1}\circ(\operatorname{id}+A_{G/P})\left(q^{[\lambda]}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}]\right)$
$\displaystyle=q^{[\lambda]}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}].$

The last term can easily be shown to be equal to
$q^{[\lambda]}\mathcal{O}_{\widetilde{w}}$ as stated in Theorem 1.1.

##  2\. Preliminaries

###  2.1. Some Lie-theoretic notations

Let $G$ be a simple and simply connected complex Lie group and $T\subset G$ a
maximal torus. Put $\mathfrak{g}:=\operatorname{Lie}(G)$ and
$\mathfrak{h}:=\operatorname{Lie}(T)$. We have the root space decomposition

$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}$

where $R$ is the set of roots associated to the pair
$(\mathfrak{g},\mathfrak{h})$ and each $\mathfrak{g}_{\alpha}$ is a one-
dimensional eigenspace with respect to the adjoint action of $\mathfrak{h}$.
Denote by $W$ the Weyl group. Fix a set $\\{\alpha_{1},\ldots,\alpha_{r}\\}$
of simple roots of $R$. Denote by $\alpha_{i}^{\vee}$ the corresponding
coroots. Define $R^{+}\subset R$ to be the set of positive roots spanned by
$\alpha_{1},\ldots,\alpha_{r}$. Define $B^{-}$ (resp. $B^{+}$) to be the Borel
subgroup of $G$ containing $T$ with Lie algebra equal to
$\mathfrak{h}\oplus\bigoplus_{\alpha\in-R^{+}}\mathfrak{g}_{\alpha}$ (resp.
$\mathfrak{h}\oplus\bigoplus_{\alpha\in R^{+}}\mathfrak{g}_{\alpha}$).

Define the affine Weyl group $W_{af}:=W\ltimes Q^{\vee}$ where
$Q^{\vee}\subset\mathfrak{h}$ is the $\mathbb{Z}$-span of $\alpha^{\vee}_{i}$,
$i=1,\ldots,r$. Typical elements of $W_{af}$ are denoted by $wt_{\lambda}$.
Define the affine simple roots
$\widetilde{\alpha}_{0},\ldots,\widetilde{\alpha}_{r}$ by
$\widetilde{\alpha}_{0}:=-\alpha_{0}+1$ and
$\widetilde{\alpha}_{i}:=\alpha_{i}$ for $i=1,\ldots,r$, where $\alpha_{0}$ is
the highest positive root.

###  2.2. Algebraic K-theory

A good reference for the following materials is [14].

Let $Y$ be a (finite-dimensional) scheme over $\mathbb{C}$ with an action of a
complex torus $T$ (which will be the maximal torus fixed in Section 2.1
throughout). Define $K_{T}(Y)$ (resp. $K^{T}(Y)$) to be the Grothendieck group
of $T$-equivariant vector bundles (resp. $T$-equivariant coherent sheaves) on
$Y$. If $Y$ is smooth and quasi-projective, they are known to be isomorphic.

Pullback and tensor product of vector bundles give rise to the pullback
operator and a ring structure on $K_{T}(Y)$ respectively. Tensor product also
defines a $K_{T}(Y)$-module structure on $K^{T}(Y)$, since for any vector
bundle $E$, $E\otimes-$ is an exact functor on the abelian category of
coherent sheaves. If $f:Y\rightarrow Z$ is a $T$-equivariant proper morphism,
we define the pushforward operator

$f_{*}:K^{T}(Y)\rightarrow K^{T}(Z)$

by

$f_{*}([\mathcal{E}]):=\sum_{i\geqslant 0}(-1)^{i}[R^{i}f_{*}(\mathcal{E})].$

In particular, if $Y$ is proper and $Z$ is a point, the corresponding operator
is denoted by $\chi_{Y}$.

There is a natural $K_{T}(pt)$-module structure on $K_{T}(Y)$ and $K^{T}(Y)$,
defined via the pullback operator associated to the structure morphism
$Y\rightarrow pt$. It is well-known that $K_{T}(pt)$ is isomorphic to the
representation ring $R(T)$ of $T$. We will adopt the latter notation
throughout the paper.

######  Remark 2.1.

In this paper, we have to work with Deligne-Mumford stacks because moduli
spaces of stable maps are not schemes in general. While the above definitions
extend to Deligne-Mumford stacks, they are not strictly necessary for the
computational aspect of this paper. We will bypass them by following the
approach explained in [31, Remark 5].

Let $\mathcal{Y}$ be a Deligne-Mumford stack arising from the moduli of stable
maps to a smooth projective variety. Consider the canonical map

$p:\mathcal{Y}\rightarrow Y$

from $\mathcal{Y}$ to its coarse moduli $Y$. By the tameness property of
$\mathcal{Y}$ (see [1]), we have

$p_{*}[\mathcal{O}_{\mathcal{Y}}]=[\mathcal{O}_{Y}].$ (2.1)

The quantum $K$-invariants considered in this paper are of the form

$\chi_{\mathcal{Y}}(\mathcal{O}^{vir}_{\mathcal{Y}}\otimes\operatorname{ev}_{1}^{*}\alpha_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\alpha_{k})$

where

  * •

$\mathcal{O}^{vir}_{\mathcal{Y}}\in K^{T}(\mathcal{Y})$ is the virtual
structure sheaf constructed in [31];

  * •

$\operatorname{ev}_{i}$ are the evaluation maps on $\mathcal{Y}$; and

  * •

$\alpha_{i}$ are some $K$-theory classes on the target space.

The key observation is that each $\operatorname{ev}_{i}$ factors through $p$,
and hence the above invariant is equal to

$\chi_{Y}(p_{*}\mathcal{O}^{vir}_{\mathcal{Y}}\otimes\operatorname{ev}_{1}^{*}\alpha_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\alpha_{k}),$

by the projection formula. For our computation, $\mathcal{Y}$ will be smooth
and of expected dimension. It follows that
$\mathcal{O}^{vir}_{\mathcal{Y}}=[\mathcal{O}_{\mathcal{Y}}]$. This allows us
to work only with the coarse moduli $Y$, by (2.1).

###  2.3. Flag varieties

A flag variety is a homogeneous space $G/P$ where $P$ is any parabolic
subgroup containing $B^{+}$. We have

$\operatorname{Lie}(P)=\operatorname{Lie}(B^{+})\oplus\bigoplus_{\alpha\in-
R_{P}^{+}}\mathfrak{g}_{\alpha}$

where $R_{P}^{+}:=R_{P}\cap R^{+}$ and $R_{P}$ is the set of roots of $P$.
Denote by $W_{P}$ the Weyl group of $P$ and by $W^{P}$ the set of minimal
length coset representatives in $W/W_{P}$. For any $v\in W^{P}$, put
$y_{v}:=vP\in G/P$. Then $\\{y_{v}\\}_{v\in W^{P}}$ is the set of $T$-fixed
points of $G/P$. Define

$\mathcal{O}_{v}:=[\mathcal{O}_{\overline{B^{-}\cdot y_{v}}}]\in K_{T}(G/P).$

######  Lemma 2.2.

$\\{\mathcal{O}_{v}\\}_{v\in W^{P}}$ is an $R(T)$-basis of $K_{T}(G/P)$.

###### Proof.

See the proof of Lemma 2.10. ∎

We recall the equivariant quantum $K$-theory of $G/P$ defined by Givental
[18]. See also the work of Lee [31] which deals with general smooth projective
varieties. Denote by $\Lambda\subset\pi_{2}(G/P)$ the semigroup of effective
curve classes. We identify $\Lambda$ with
$\bigoplus_{i=1}^{r}\mathbb{Z}_{\geqslant
0}\langle\alpha_{i}^{\vee}\rangle\subset Q^{\vee}$ via the dual of the
composition of three isomorphisms

$\left(Q^{\vee}/Q^{\vee}_{P}\right)^{*}\xrightarrow{\rho~{}\mapsto
L_{\rho}}\operatorname{Pic}(G/P)\xrightarrow{c_{1}}H^{2}(G/P)\simeq\pi_{2}(G/P)^{*}$
(2.2)

where

  * •

$Q^{\vee}_{P}:=\operatorname{Span}_{\mathbb{Z}}\\{\alpha_{i}^{\vee}|~{}\alpha_{i}\in
R_{P}\\}$;

  * •

$L_{\rho}:=G\times_{P}\mathbb{C}_{-\rho}$; and

  * •

$\mathbb{C}_{-\rho}$ is the one-dimensional representation of weight $-\rho$
on which $P$ acts by forgetting the semi-simple and unipotent parts.

We have, as abelian groups,

$QK_{T}(G/P):=K_{T}(G/P)\otimes\mathbb{Z}[[\Lambda]]$

where $\mathbb{Z}[[\Lambda]]$ is the formal completion of the group ring
$\mathbb{Z}[\Lambda]$.

######  Remark 2.3.

The reason for enlarging the standard coefficient ring $\mathbb{Z}[\Lambda]$
for quantum cohomology is to ensure that the ring product we are going to
define is well-defined. It turns out that this is unnecessary by Corollary
1.2.

What is non-trivial is the definition of the quantum $K$-product $\star$ on
$QK_{T}(G/P)$. For any $\beta\in\Lambda$ and $\gamma_{1},\ldots,\gamma_{k}\in
K_{T}(G/P)$, define

$\operatorname{KGW}^{\beta}(\gamma_{1},\ldots,\gamma_{k}):=\chi_{\overline{\mathcal{M}}_{0,k}(G/P,\beta)}(\operatorname{ev}_{1}^{*}\gamma_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\gamma_{k})\in
R(T)$

where $\overline{\mathcal{M}}_{0,k}(G/P,\beta)$ is the Deligne-Mumford moduli
stack of genus zero $k$-pointed stable maps to $G/P$ representing $\beta$.
Clearly, we can extend $\operatorname{KGW}$ to a linear map
$(QK_{T}(G/P))^{\otimes k}\rightarrow R(T)\otimes\mathbb{Z}[[\Lambda]]$ by
linearity. Take an $R(T)$-basis $\\{e_{i}\\}_{i\in I}$ of $K_{T}(G/P)$ (the
Schubert basis, for example) and denote by $\\{g^{ij}\\}_{i,j\in I}$ the
inverse of the matrix $\\{\chi_{G/P}(e_{i}\otimes e_{j})\\}_{i,j\in I}$. It is
well-known that the latter matrix is indeed invertible. Define a linear
operator

$A_{G/P}:QK_{T}(G/P)\rightarrow QK_{T}(G/P)$

by

$A_{G/P}(\gamma):=\sum_{i,j\in
I}\sum_{\beta\in\Lambda\setminus\\{0\\}}q^{\beta}g^{ij}\operatorname{KGW}^{\beta}(\gamma,e_{i})e_{j}.$

We have, for any $\gamma_{1},\gamma_{2}\in QK_{T}(G/P)$,

$\gamma_{1}\star\gamma_{2}:=\sum_{i,j\in
I}\sum_{\beta\in\Lambda}q^{\beta}g^{ij}\operatorname{KGW}^{\beta}(\gamma_{1},\gamma_{2},e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$

By [18] or [31], $\star$ defines a ring structure on $QK_{T}(G/P)$.

###  2.4. Affine Grassmannian

There are many models for the affine Grassmannian $Gr_{G}$. In this paper, we
work with Pressley-Segal’s version [36].

Define $H:=L^{2}(S^{1};\mathfrak{g})$ to be the Hilbert space of
$L^{2}$-functions on $S^{1}$ with values in $\mathfrak{g}$. We have an
orthogonal decomposition $H=H_{+}\oplus H_{-}$ where $H_{+}$ (resp. $H_{-}$)
consists of functions whose negative (resp. non-negative) Fourier coefficients
are zero. Let $\operatorname{pr}_{\pm}:H\rightarrow H_{\pm}$ denote the
orthogonal projections. Define

$Gr(H):=\\{W\subset H\text{ closed
subspaces}|~{}\operatorname{pr}_{+}|_{W}\text{ is Fredholm and
}\operatorname{pr}_{-}|_{W}\text{ is Hilbert-Schmidt}\\}.$

It is proved in [36] that $Gr(H)$ is a complex Hilbert manifold.

Fix a maximal compact subgroup $K$ of $G$. Define

$\displaystyle L_{sm}G$ $\displaystyle:=\\{\text{smooth free loops in }G\\}$
$\displaystyle L_{pol}G$ $\displaystyle:=\\{\text{polynomial free loops in
}G\\}$ $\displaystyle\Omega_{sm}K$ $\displaystyle:=\\{\text{smooth based loops
in }K\\}$ $\displaystyle\Omega_{pol}K$ $\displaystyle:=\\{\text{polynomial
based loops in }K\\}.$

Then $L_{sm}G$ (resp. $\Omega_{sm}K$) is an infinite-dimensional complex
(resp. real) Fréchet Lie group in the $C^{\infty}$-topology. Consider the
following action on $H$ by $L_{sm}G$:

$(\varphi\cdot f)(z):=\operatorname{Ad}(\varphi(z))f(z)\quad\varphi\in
L_{sm}G,~{}f\in H$

where $\operatorname{Ad}$ is the adjoint action. This action induces an
$L_{sm}G$-action on $Gr(H)$ with respect to which the stabilizer of $H_{+}\in
Gr(H)$ is equal to $L^{0}_{sm}G$, the subgroup of $L_{sm}G$ consisting of
$\varphi$ which extend to holomorphic functions defined on the unit disk.

######  Theorem 2.4.

[36, Theorem 8.6.2] There exists a diffeomorphism

$L_{sm}G\cdot H_{+}\simeq\Omega_{sm}K$

under which the sub-orbit $L_{pol}G\cdot H_{+}$ corresponds to
$\Omega_{pol}K$.

From now on, we identify the orbit $L_{sm}G\cdot H_{+}$ (resp. $L_{pol}G\cdot
H_{+}$) with $\Omega_{sm}K$ (resp. $\Omega_{pol}K$) via the above
diffeomorphism . Since $G$ is assumed to be simply connected, $\Omega_{sm}K$
is connected and so lies in the connected component $Gr(H)^{o}$ of $Gr(H)$
containing $H_{+}$. For any natural number $n$, define

$Gr^{(n)}(H):=\\{W\in Gr(H)^{o}|~{}z^{n}H_{+}\subseteq W\subseteq
z^{-n}H_{+}\\}$

and

$\Omega_{pol}^{(n)}K:=\Omega_{pol}K\cap Gr^{(n)}(H).$

Notice that $Gr^{(n)}(H)$ is a submanifold of $Gr(H)^{o}$ biholomorphic to the
type-A Grassmannian
$Gr(n\cdot\dim_{\mathbb{C}}\mathfrak{g},2n\cdot\dim_{\mathbb{C}}\mathfrak{g})$
and $\Omega_{pol}^{(n)}K$ is a possibly singular closed subvariety of
$Gr^{(n)}(H)$.

######  Theorem 2.5.

[36, Theorem 8.3.3] $\Omega_{pol}K=\bigcup_{n=0}^{\infty}\Omega_{pol}^{(n)}K$.

Consider the action on $\Omega_{pol}K$ by the maximal torus $T\subset G$. It
is easy to see that the fixed-point set $(\Omega_{pol}K)^{T}$ is equal to
$\\{x_{\mu}\\}_{\mu\in Q^{\vee}}$ where $x_{\mu}$ is the cocharacter of $T$
associated to any element $\mu\in Q^{\vee}$. Define

$\mathcal{B}^{0,-}_{sm}:=\\{\varphi\in L^{0}_{sm}G|~{}\varphi(0)\in B^{-}\\}.$

(By abuse of notation, the holomorphic extension of any $\varphi\in
L^{0}_{sm}G$ is denoted by the same symbol.)

######  Theorem 2.6.

[36, Theorem 8.6.3]

  1. (1)

(Bruhat decomposition) We have

$\Omega_{pol}K=\bigcup_{\mu\in Q^{\vee}}\mathcal{B}_{sm}^{0,-}\cdot x_{\mu}.$

  2. (2)

For any $\mu\in Q^{\vee}$, the orbit $\mathcal{B}_{sm}^{0,-}\cdot x_{\mu}$ is
biholomorphic to a complex affine space.

######  Definition 2.7.

Define the affine Grassmannian $Gr_{G}:=\Omega_{pol}K$.

Now we define, following [27], the $K$-homology $K^{T}(Gr_{G})$ of $Gr_{G}$.
Notice that the definition does not follow directly from Section 2.2 where we
only deal with finite-dimensional schemes. By Theorem 2.5, $Gr_{G}$ is the
union of the chain of projective varieties

$\Omega_{pol}^{(0)}K\subset\Omega_{pol}^{(1)}K\subset\Omega_{pol}^{(2)}K\subset\cdots.$

This chain induces a direct system of $R(T)$-modules

$K^{T}(\Omega_{pol}^{(0)}K)\rightarrow K^{T}(\Omega_{pol}^{(1)}K)\rightarrow
K^{T}(\Omega_{pol}^{(2)}K)\rightarrow\cdots.$

######  Definition 2.8.

Define

$K^{T}(Gr_{G}):=\varinjlim_{n}K^{T}(\Omega_{pol}^{(n)}K).$

Denote by $W_{af}^{-}$ the set of minimal length coset representatives in
$W_{af}/W$. Notice that the map $W_{af}^{-}\rightarrow Q^{\vee}$ sending
$wt_{\lambda}$ to $w(\lambda)$ is bijective.

######  Definition 2.9.

  1. (1)

Let $\mu\in Q^{\vee}$. Define

$\mathcal{O}_{\mu}:=[\mathcal{O}_{x_{\mu}}]\in K^{T}(Gr_{G}).$

  2. (2)

Let $wt_{\lambda}\in W_{af}^{-}$. Define

$\mathcal{O}_{wt_{\lambda}}:=[\mathcal{O}_{\overline{\mathcal{B}^{0,-}_{sm}\cdot
x_{w(\lambda)}}}]\in K^{T}(Gr_{G})$

where $\overline{\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}}$ is the Zariski
closure of $\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}$ taken in
$\Omega_{pol}^{(n)}K$ for some large $n$.

######  Lemma 2.10.

  1. (1)

$\\{\mathcal{O}_{wt_{\lambda}}\\}_{wt_{\lambda}\in W_{af}^{-}}$ is an
$R(T)$-basis of $K^{T}(Gr_{G})$.

  2. (2)

There exists a monomorphism

$K^{T}(Gr_{G})\hookrightarrow\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$ (2.3)

which fixes every $\mathcal{O}_{\mu}$.

###### Proof.

(1) follows from an argument of Kumar [27] based on the following two standard
results:

  1. (i)

(The excision sequence) If we have
$U\xhookrightarrow{i}X\xhookleftarrow{j}X\setminus U$ where $X$ is projective
and $U$ is open, then the sequence

$K^{T}(X\setminus
U)\xrightarrow{j_{*}}K^{T}(X)\xrightarrow{i^{*}}K^{T}(U)\rightarrow 0$ (2.4)

is exact.

  2. (ii)

(The Thom isomorphism: a special case) If $T$ acts on a vector space
$\mathbb{C}^{r}$ linearly, then $K^{T}(\mathbb{C}^{r})$ is freely generated by
$[\mathcal{O}_{\mathbb{C}^{r}}]$.

Proofs of (i) and (ii) can be found in [14]. To prove (2), it suffices to show
that every $\mathcal{O}_{wt_{\lambda}}$ is an
$\operatorname{Frac}(R(T))$-linear combination of $\mathcal{O}_{w(\lambda)}$
and some other $\mathcal{O}_{w^{\prime}t_{\lambda^{\prime}}}$ with
$\ell(w^{\prime}t_{\lambda^{\prime}})<\ell(wt_{\lambda})$. This follows from
(2.4) and a local computation in $K^{T}(\mathbb{C}^{\ell(wt_{\lambda})})$. ∎

######  Definition 2.11.

Define an $R(T)$-algebra structure
$\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}$
on $\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$ by

$\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}}:=\mathcal{O}_{\mu_{1}+\mu_{2}}\quad\mu_{1},\mu_{2}\in
Q^{\vee}.$ (2.5)

######  Lemma 2.12.

Via the monomorphism (2.3), $K^{T}(Gr_{G})$ is a sub-$R(T)$-algebra.

###### Proof.

This is proved by Lam-Schilling-Shimozono [29]. Notice that they first defined
an $R(T)$-algebra structure on $K^{T}(Gr_{G})$ and verified (2.5).

Alternatively, we show that it follows from our proof of Theorem 1.1, although
the statement of this theorem assumes the lemma we are proving. Take
$P=B^{+}$. In Section 4.1, we will construct an $R(T)$-algebra homomorphism

$\Phi:\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle\rightarrow
QK_{T}(G/B^{+})[\Lambda^{-1}]\otimes\operatorname{Frac}(R(T)).$

In Section 4.2, 4.3 and 4.4, we will show

$\Phi(\mathcal{O}_{wt_{\lambda}})=q^{\lambda}\mathcal{O}_{w}$

where $\mathcal{O}_{wt_{\lambda}}$ is regarded as an element of the domain of
$\Phi$ via (2.3). This implies that $\Phi$ sends an $R(T)$-basis of
$K^{T}(Gr_{G})$ injectively into an $R(T)$-basis of
$K_{T}(G/B^{+})\otimes\mathbb{Z}[\Lambda^{\pm}]$, and hence

$K^{T}(Gr_{G})=\Phi^{-1}(QK_{T}(G/B^{+})[\Lambda^{-1}])$

which is clearly a sub-$R(T)$-algebra of $\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$. ∎

##  3\. The key moduli

###  3.1. The G/P-fibration

The following theorem is the starting point of everything.

######  Theorem 3.1.

[36, Theorem 8.10.2] For any complex manifold $\Gamma$, there exists a
bijection between

  1. (1)

the set of holomorphic maps $\Gamma\rightarrow\Omega_{sm}K$; and

  2. (2)

the set of isomorphism classes of holomorphic principal $G$-bundles over
$\mathbb{P}^{1}\times\Gamma$ with a trivialization over
$(\mathbb{P}^{1}\setminus\\{|z|\leqslant 1\\})\times\Gamma$.

We will need the $G/P$-bundle associated to the bundle in (2). To simplify the
exposition on how this bundle is constructed, we introduce some Banach Lie
groups as in [13]. Define

$\displaystyle D_{0}$ $\displaystyle:=\\{z\in\mathbb{C}|~{}|z|\leqslant 2\\}$
$\displaystyle D_{\infty}$
$\displaystyle:=\\{z\in\mathbb{C}\cup\\{\infty\\}|~{}1/2\leqslant|z|\\}$
$\displaystyle A$ $\displaystyle:=D_{0}\cap D_{\infty},$

and

$\displaystyle\mathcal{G}$ $\displaystyle:=\\{\varphi:A\rightarrow
G|~{}\varphi\text{ is continuous and }\varphi|_{\mathring{A}}\text{ is
holomorphic}\\}$ $\displaystyle\mathcal{G}^{0}$
$\displaystyle:=\\{\varphi:D_{0}\rightarrow G|~{}\varphi\text{ is continuous
and }\varphi|_{\mathring{D}_{0}}\text{ is holomorphic}\\}$
$\displaystyle\mathcal{G}^{\infty}$
$\displaystyle:=\\{\varphi:D_{\infty}\rightarrow G|~{}\varphi\text{ is
continuous and }\varphi|_{\mathring{D}_{\infty}}\text{ is holomorphic}\\}.$

These groups are complex Banach Lie groups in the $C^{0}$-topology. Moreover,
$\mathcal{G}^{0}$ and $\mathcal{G}^{\infty}$ naturally embed into
$\mathcal{G}$ as subgroups in the sense of [9].

Define $\widetilde{P(G/P)}$ to be the pushout of the diagram

$\mathring{D}_{0}\times\mathcal{G}\times
G/P$$\mathring{D}_{0}\times\mathcal{G}\times
G/P$$\mathring{A}\times\mathcal{G}\times G/P$inclusion
$~{}(z,\phi,y)\mapsto(z^{-1},\phi,\phi(z)\cdot y)$

.

We call the left (resp. right) copy $\mathring{D}_{0}\times\mathcal{G}\times
G/P$ the $0$-chart (resp. $\infty$-chart). We have a map

$\widetilde{\pi}:\widetilde{P(G/P)}\rightarrow\mathbb{P}^{1}\times\mathcal{G}$

defined by forgetting the factor $G/P$ in each of these charts.

Define a left $\mathcal{G}^{\infty}$-action and a right
$\mathcal{G}^{0}$-action on $\widetilde{P(G/P)}$ by

$\psi^{\infty}\cdot(z,\phi,y)\cdot\psi^{0}:=\left\\{\begin{array}[]{ll}(z,\psi^{\infty}\phi\psi^{0},\psi^{0}(z)^{-1}\cdot
y)&0\text{-chart}\\\ (z,\psi^{\infty}\phi\psi^{0},\psi^{\infty}(z^{-1})\cdot
y)&\infty\text{-chart}\\\ \end{array}\right.$

for any $\psi^{0}\in\mathcal{G}^{0}$ and
$\psi^{\infty}\in\mathcal{G}^{\infty}$. It is easy to see that these actions
are free and commute with each other. Moreover, $\widetilde{\pi}$ is
equivariant where the action on $\mathbb{P}^{1}$ is assumed to be trivial.
Define

$P(G/P):=\widetilde{P(G/P)}/\mathcal{G}^{0}$

and

$\pi:P(G/P)\rightarrow\mathbb{P}^{1}\times(\mathcal{G}/\mathcal{G}^{0})$

to be the map induced by $\widetilde{\pi}$. It is straightforward to verify
that

  1. (1)

the left $\mathcal{G}^{\infty}$-action on $\widetilde{P(G/P)}$ induces a left
$\mathcal{G}^{\infty}$-action on $P(G/P)$;

  2. (2)

$\pi$ is a $\mathcal{G}^{\infty}$-equivariant $G/P$-fibration; and

  3. (3)

the $\infty$-chart of $\widetilde{P(G/P)}$ induces an $\infty$-chart of
$P(G/P)$ in the obvious sense.

Observe that the above construction works not only for $G/P$ but also any
$G$-spaces. In particular, if we take the $G$-equivariant line bundle
$L_{\rho}=G\times_{P}\mathbb{C}_{-\rho}$ associated to any
$\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$ (see Section 2.3), we obtain a line
bundle $\mathcal{L}_{\rho}$ on $P(G/P)$ which restricts to $L_{\rho}$ on each
fiber of $\pi$. Line bundles of this form will be useful in the next
subsection.

###  3.2. Definition of the moduli

Following [13], we introduce two moduli spaces
$\overline{\mathcal{M}}(\mu,\beta)$ and
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ which will be used for the
definition and computation of the $R(T)$-algebra homomorphism $\Phi$ stated in
Theorem 1.1 respectively. Recall the fibration $P(G/P)$ defined in the last
subsection.

Let $\mu\in Q^{\vee}$. We have the associated cocharacter $x_{\mu}$ of $T$
which is naturally an element of $\mathcal{G}$. By abuse of notation, the
corresponding point in $\mathcal{G}/\mathcal{G}^{0}$ is denoted by the same
symbol. Define

$P_{{\mu}}(G/P):=P(G/P)|_{\mathbb{P}^{1}\times\\{x_{\mu}\\}}$

and

$\pi_{\mu}:P_{{\mu}}(G/P)\rightarrow\mathbb{P}^{1}$

to be the map induced by $\pi$. By [13, Lemma 3.2], it is a smooth projective
variety. Define $D_{\mu}:=\pi_{\mu}^{-1}(\infty)$ and
$\iota_{\mu}:D_{\mu}\hookrightarrow P_{{\mu}}(G/P)$ to be the inclusion.

######  Definition 3.2.

Let $\beta\in\pi_{2}(G/P)$.

  1. (1)

Define

$\overline{\mathcal{M}}(\mu,\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,1}(P_{{\mu}}(G/P),\widetilde{\beta})\times_{(\operatorname{ev}_{1},\iota_{\mu})}D_{\mu}$

where $\widetilde{\beta}$ runs over all classes in $\pi_{2}(P_{{\mu}}(G/P))$
satisfying

    1. (i)

$(\pi_{\mu})_{*}\widetilde{\beta}=[\mathbb{P}^{1}]\in\pi_{2}(\mathbb{P}^{1})$;
and

    2. (ii)

$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.

(The line bundles $\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section
3.1 and Section 2.3 respectively.)

  2. (2)

Define

$\operatorname{ev}:\overline{\mathcal{M}}(\mu,\beta)\rightarrow D_{\mu}\simeq
G/P$

to be the morphism induced by the evaluation map $\operatorname{ev}_{1}$ on
$\overline{\mathcal{M}}_{0,1}(P_{{\mu}}(G/P),\widetilde{\beta})$.

Next we define $\overline{\mathcal{M}}(wt_{\lambda},\beta)$. Define

$\mathcal{B}^{0,-}:=\\{\varphi\in\mathcal{G}^{0}|~{}\varphi(0)\in B^{-}\\}.$

For any affine simple root $\widetilde{\alpha}_{i}$, $i=0,\ldots,r$ (see
Section 2.1), there exists a unique connected subgroup
$\mathcal{P}_{\widetilde{\alpha}_{i}}$ of $\mathcal{G}$ with
$\mathcal{B}^{0,-}\subset\mathcal{P}_{\widetilde{\alpha}_{i}}$ such that

$\operatorname{Lie}(\mathcal{P}_{\widetilde{\alpha}_{i}})=\left\\{\begin{array}[]{ll}\operatorname{Lie}(\mathcal{B}^{0,-})\oplus
z^{-1}\mathfrak{g}_{-\alpha_{0}}&i=0\\\\[10.00002pt]
\operatorname{Lie}(\mathcal{B}^{0,-})\oplus\mathfrak{g}_{\alpha_{i}}&i=1,\ldots,r\\\
\end{array}\right..$

For any $wt_{\lambda}\in W_{af}^{-}$, choose a reduced word decomposition
$(i_{1},\ldots,i_{\ell(wt_{\lambda})})$ of it. Define the associated Bott-
Samelson variety

$\Gamma_{wt_{\lambda}}:=\mathcal{P}_{\widetilde{\alpha}_{i_{1}}}\times_{\mathcal{B}^{0,-}}\cdots\times_{\mathcal{B}^{0,-}}\mathcal{P}_{\widetilde{\alpha}_{i_{\ell(wt_{\lambda})}}}/\mathcal{B}^{0,-}.$

It is easy to see that $\Gamma_{wt_{\lambda}}$ is a smooth projective variety
with a structure of iterated $\mathbb{P}^{1}$-bundles. Define a holomorphic
map

$f_{wt_{\lambda}}:\Gamma_{wt_{\lambda}}\rightarrow\mathcal{G}/\mathcal{G}^{0}$

by

$f_{wt_{\lambda}}([\varphi_{1}:\cdots:\varphi_{\ell(wt_{\lambda})}]):=\varphi_{1}\cdots\varphi_{\ell(wt_{\lambda})}\mathcal{G}^{0}.$

Define

$P_{wt_{\lambda}}(G/P):=(\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}})\times_{(\operatorname{id}\times
f_{wt_{\lambda}},\pi)}P(G/P)$

and

$\pi_{wt_{\lambda}}:P_{wt_{\lambda}}(G/P)\rightarrow\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}}$

to be the map induced by $\pi$. By [13, Lemma 3.2], $P_{wt_{\lambda}}(G/P)$ is
a smooth projective variety. Define
$D_{wt_{\lambda}}:=\pi_{wt_{\lambda}}^{-1}(\\{\infty\\}\times\Gamma_{wt_{\lambda}})$
and $\iota_{wt_{\lambda}}:D_{wt_{\lambda}}\hookrightarrow
P_{wt_{\lambda}}(G/P)$ to be the inclusion. Then $D_{wt_{\lambda}}$ is a
smooth divisor of $P_{wt_{\lambda}}(G/P)$ and isomorphic to
$\Gamma_{wt_{\lambda}}\times G/P$ via the $\infty$-chart of
$P_{wt_{\lambda}}(G/P)$.

######  Definition 3.3.

Let $\beta\in\pi_{2}(G/P)$.

  1. (1)

Define

$\overline{\mathcal{M}}(wt_{\lambda},\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,1}(P_{wt_{\lambda}}(G/P),\widetilde{\beta})\times_{(\operatorname{ev}_{1},\iota_{wt_{\lambda}})}D_{wt_{\lambda}}$

where $\widetilde{\beta}$ runs over all classes in
$\pi_{2}(P_{wt_{\lambda}}(G/P))$ satisfying

    1. (i)

$(\pi_{wt_{\lambda}})_{*}\widetilde{\beta}=[\mathbb{P}^{1}\times\\{pt\\}]\in\pi_{2}(\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}})$;
and

    2. (ii)

$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.

  2. (2)

Define

$\operatorname{ev}:\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow G/P$

to be the composite

$\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow
D_{wt_{\lambda}}\simeq\Gamma_{wt_{\lambda}}\times G/P\rightarrow G/P$

where the first arrow is induced by the evaluation map $\operatorname{ev}_{1}$
on $\overline{\mathcal{M}}_{0,1}(P_{wt_{\lambda}}(G/P),\widetilde{\beta})$ and
the second arrow is the canonical projection.

In order to compute $\Phi$, we have to establish some geometric properties of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. This will be done in Section
3.5. The intermediate subsections 3.3 and 3.4 will serve as preparations.

###  3.3. An extra torus action on the moduli

We define an algebraic $\mathbb{C}^{\times}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. To avoid confusion with other
actions, we will introduce the subscript “$t$” for the action.

The desired action is defined in several steps:

  1. (1)

Define an $S^{1}_{t}$-action on $\mathcal{G}$ by

$(t\cdot\varphi)(z):=\varphi(tz)\quad t\in
S^{1}_{t},~{}\varphi\in\mathcal{G}.$

Observe that this action preserves the subgroups $\mathcal{G}^{0}$,
$\mathcal{G}^{\infty}$, $\mathcal{B}^{0,-}$ and
$\mathcal{P}_{\widetilde{\alpha}_{i}}$.

  2. (2)

Define an $S^{1}_{t}$-action on $\widetilde{P(G/P)}$ by

$t\cdot(z,\phi,y)=\left\\{\begin{array}[]{cl}(t^{-1}z,t\cdot\phi,y)&0\text{-chart}\\\
(tz,t\cdot\phi,y)&\infty\text{-chart}\end{array}\right..$

It is straightforward to check that this action descends to an
$S^{1}_{t}$-action on $P(G/P)$ which satisfies the following properties:

    1. (i)

it is compatible with the $\mathcal{G}^{\infty}$-action in the sense that

$t\cdot(\psi^{\infty}\cdot x)=(t\cdot\psi^{\infty})\cdot(t\cdot x)$

for any $t\in S^{1}_{t}$, $\psi^{\infty}\in\mathcal{G}^{\infty}$ and $x\in
P(G/P)$; and

    2. (ii)

the map
$\pi:P(G/P)\rightarrow\mathbb{P}^{1}\times(\mathcal{G}/\mathcal{G}^{0})$ is
equivariant where $S^{1}_{t}$ acts on $\mathbb{P}^{1}$ by

$t\cdot z=\left\\{\begin{array}[]{cl}t^{-1}z&0\text{-chart}\\\
tz&\infty\text{-chart}\end{array}\right..$

  3. (3)

Consider the $S^{1}_{t}$-action on $\Gamma_{wt_{\lambda}}$ induced by the one
defined in (1). Then $f_{wt_{\lambda}}$ is equivariant.

  4. (4)

It follows that there is an induced $S^{1}_{t}$-action on
$P_{wt_{\lambda}}(G/P)$. It is not hard to show that this $S^{1}_{t}$-action
extends to a unique algebraic $\mathbb{C}^{\times}_{t}$-action. See Remark 3.4
below.

  5. (5)

Clearly, $D_{wt_{\lambda}}$ is $\mathbb{C}^{\times}_{t}$-invariant, and hence
the $\mathbb{C}^{\times}_{t}$-action in (4) induces a
$\mathbb{C}^{\times}_{t}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$.

######  Remark 3.4.

Every $S^{1}$-action on a smooth projective variety by biholomorphisms extends
to a unique holomorphic $\mathbb{C}^{\times}$-action. In this paper, we
require the latter action to be algebraic. One way of showing this is to
construct a lift of the given $S^{1}$-action on an ample line bundle. Observe
that the spaces in question, including $\Gamma_{wt_{\lambda}}$ and
$P_{wt_{\lambda}}(G/P)$, have a structure of iterated fibrations such that the
fibers at each step are Fano. Thus, it suffices to deal with the following
situation. Let $F\rightarrow E\xrightarrow{\pi}B$ be a fibration with $F$
being Fano. Suppose $S^{1}$ acts on $E$ and $B$ such that $\pi$ is equivariant
and $B$ admits an equivariant ample line bundle $\mathcal{L}_{B}$. For
sufficiently large $N$, the line bundle
$\mathcal{L}_{E}:=\omega_{E/B}^{\vee}\otimes\pi^{*}\left(\mathcal{L}_{B}^{\otimes
N}\right)$ is ample. Then $S^{1}$ acts on $\mathcal{L}_{E}$ naturally because
$\omega_{E/B}^{\vee}$ is formed out of the vertical tangent bundle of $\pi$
and $\pi$ is equivariant.

###  3.4. Constant sections

Let $\mu\in Q^{\vee}$. Recall the point
$x_{\mu}\in\mathcal{G}/\mathcal{G}^{0}$. Define

$P_{\mu}:=\\{g\in G|~{}g\cdot x_{\mu}=x_{\mu}\\}.$

######  Lemma 3.5.

$P_{\mu}$ is a parabolic subgroup with Lie algebra

$\operatorname{Lie}(P_{\mu})=\mathfrak{h}\oplus\bigoplus_{\alpha(\mu)\leqslant
0}\mathfrak{g}_{\alpha}.$ (3.1)

###### Proof.

First notice that $P_{\mu}$ is an algebraic subgroup of $G$. It is clear that
the RHS of (3.1) is a parabolic subalgebra and so defines a parabolic subgroup
$P^{\prime}$. Then $P^{\prime}\subseteq P_{\mu}$, and hence $P_{\mu}$ is
connected. Suppose $P^{\prime}\subsetneq P_{\mu}$. Then there exists
$v\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$ for some $\alpha\in R$ with
$\alpha(\mu)>0$ such that $g:=\exp(v)\in P_{\mu}$. This implies that the
holomorphic function $z\mapsto x_{\mu}(z^{-1})gx_{\mu}(z)$ extends to a
holomorphic function on $\mathbb{P}^{1}$. Since $G$ is affine, this function
is constant but this is impossible. ∎

Recall $P_{{\mu}}(G/P)=P(G/P)|_{\mathbb{P}^{1}\times\\{x_{\mu}\\}}$. Observe
it can also be defined as the pushout of the diagram

$\mathbb{C}\times G/P$$\mathbb{C}\times G/P$$\mathbb{C}^{\times}\times
G/P$inclusion $~{}(z,y)\mapsto(z^{-1},x_{\mu}(z)\cdot y)$

.

Notice $x_{\mu}\in(\mathcal{G}/\mathcal{G}^{0})^{S^{1}_{t}}$, and hence the
$S^{1}_{t}$-action preserves $P_{{\mu}}(G/P)$. One checks easily that the
induced action on $P_{{\mu}}(G/P)$, which actually extends to a
$\mathbb{C}^{\times}_{t}$-action, reads

$t\cdot(z,y)=\left\\{\begin{array}[]{cl}(t^{-1}z,x_{\mu}(t)\cdot
y)&0\text{-chart}\\\ (tz,y)&\infty\text{-chart}\end{array}\right..$

Consider the $\mathbb{C}^{\times}$-action on $G/P$ induced by the cocharacter
$x_{\mu}:\mathbb{C}^{\times}\rightarrow T$. We will write
$\mathbb{C}^{\times}_{\mu}$ in place of $\mathbb{C}^{\times}$ in this context.
Let $y\in G/P$. Since $G/P$ is complete, the morphism
$\mathbb{C}^{\times}\rightarrow G/P:z\mapsto x_{\mu}(z^{-1})\cdot y$ extends
to a morphism defined on $\mathbb{C}$.

######  Definition 3.6.

Let $y\in G/P$. Define a section $u_{y}$ of $P_{\mu}(G/P)$ by

$u_{y}(z):=\left\\{\begin{array}[]{cl}(z,x_{\mu}(z^{-1})\cdot
y)&0\text{-chart}\\\ (z,y)&\infty\text{-chart}\end{array}\right..$

Any sections of the form $u_{y}$ are called constant sections (meaning
constant in the $\infty$-chart).

######  Lemma 3.7.

Let $u:\mathbb{P}^{1}\rightarrow P_{{\mu}}(G/P)$ be a holomorphic section.
Suppose for any $t\in\mathbb{C}^{\times}_{t}$, there exists
$\phi\in\operatorname{Aut}(\mathbb{P}^{1})$ such that for any
$z\in\mathbb{P}^{1}$,

$t\cdot u(z)=u(\phi(z)).$

Then $u$ is a constant section.

###### Proof.

Restricting $u$ to the $0$-chart and $\infty$-chart, we get two maps
$u_{0},u_{\infty}:\mathbb{C}\rightarrow G/P$ satisfying

$u_{\infty}(z^{-1})=x_{\mu}(z)\cdot u_{0}(z)\quad\text{for any
}z\in\mathbb{C}^{\times}.$

The given condition implies $u_{\infty}$ is constantly equal to a point $y\in
G/P$. This forces $u\equiv u_{y}$. ∎

Denote by $F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$ the unique component of
$(G/P)^{\mathbb{C}^{\times}_{\mu}}$ whose normal bundle has only positive
weights. The superscript “attr” will be explained in Definition 3.11.

######  Lemma 3.8.

$P_{\mu}$ preserves and acts transitively on
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$.

###### Proof.

It suffices to look at the infinitesimal action. Let $y\in
F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$. Notice that $T_{y}(G/P)$ is a
direct sum of weight spaces (with respect to the
$\mathbb{C}^{\times}_{\mu}$-action) of non-negative weights and the set of
these weights (counted with multiplicities) is a subset of the set of weights
of the $\mathbb{C}^{\times}_{\mu}$-module $\mathfrak{g}$. By definition,
$\operatorname{Lie}(P_{\mu})$ contains all non-positive weights of
$\mathfrak{g}$. It follows that the weights contributed by the infinitesimal
action of $P_{\mu}$ are precisely all the zero weights. ∎

######  Definition 3.9.

Define $\beta_{\mu}\in\pi_{2}(G/P)$ to be the unique element such that

$\deg(u^{*}\mathcal{L}_{\rho})=\langle\beta_{\mu},c_{1}(L_{\rho})\rangle$

for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$ where $u$ is the constant section
of $P_{{\mu}}(G/P)$ corresponding to a point in
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$. (The line bundles
$\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section 3.1 and Section 2.3
respectively.)

###  3.5. Some properties of the moduli

######  Proposition 3.10.

[13, Proposition 4.5] The stack $\overline{\mathcal{M}}(wt_{\lambda},\beta)$
is smooth and of expected dimension.

In what follows, we prove some further properties of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. More precisely, we determine the
set of $\beta$ for which
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$ and show that
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible for these $\beta$.
Our approach is to apply Oprea’s stacky version [34] of a theorem of
Białynicki-Birula [8] to the $\mathbb{C}^{\times}_{t}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ defined in Section 3.3 and study
the geometry of a particular fixed-point component. It turns out that this
component is determined by $\overline{\mathcal{M}}_{0,2}(G/P,\beta^{\prime})$
for some other $\beta^{\prime}$ and
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$ defined in Section 3.4.

######  Definition 3.11.

Let $\mathbb{C}^{\times}$ act on a smooth Deligne-Mumford stack $\mathcal{X}$.
A component $\mathcal{F}$ of $\mathcal{X}^{\mathbb{C}^{\times}}$ is said to be
attractive if for a (and hence any) geometric point $x\in\mathcal{F}$, the
weights (more precisely, the orbi-weights) of the tangent space
$T_{x}\mathcal{X}$ with respect to the $\mathbb{C}^{\times}$-action are all
non-negative.

######  Theorem 3.12.

[8] Let $X$ be a smooth quasi-projective variety with a
$\mathbb{C}^{\times}$-action and $F$ an attractive component of
$X^{\mathbb{C}^{\times}}$. There exists a unique
$\mathbb{C}^{\times}$-invariant open subscheme $U$ of $X$ containing $F$ which
is isomorphic to a $\mathbb{C}^{\times}$-equivariant affine fibration over
$F$.

For other components of $X^{\mathbb{C}^{\times}}$, there are similar affine
fibrations which are in general locally closed subschemes. Białynicki-Birula
also showed that if $X$ is proper, these subschemes form a decomposition of
$X$. His result has been generalized by Oprea [34] to Deligne-Mumford stacks.
For our purpose, we only need the following application.

######  Theorem 3.13.

Let $\mathcal{X}$ be a non-empty proper smooth Deligne-Mumford stack with a
$\mathbb{C}^{\times}$-action. Suppose $\mathcal{X}$ admits a
$\mathbb{C}^{\times}$-equivariant étale atlas. Then
$\mathcal{X}^{\mathbb{C}^{\times}}$ has an attractive component. It is unique
if and only if $\mathcal{X}$ is irreducible.

Denote by $F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ the
unique attractive component of
$\Gamma_{wt_{\lambda}}^{\mathbb{C}^{\times}_{t}}$.

######  Lemma 3.14.

$f_{wt_{\lambda}}(F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}}))\subseteq
G\cdot x_{w(\lambda)}$.

###### Proof.

Let $\gamma\in\Gamma_{wt_{\lambda}}$ be the unique point such that
$f_{wt_{\lambda}}(\gamma)=x_{w(\lambda)}$. The result follows from the
observations that $\mathcal{B}^{0,-}\cdot\gamma$ is open and the weights of
$\operatorname{Lie}(B^{-})$ (resp.
$\operatorname{Lie}(\mathcal{B}^{0,-})/\operatorname{Lie}(B^{-})$) are all
zero (resp. positive). ∎

By Lemma 3.8,
$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ is
well-defined. Consider the diagram

$F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$$G\cdot
x_{w(\lambda)}\simeq
G/P_{w(\lambda)}$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$$G/P$$f$$\pi$$j$
(3.2)

where

  * •

$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ and $P_{w(\lambda)}$ are
defined in Section 3.4;

  * •

$f$ is the restriction of $f_{wt_{\lambda}}$ to
$F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ (it does land in
$G\cdot x_{w(\lambda)}$ by Lemma 3.14);

  * •

$\pi$ is the canonical projection; and

  * •

$j$ is the unique $G$-equivariant map extending the inclusion
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)\hookrightarrow G/P$.

######  Definition 3.15.

Define a smooth variety

$F_{wt_{\lambda}}:=F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})\times_{(f,\pi)}(G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P))$

and $h_{wt_{\lambda}}:F_{wt_{\lambda}}\rightarrow G/P$ to be the composite

$F_{wt_{\lambda}}\xrightarrow{f^{\prime}}G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)\xrightarrow{j}G/P$

where $f^{\prime}$ is induced by $f$ in the fiber product.

The role of $F_{wt_{\lambda}}$ is to parametrize a class of holomorphic
sections of $P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times
F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})}$. Notice
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ parametrizes a component of
the space of constant sections of $P_{{w(\lambda)}}(G/P)$. Indeed,
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ is attractive so we have

$\displaystyle\lim_{z\to 0}x_{\mu}(z^{-1})\cdot y\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)~{}\Longrightarrow~{}y\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P).$ (3.3)

Since $P(G/P)|_{\mathbb{P}^{1}\times(G\cdot x_{w(\lambda)})}\simeq
G\times_{P_{w(\lambda)}}P_{{w(\lambda)}}(G/P)$, we see that $F_{wt_{\lambda}}$
parametrizes the pullbacks of the $G$-translates of these constant sections.

######  Definition 3.16.

Let $\beta\in\pi_{2}(G/P)$. Define a morphism

$F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})\rightarrow\overline{\mathcal{M}}(wt_{\lambda},\beta)$
(3.4)

as follows.

  * •

Every point of the domain of (3.4) is of the form $(\gamma,[g:y],u)$ where

    1. (a)

$\gamma\in F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$;

    2. (b)

$[g:y]\in
G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$; and

    3. (c)

$u\in\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$

such that $f_{wt_{\lambda}}(\gamma)=g\cdot x_{w(\lambda)}$ and
$u(z_{1})=g\cdot y$, where $z_{1}$ is the first marked point on the domain of
$u$.

  * •

We send this point to $u_{1}\\#u_{2}$ where

    1. (i)

$u_{1}$ is the section of
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{\gamma\\}}\simeq
P(G/P)|_{\mathbb{P}^{1}\times\\{g\cdot x_{w(\lambda)}\\}}$ which is the
$g$-translate of the constant section of $P_{{w(\lambda)}}(G/P)$ corresponding
to $y$; and

    2. (ii)

$u_{2}$ is just $u$ but regarded as a stable map to the fiber of
$P_{wt_{\lambda}}(G/P)$ over $(\infty,\gamma)$.

  * •

If $\beta=\beta_{w(\lambda)}$, the domain of (3.4) is understood to be
$F_{wt_{\lambda}}$. In this case, the morphism is defined in a similar way.

It is easy to see that morphism (3.4) is injective. One can show that it is
even a closed immersion. But we will not use this fact.

######  Proposition 3.17.

Suppose $\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$. Then
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$ has a
unique attractive component. Set-theoretically, it is equal to the image of
morphism (3.4).

Recall $\Lambda\subset\pi_{2}(G/P)$ is the semigroup of effective curve
classes in $G/P$ and $\beta_{w(\lambda)}\in\pi_{2}(G/P)$ is defined in
Definition 3.9.

######  Corollary 3.18.

The set
$\\{\beta\in\pi_{2}(G/P)|~{}\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset\\}$
is equal to $\beta_{w(\lambda)}+\Lambda$.

###### Proof.

If $\beta=\beta_{w(\lambda)}$, then the domain of (3.4) is $F_{wt_{\lambda}}$
which is clearly non-empty. If $\beta\neq\beta_{w(\lambda)}$, then the stack
$\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$ is non-empty if
and only if $\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})$. ∎

######  Corollary 3.19.

For any $\beta\in\beta_{w(\lambda)}+\Lambda$,
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible.

###### Proof.

This follows from Proposition 3.17 and Theorem 3.13. Notice that
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ does admit a
$\mathbb{C}^{\times}_{t}$-equivariant étale atlas, provided we reparametrize
the torus $\mathbb{C}^{\times}_{t}$. (So we actually apply Theorem 3.13 to
this reparametrized action, but this will not affect our arguments.) See
Remark 4.7. ∎

####  Proof of Proposition 3.17.

First observe that the domain of (3.4) is irreducible, by a result of Kim-
Pandharipande [24], and that this morphism sends every point into
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$. It
follows that the image of (3.4) is contained in a unique component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$. Let
$\mathcal{F}$ be an attractive component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$ which
exists by Theorem 3.13. Let $u\in\mathcal{F}$. Observe that $u$ lies over a
point $p$ in a component $F$ of
$\Gamma_{wt_{\lambda}}^{\mathbb{C}^{\times}_{t}}$.

We claim $F=F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$.
Suppose the contrary. Since the $T$-action commutes with the
$\mathbb{C}^{\times}_{t}$-action (see Section 3.3(2)(i)), $T$ preserves
$\mathcal{F}$, and hence we can replace $u$ with another
$\overline{u}\in\mathcal{F}$ which is also a $T$-fixed point of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. Then $\overline{u}$ lies over a
$T$-fixed point $\overline{p}$ of $F$. Since $F\neq
F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$, the tangent space
$T_{\overline{p}}\Gamma_{wt_{\lambda}}$ contains a weight vector $v$ of
negative weight. Consider $\mathcal{B}^{0,-}\cdot\overline{p}$, the
$\mathcal{B}^{0,-}$-orbit passing through $\overline{p}$. Then $v\not\in
T_{\overline{p}}(\mathcal{B}^{0,-}\cdot\overline{p})$. By [13, Proposition
4.5], $\overline{u}$ is still unobstructed when it is regarded as a stable map
to
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times(\mathcal{B}^{0,-}\cdot\overline{p})}$
(a smooth $\mathcal{B}^{0,-}$-equivariant compactification of
$\mathcal{B}^{0,-}\cdot\overline{p}$ is not required since $\overline{u}$ is
$T$-invariant). Therefore, $v$ lifts to a weight vector in the tangent space
$T_{\overline{u}}\overline{\mathcal{M}}(wt_{\lambda},\beta)$ which has the
same weight as $v$. By assumption, the weight is negative, a contradiction.

By Lemma 3.14,
$f_{wt_{\lambda}}(F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}}))\subseteq
G\cdot x_{w(\lambda)}$ and

$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{p\\}}\simeq
P(G/P)|_{\mathbb{P}^{1}\times{\\{f_{wt_{\lambda}}(p)}\\}}\simeq
P_{{w(\lambda)}}(G/P)$

as $\mathbb{C}^{\times}_{t}$-varieties. Since the rest of the proof relies
only on constructing some deformation vector fields of $u$ in
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{p\\}}$, we may assume
$f_{wt_{\lambda}}(p)=x_{w(\lambda)}$ so that $u$ is a stable map to
$P_{{w(\lambda)}}(G/P)$ which represents a section class and is a
$\mathbb{C}^{\times}_{t}$-fixed point in the moduli. Write
$u=u_{0}\\#u_{s}\\#u_{\infty}$ where $u_{s}$ is a section and $u_{0}$ (resp.
$u_{\infty}$) is a stable map to the fiber of $P_{{w(\lambda)}}(G/P)$ over $0$
(resp. $\infty$). By Lemma 3.7, $u_{s}$ is the constant section corresponding
to a point $y\in G/P$.

We first reduce the situation to the case
$y\in(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. More precisely, we show that
there exists another stable map $u^{\prime}\in\mathcal{F}$ such that if we
write $u^{\prime}=u^{\prime}_{0}\\#u^{\prime}_{s}\\#u^{\prime}_{\infty}$ as
before, then $u^{\prime}_{s}$ is the constant section corresponding to a point
in $(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. For any
$\eta\in\mathbb{C}^{\times}$, define $y_{\eta}:=x_{\mu}(\eta^{-1})\cdot y$.
Let $u_{y_{\eta}}$ be the constant section of $P_{w(\lambda)}(G/P)$
corresponding to $y_{\eta}$. Define a morphism

$\begin{array}[]{ccc}\mathbb{C}^{\times}&\rightarrow&\overline{\mathcal{M}}(wt_{\lambda},\beta)\\\\[5.0pt]
\eta&\mapsto&u_{0}\\#u_{y_{\eta}}\\#(x_{\mu}(\eta^{-1})\cdot
u_{\infty})\end{array}$ (3.5)

By [17, Proposition 6], after a base change
$\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times}$, the above morphism
extends to a morphism defined on $\mathbb{C}$. The stable map at $\eta=0$ will
be our $u^{\prime}$.

From now on, we assume $y\in(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. We show

  1. (1)

$u_{0}$ does not exist; and

  2. (2)

$y\in F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$.

For (1), recall $\mathbb{C}^{\times}_{t}$ acts on $\mathbb{P}^{1}$, the base
of $P_{{w(\lambda)}}(G/P)$, in the following way:

$t\cdot z=\left\\{\begin{array}[]{cl}t^{-1}z&0\text{-chart}\\\
tz&\infty\text{-chart}\end{array}\right..$

Take a weight vector $\zeta\in H^{0}(\mathbb{P}^{1};T\mathbb{P}^{1})$ such
that $\zeta(0)\neq 0$. Then it has weight $-1$. It is easy to show that there
exists a weight vector $\zeta^{\prime}$ in
$H^{0}(u^{*}TP_{{w(\lambda)}}(G/P))$ which is non-tangential to $u$ such that
$\zeta^{\prime}|_{u_{s}}$ projects to $\zeta$. It follows that
$\zeta^{\prime}$ defines a weight vector in
$T_{u}\overline{\mathcal{M}}(wt_{\lambda},\beta)$ of weight $-1$, a
contradiction.

For (2), notice that $T_{y}(G/P)$ is isomorphic, as
$\mathbb{C}^{\times}_{w(\lambda)}$-modules, to a direct sum of weight spaces
of the form $\mathbb{C}_{\alpha(w(\lambda))}$ where $\alpha\in R$. If
$y\not\in F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$, then $T_{y}(G/P)$
contains a weight vector $v\in\mathbb{C}_{\alpha(w(\lambda))}$ for some
$\alpha$ with $\alpha(w(\lambda))<0$. Define a vector field $\xi\in
H^{0}(\mathbb{P}^{1};u_{s}^{*}TP_{{w(\lambda)}}(G/P))$ by

$\xi(z)=\left\\{\begin{array}[]{cl}v&0\text{-chart}\\\
z^{-\alpha(w(\lambda))}v&\infty\text{-chart}\end{array}\right..$

Since $\xi(\infty)=0$, we can extend $\xi$ trivially to a deformation vector
field $\xi^{\prime}$ of $u$. (Recall we have proved that $u_{0}$ does not
exist.) It is clear that $\xi^{\prime}$ is non-tangential to $u$, and hence it
defines a weight vector in $T_{u}\overline{\mathcal{M}}(wt_{\lambda},\beta)$
of weight $\alpha(w(\lambda))<0$, a contradiction.

Thus, every $u\in\mathcal{F}$ is contained in the set-theoretic image of
morphism (3.4) after passing to the limit of morphism (3.5). By (3.3), $u$
actually lies in the image set before passing to the limit. The proof of
Proposition 3.17 is complete. $\square$

##  4\. Proof of the main theorem

###  4.1. Construction of the homomorphism

######  Definition 4.1.

Define an $R(T)$-linear map

$\Phi:\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle\rightarrow
QK_{T}(G/P)[\Lambda^{-1}]\otimes\operatorname{Frac}(R(T))$

by

$\Phi(\mathcal{O}_{\mu}):=\sum_{i,j\in
I}\sum_{\beta\in\pi_{2}(G/P)}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(\mu,\beta)}(\mathcal{O}_{\overline{\mathcal{M}}(\mu,\beta)}^{vir}\otimes\operatorname{ev}^{*}e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j}),$

where $\mathcal{O}_{\overline{\mathcal{M}}(\mu,\beta)}^{vir}\in
K^{T}(\overline{\mathcal{M}}(\mu,\beta))$ is the virtual structure sheaf
constructed in [31]. See Section 2.3 for the definition of $\\{e_{i}\\}_{i\in
I}$, $\\{g^{ij}\\}_{i,j\in I}$ and $A_{G/P}$, and Section 3.2 for the
definition of $\overline{\mathcal{M}}(\mu,\beta)$.

In order for $\Phi$ to be well-defined, we must verify

######  Lemma 4.2.

$\Phi(\mathcal{O}_{\mu})$ lands in $QK_{T}(G/P)[\Lambda^{-1}]$.

###### Proof.

This follows from Corollary 3.18 since
$\overline{\mathcal{M}}(\mu,\beta)\subseteq\overline{\mathcal{M}}(wt_{\lambda},\beta)$
for some $wt_{\lambda}\in W_{af}^{-}$. ∎

######  Proposition 4.3.

$\Phi$ is an $R(T)$-algebra homomorphism.

###### Proof.

The proof relies heavily on Appendix A. Let $\mu_{1},\mu_{2}\in Q^{\vee}$. We
have

$\Phi(\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}})=\Phi(\mathcal{O}_{\mu_{1}+\mu_{2}})=\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}+\mu_{2}}(e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$

where $\operatorname{PKGW}$ is defined in Definition A.2. By Proposition A.3,
the last expression is equal to

$\displaystyle\sum_{i,j\in I}\sum_{i^{\prime},j^{\prime}\in
I}g^{ij}g^{i^{\prime}j^{\prime}}\operatorname{PKGW}_{\mu_{1}}\left(e_{i},(\operatorname{id}+A_{G/P})^{-1}(e_{i^{\prime}})\right)\operatorname{PKGW}_{\mu_{2}}(e_{j^{\prime}})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
$\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}})\right)(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$

Applying Proposition A.3 again, to the splitting $\mu_{1}=\mu_{1}+0$, the last
expression is equal to

$\displaystyle\sum_{i,j\in I}\sum_{i^{\prime},j^{\prime}\in
I}g^{ij}g^{i^{\prime}j^{\prime}}\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),(\operatorname{id}+A_{G/P})^{-1}(e_{i^{\prime}})\right)\operatorname{PKGW}_{\mu_{1}}(e_{j^{\prime}})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
$\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),\Phi(\mathcal{O}_{\mu_{1}})\right)(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$

But we have (cf. [13, Lemma 3.7])

$\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),\Phi(\mathcal{O}_{\mu_{1}})\right)=\sum_{\beta\in\Lambda}q^{\beta}\operatorname{KGW}^{\beta}(\Phi(\mathcal{O}_{\mu_{1}}),\Phi(\mathcal{O}_{\mu_{2}}),e_{i}).$

Therefore,

$\Phi(\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}})=\Phi(\mathcal{O}_{\mu_{1}})\star\Phi(\mathcal{O}_{\mu_{2}})$

as desired. ∎

###  4.2. Step 1 of the computation: reducing to the initial term

The main result of this subsection is Proposition 4.8. Let $wt_{\lambda}\in
W_{af}^{-}$. Recall the map
$f_{wt_{\lambda}}:\Gamma_{wt_{\lambda}}\rightarrow\mathcal{G}/\mathcal{G}^{0}$
defined in Section 3.2. By abuse of notation, we also denote by
$f_{wt_{\lambda}}$ the composite

$\Gamma_{wt_{\lambda}}\xrightarrow{f_{wt_{\lambda}}}\mathcal{G}/\mathcal{G}^{0}\rightarrow
L_{sm}G/L_{sm}^{0}G\xrightarrow{\sim}\Omega_{sm}K$

where the second arrow is the obvious map and the third is the diffeomorphism
in Theorem 2.4.

######  Lemma 4.4.

$f_{wt_{\lambda}}$ is a $\mathcal{B}^{0,-}$-equivariant resolution of
$\overline{\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}}$. In particular, the
image of $f_{wt_{\lambda}}$ lies in $Gr_{G}=\Omega_{pol}K$.

###### Proof.

First notice that for any $n\in\mathbb{N}$ and $i=0,\ldots,r$, we have

$\mathcal{P}_{\widetilde{\alpha}_{i}}\cdot Gr^{(n)}(H)\subseteq
Gr^{(n+1)}(H).$

It follows that $f_{wt_{\lambda}}$ lands in $Gr^{(N)}(H)$ for sufficiently
large $N$. In particular, $f_{wt_{\lambda}}$ is algebraic. Since
$wt_{\lambda}\in W_{af}^{-}$ and the word defining $\Gamma_{wt_{\lambda}}$ is
reduced, there exists a unique point $\gamma\in\Gamma_{wt_{\lambda}}$ such
that $f_{wt_{\lambda}}(\gamma)=x_{w(\lambda)}$. Moreover, the orbit
$\mathcal{B}^{0,-}\cdot\gamma$ is open and
$f_{wt_{\lambda}}|_{\mathcal{B}^{0,-}\cdot\gamma}$ is bijective onto
$\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}$. The rest of the proof is clear.
∎

######  Lemma 4.5.

$(f_{wt_{\lambda}})_{*}[\mathcal{O}_{\Gamma_{wt_{\lambda}}}]=[\mathcal{O}_{\overline{\mathcal{B}^{0,-}_{sm}\cdot
x_{w(\lambda)}}}]=\mathcal{O}_{wt_{\lambda}}\in K^{T}(Gr_{G})$.

###### Proof.

This follows from the fact that affine Schubert varieties have rational
singularities. A proof can be found in [26, Theorem 8.2.2]. Although our
definition of these varieties is a priori different from the one in loc. cit.,
the arguments there apply well to our case. ∎

Recall $\mathcal{O}_{wt_{\lambda}}$ is regarded as an element of the domain of
$\Phi$ via (2.3).

######  Lemma 4.6.

We have

$\Phi(\mathcal{O}_{wt_{\lambda}})=\sum_{i,j\in
I}\sum_{\beta\in\beta_{w(\lambda)}+\Lambda}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}(\operatorname{ev}^{*}e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$

where $\beta_{w(\lambda)}$ is defined in Definition 3.9.

###### Proof.

By the classical localization formula,

$[\mathcal{O}_{\Gamma_{wt_{\lambda}}}]=\sum_{\gamma\in\Gamma_{wt_{\lambda}}^{T}}\frac{1}{\Lambda_{-1}((T_{\gamma}\Gamma_{wt_{\lambda}})^{\vee})}[\mathcal{O}_{\gamma}]\in
K^{T}(\Gamma_{wt_{\lambda}})$

where $\Lambda_{-1}(V):=\sum_{i\geqslant 0}(-1)^{i}[\Lambda^{i}V]\in R(T)$ for
any $T$-module $V$. Applying $(f_{wt_{\lambda}})_{*}$ to both sides of the
last equation and using Lemma 4.5, we get

$\mathcal{O}_{wt_{\lambda}}=\sum_{\mu\in
Q^{\vee}}\left(\sum_{\gamma\in\Gamma_{wt_{\lambda}}^{T}\cap
f_{wt_{\lambda}}^{-1}(\mu)}\frac{1}{\Lambda_{-1}((T_{\gamma}\Gamma_{wt_{\lambda}})^{\vee})}\right)\mathcal{O}_{\mu}.$

The rest follows from a parallel argument used in the proof of [13, Lemma 3.9]
which deals with the case of quantum cohomology. In our case, we need the
$K$-theoretic version of virtual localization formula in [19]. See [37] for
the explicit formula and its proof. ∎

######  Remark 4.7.

Some care needs to be taken when we apply the virtual localization formula:
The proof of this formula given in [37] assumes an extra condition which, by
[23, Proposition 5.13], is satisfied if our stack admits a $T$-equivariant
étale atlas of finite type. According to the remark following that
proposition, which cites [2, Theorem 4.3], every separated Deligne-Mumford
stack of finite type with a $T$-action admits such an atlas, after possibly
reparametrizing $T$. Notice that such reparametrization will not affect the
argument in the proof of Lemma 4.6. Alternatively, the existence of the
required atlas for our particular stack follows from [34, Corollary 4].

Define

$\varphi:=\sum_{i,j\in
I}\sum_{\beta\in\beta_{w(\lambda)}+\Lambda}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}(\operatorname{ev}^{*}e_{i})e_{j}.$

Notice the absence of $(\operatorname{id}+A_{G/P})^{-1}$. Write
$\varphi=\varphi_{0}+\varphi_{+}$ where $\varphi_{0}$ (resp. $\varphi_{+}$) is
the expression contributed by $\beta=\beta_{w(\lambda)}$ (resp.
$\beta\neq\beta_{w(\lambda)}$).

######  Proposition 4.8.

$\Phi(\mathcal{O}_{wt_{\lambda}})=\varphi_{0}$.

###### Proof.

By Lemma 4.10 below, we have $\varphi_{+}=A_{G/P}(\varphi_{0})$, and hence

$\Phi(\mathcal{O}_{wt_{\lambda}})=(\operatorname{id}+A_{G/P})^{-1}(\varphi_{0}+\varphi_{+})=(\operatorname{id}+A_{G/P})^{-1}\circ(\operatorname{id}+A_{G/P})(\varphi_{0})=\varphi_{0}.$

∎

Before proving Lemma 4.10 which is used in the proof of Proposition 4.8, we
first prove another lemma. Recall morphism (3.4). Its domain is
$F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$
if $\beta\neq\beta_{w(\lambda)}$ and $F_{wt_{\lambda}}$ if
$\beta=\beta_{w(\lambda)}$.

######  Lemma 4.9.

We have

$\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]=\left\\{\begin{array}[]{cc}(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]&\beta\neq\beta_{w(\lambda)}\\\\[10.00002pt]
(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]&\beta=\beta_{w(\lambda)}\end{array}\right.$

where $\operatorname{ev}^{\prime}_{2}$ is induced by the evaluation map on
$\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$ at the second
marked point.

###### Proof.

By Remark 2.1, it suffices to verify the corresponding equality for their
coarse moduli. Let $M$ and $F$ be the coarse moduli of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ and the domain of (3.4). They are
projective varieties with only finite quotient singularities. Since
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible by Corollary 3.19
and the canonical morphism
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow M$ is surjective, $M$
is also irreducible. Similarly, $F$ is irreducible. By the definition of
coarse moduli, morphism (3.4) induces a unique morphism $\iota:F\rightarrow M$
which is equal to (3.4) set-theoretically. The uniqueness implies that $\iota$
is $T$-equivariant.

Denote by $M^{sm}$ the smooth locus of $M$. It is easy to see that
$\iota^{-1}(M^{sm})\neq\emptyset$ (look at chains of embedded spheres) and
$\iota$ maps $\iota^{-1}(M^{sm})$ bijectively onto an attractive component
$F^{\prime}$ of $(M^{sm})^{\mathbb{C}^{\times}_{t}}$. Since
$\iota^{-1}(M^{sm})$ is reduced, $\iota|_{\iota^{-1}(M^{sm})}$ factors through
the inclusion $F^{\prime}\hookrightarrow M^{sm}$. There exists a non-empty
open subscheme $V\subseteq\iota^{-1}(M^{sm})$ such that $\iota|_{V}$ is smooth
over $F^{\prime}$. Since $\iota$ is injective, $\iota|_{V}$ is étale over
$F^{\prime}$ and hence an isomorphism onto its image.

By Theorem 3.12, there exists a $\mathbb{C}^{\times}_{t}$-invariant open
subscheme $U\subseteq M^{sm}$ containing $F^{\prime}$ and an affine fibration
$U\rightarrow F^{\prime}$. The latter morphism induces, via the morphism
$\iota|_{V}:V\xrightarrow{\sim}\iota(V)\subseteq F^{\prime}$, a rational map
$\phi:M\dashrightarrow F$. Since $\iota$ is $T$-equivariant, $U$ is unique and
the $T$-action commutes with the $\mathbb{C}^{\times}_{t}$-action (see Section
3.3(2)(i)), it follows that everything is $T$-equivariant.

By resolving the indeterminacy locus of $\phi$, we obtain a smooth irreducible
projective variety $Z$ and morphisms $\nu_{M}:Z\rightarrow M$ and
$\nu_{F}:Z\rightarrow F$ such that $\nu_{M}$ is birational and
$\phi\circ\nu_{M}=\nu_{F}$. Since equivariant resolutions of singularities
exist (see e.g. [25]), we may assume $Z$ has a $T$-action and $\nu_{M}$,
$\nu_{F}$ are $T$-equivariant. Define
$\operatorname{ev}_{F}:=\operatorname{ev}_{2}^{\prime}$ if
$\beta\neq\beta_{w(\lambda)}$ and $\operatorname{ev}_{F}:=h_{wt_{\lambda}}$
otherwise. By the fact that $\operatorname{ev}$ is
$\mathbb{C}^{\times}_{t}$-invariant, we have
$\operatorname{ev}=\operatorname{ev}_{F}\circ\phi$, and hence
$\operatorname{ev}\circ\nu_{M}=\operatorname{ev}_{F}\circ\nu_{F}$, giving

$\operatorname{ev}_{*}(\nu_{M})_{*}[\mathcal{O}_{Z}]=(\operatorname{ev}_{F})_{*}(\nu_{F})_{*}[\mathcal{O}_{Z}]\in
K_{T}(G/P).$

Since $Z$ has only finite quotient singularities and singularities of this
kind are rational, by [40], we have
$(\nu_{M})_{*}[\mathcal{O}_{Z}]=[\mathcal{O}_{M}]$. To conclude the proof, it
suffices to show $(\nu_{F})_{*}[\mathcal{O}_{Z}]=[\mathcal{O}_{F}]$. This
follows from [12, Theorem 3.1], given the following conditions:

  1. (1)

$\nu_{F}$ is surjective and $T$-equivariant;

  2. (2)

$Z$ and $F$ are projective with rational singularities; and

  3. (3)

the general fiber of $\nu_{F}$ is rational.

Condition (1) is obvious. To verify (2), we use the above cited result [40].
For (3), take a non-empty open subscheme $W\subseteq V$ such that
$\nu_{F}|_{\nu_{F}^{-1}(W)}$ is smooth. Since $Z$ contains an open dense
subscheme $U^{\prime}$ such that $\nu_{F}|_{U^{\prime}}$ is an affine
fibration over $W$, it follows that every geometric fiber of
$\nu_{F}|_{\nu_{F}^{-1}(W)}$ is connected and contains the affine space as an
open subscheme, i.e. it is rational. ∎

######  Lemma 4.10.

$\varphi_{+}=A_{G/P}(\varphi_{0})$.

###### Proof.

By the projection formula,

$\varphi_{+}=\sum_{\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})}q^{\beta}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]$

which is equal to
$\sum_{\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})}q^{\beta}(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]$
by Lemma 4.9.

Denote by $\operatorname{ev}_{1}$ and $\operatorname{ev}_{2}$ the evaluation
maps on $\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$. Since
$\operatorname{ev}_{1}$ is flat, we have

$(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]=(\operatorname{ev}_{2})_{*}(\operatorname{ev}_{1})^{*}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}],$
(4.1)

by the base change formula. Summing up (4.1) over all
$\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})$, weighted by
$q^{\beta}$, we get
$\varphi_{+}=A_{G/P}(q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}])$.
By Lemma 4.9 applied to $\beta=\beta_{w(\lambda)}$, we get
$\varphi_{0}=q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]$.
The result follows. ∎

###  4.3. Step 2 of the computation: determining the initial term

By Proposition 4.8, it suffices to determine $\varphi_{0}$. By Lemma 4.9
applied to $\beta=\beta_{w(\lambda)}$, we have

$\varphi_{0}=q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]$
(4.2)

where $F_{wt_{\lambda}}$ and $h_{wt_{\lambda}}$ are defined in Definition
3.15. Since $F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ is
$B^{-}$-invariant and
$f=f_{wt_{\lambda}}|_{F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})}$
is $B^{-}$-equivariant, $f$ is birational onto the Schubert variety
$\overline{B^{-}\cdot x_{w(\lambda)}}\subseteq G\cdot x_{w(\lambda)}\simeq
G/P_{w(\lambda)}$. Since Schubert varieties have rational singularities (see
Lemma 4.5), we have, by the base change formula,

$(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]=j_{*}[\mathcal{O}_{\pi^{-1}(\overline{B^{-}\cdot
x_{w(\lambda)}})}].$ (4.3)

See (3.2) for the definition of $j$ and $\pi$.

Let us deal with the case $P=B^{+}$ first. Define $y_{w}^{\prime}:=wB^{+}\in
G/B^{+}$ and $B_{w}:=wBw^{-1}$.

######  Lemma 4.11.

  1. (1)

We have $y_{w}^{\prime}\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$ and $B_{w}\subseteq
P_{w(\lambda)}$.

  2. (2)

There exists a $G$-equivariant isomorphism

$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
G/B_{w}$

under which $\pi$ and $j$ (see diagram (3.2)) are identified with the
projection $G/B_{w}\rightarrow G/P_{w(\lambda)}$ and the isomorphism
$G/B_{w}\xrightarrow{\sim}G/B^{+}:gB_{w}\mapsto gwB^{+}$ respectively.

  3. (3)

$j$ maps $\pi^{-1}(\overline{B^{-}\cdot x_{w(\lambda)}})$ isomorphically onto
$\overline{B^{-}\cdot y_{w}^{\prime}}$.

###### Proof.

(1) is proved by looking at the weight spaces:

$T_{y_{w}^{\prime}}(G/B^{+})\simeq\bigoplus_{\alpha\in-
wR^{+}}\mathfrak{g}_{\alpha}\quad\text{ and
}\quad\operatorname{Lie}(B_{w})=\mathfrak{h}\oplus\bigoplus_{\alpha\in
wR^{+}}\mathfrak{g}_{\alpha}.$

Since $wt_{\lambda}\in W_{af}^{-}$, $\lambda$ is anti-dominant, and hence
$\pm\alpha(w(\lambda))\geqslant 0$ for any $\alpha\in\mp wR^{+}$.

To prove (2), recall (Lemma 3.8) $P_{w(\lambda)}$ acts transitively on
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$. Hence, by (1), we have
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
P_{w(\lambda)}/B_{w}$ so that

$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
G\times_{P_{w(\lambda)}}(P_{w(\lambda)}/B_{w})\simeq G/B_{w}.$

The rest of the proof is clear.

For (3), we use the identifications in (2). Denote by $C$ the dominant Weyl
chamber. It suffices to show that $wC$ has the smallest length (with respect
to $C$) among other chambers which contain $-w(\lambda)$. It amounts to
showing

$\alpha(w(\lambda))=0\text{ and }\alpha\in R^{+}\Longrightarrow
w^{-1}\alpha\in R^{+}.$ (4.4)

This requires the assumption $wt_{\lambda}\in W_{af}^{-}$. Denote by $\Delta$
the dominant alcove. By definition, the alcove
$wt_{\lambda}(\Delta)=w(\lambda)+w(\Delta)$ has the smallest length (with
respect to $\Delta$) among other alcoves which contain $w(\lambda)$. If
$\alpha(w(\lambda))=0$, then $w(\lambda)+w(\Delta)$ and $\Delta$ lie in the
same side with respect to the wall $\\{\alpha=0\\}$. This proves (4.4). ∎

For general $P$, we have the commutative diagram

$G\cdot
x_{w(\lambda)}$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$$G/B^{+}$$G/P$$\simeq$$\pi_{G/B^{+}}$$\pi_{G/P}$$p$$j_{G/B^{+}}$$j_{G/P}$

where the horizontal arrows are some canonical projections. By Lemma 4.11 and
the fact that the upper horizontal arrow has rational fibers, we have

$(j_{G/P})_{*}[\mathcal{O}_{(\pi_{G/P})^{-1}(\overline{B^{-}\cdot
x_{w(\lambda)}})}]=p_{*}[\mathcal{O}_{\overline{B^{-}\cdot y_{w}^{\prime}}}].$
(4.5)

Denote by $\widetilde{w}\in W/W_{P}$ the minimal length coset representative
of $wW_{P}$. Notice $p(\overline{B^{-}\cdot
y_{w}^{\prime}})=\overline{B^{-}\cdot y_{\widetilde{w}}}$ but the dimension of
some fibers of $p|_{\overline{B^{-}\cdot y_{w}^{\prime}}}$ may be positive.
Choose Bott-Samelson resolutions
$\Gamma^{\prime}\rightarrow\overline{B^{-}\cdot y_{w}^{\prime}}$ and
$\Gamma\rightarrow\overline{B^{-}\cdot y_{\widetilde{w}}}$ such that there is
a map $p^{\prime}:\Gamma^{\prime}\rightarrow\Gamma$ defined by forgetting last
few factors of $\Gamma^{\prime}$ which fits into the commutative diagram

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By the facts that $p^{\prime}$ has rational fibers and Schubert varieties have
rational singularities, we have

$p_{*}[\mathcal{O}_{\overline{B^{-}\cdot
y_{w}^{\prime}}}]=[\mathcal{O}_{\overline{B^{-}\cdot y_{\widetilde{w}}}}].$
(4.6)

Alternatively, (4.6) follows from [12, Theorem 3.1].

###  4.4. Proof of Theorem 1.1

We start with summarizing what we have done in the previous subsections. We
defined an $R(T)$-linear map $\Phi$ in Definition 4.1 and proved in
Proposition 4.3 that it is an $R(T)$-algebra homomorphism. A priori, $\Phi$
was defined in terms of the localization basis
$\\{\mathcal{O}_{\mu}\\}_{\mu\in Q^{\vee}}$. But by Lemma 4.6, it can also be
defined in the same style in terms of the affine Schubert basis
$\\{\mathcal{O}_{wt_{\lambda}}\\}_{wt_{\lambda}\in W_{af}^{-}}$. By
Proposition 4.8, (4.2), (4.3), (4.5) and (4.6), we have

$\Phi(\mathcal{O}_{wt_{\lambda}})=q^{\beta_{w(\lambda)}}[\mathcal{O}_{\overline{B^{-}\cdot
y_{\widetilde{w}}}}].$

It remains to show that $\beta_{w(\lambda)}$ corresponds to
$\lambda+Q^{\vee}_{P}$ via the dual of isomorphism (2.2). By making use of the
canonical projection $G/B^{+}\rightarrow G/P$, we may assume $P=B^{+}$. Let
$\rho\in(Q^{\vee})^{*}$. We have to show
$\deg(u^{*}\mathcal{L}_{\rho})=\rho(\lambda)$ for the constant section $u$ of
$P_{w(\lambda)}(G/B^{+})$ corresponding to a point of
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$. By Lemma 4.11, we can
take that point to be $y^{\prime}_{w}:=wB^{+}$. Recall
$L_{\rho}=G\times_{B^{+}}\mathbb{C}_{-\rho}$ so that
$(L_{\rho})_{y^{\prime}_{w}}\simeq\mathbb{C}_{-w\rho}$ as $T$-modules, and
hence $(L_{\rho})_{y^{\prime}_{w}}\simeq\mathbb{C}_{-\rho(\lambda)}$ as
$\mathbb{C}^{\times}_{w(\lambda)}$-modules. Therefore,
$u^{*}\mathcal{L}_{\rho}\simeq\mathcal{O}_{\mathbb{P}^{1}}(\rho(\lambda))$ as
desired. The proof of Theorem 1.1 is complete.

##  Appendix A A K-theoretic degeneration formula

Let $\mu\in Q^{\vee}$. Recall the $G/P$-fibration $P_{\mu}(G/P)$ over
$\mathbb{P}^{1}$ and the projection
$\pi_{\mu}:P_{\mu}(G/P)\rightarrow\mathbb{P}^{1}$ defined in Section 3.2. Let
$k\in\mathbb{N}$. Fix some points $z_{1},\ldots,z_{k}\in\mathbb{P}^{1}$. For
each $i=1,\ldots,k$, define $D_{\mu,i}:=\pi_{\mu}^{-1}(z_{i})$ and
$\iota_{\mu,i}:D_{\mu,i}\hookrightarrow P_{\mu}(G/P)$ to be the inclusion.

######  Definition A.1.

Let $\beta\in\pi_{2}(G/P)$.

  1. (1)

Define

$\overline{\mathcal{M}}_{k}(\mu,\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,k}(P_{{\mu}}(G/P),\widetilde{\beta})\times_{(\vec{\operatorname{ev}},\iota_{\mu,1}\times\cdots\times\iota_{\mu,k})}\left(D_{\mu,1}\times\cdots\times
D_{\mu,k}\right)$

where $\widetilde{\beta}$ runs over all classes in $\pi_{2}(P_{{\mu}}(G/P))$
satisfying

    1. (i)

$(\pi_{\mu})_{*}\widetilde{\beta}=[\mathbb{P}^{1}]\in\pi_{2}(\mathbb{P}^{1})$;
and

    2. (ii)

$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.

(The line bundles $\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section
3.1 and Section 2.3 respectively.)

  2. (2)

By abuse of notation, define

$\operatorname{ev}_{i}:\overline{\mathcal{M}}_{k}(\mu,\beta)\rightarrow
D_{\mu,i}\simeq G/P$

to be the morphism induced by the evaluation map $\operatorname{ev}_{i}$ on
$\overline{\mathcal{M}}_{0,k}(P_{{\mu}}(G/P),\widetilde{\beta})$.

######  Definition A.2.

  1. (1)

Let $\gamma_{1},\ldots,\gamma_{k}\in K_{T}(G/P)$ and $\beta\in\pi_{2}(G/P)$.
Define

$\operatorname{PKGW}^{\beta}_{\mu}(\gamma_{1},\ldots,\gamma_{k}):=\chi_{\overline{\mathcal{M}}_{k}(\mu,\beta)}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{k}(\mu,\beta)}\otimes\operatorname{ev}_{1}^{*}\gamma_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\gamma_{k}\right)\in
R(T)$

where $\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{k}(\mu,\beta)}$ is the
virtual structure sheaf constructed in [31].

  2. (2)

Let $\gamma_{1},\ldots,\gamma_{k}\in K_{T}(G/P)$. Define

$\operatorname{PKGW}_{\mu}(\gamma_{1},\ldots,\gamma_{k}):=\sum_{\beta\in\pi_{2}(G/P)}q^{\beta}\operatorname{PKGW}^{\beta}_{\mu}(\gamma_{1},\ldots,\gamma_{k})\in
R(T)\otimes\mathbb{Z}[[\Lambda,\Lambda^{-1}].$

(By Corollary 3.18, the ring $\mathbb{Z}[[\Lambda,\Lambda^{-1}]$ is large
enough in order for $\operatorname{PKGW}_{\mu}$ to be well-defined.)

Recall the notations $\\{e_{i}\\}_{i\in I}$, $\\{g^{ij}\\}_{i,j\in I}$ and
$A_{G/P}$ defined in Section 2.4. The following proposition is a $K$-theoretic
degeneration formula which we need in order to prove Proposition 4.3.

######  Proposition A.3.

For any $\mu_{1},\mu_{2}\in Q^{\vee}$ and tuples
$\vec{\gamma}^{(1)}=(\gamma^{(1)}_{1},\ldots,\gamma^{(1)}_{k_{1}})$,
$\vec{\gamma}^{(2)}=(\gamma^{(2)}_{1},\ldots,\gamma^{(2)}_{k_{2}})$ of
elements of $K_{T}(G/P)$,

$\operatorname{PKGW}_{\mu_{1}+\mu_{2}}(\vec{\gamma}^{(1)},\vec{\gamma}^{(2)})=\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}}\left(\vec{\gamma}^{(1)},(\operatorname{id}+A_{G/P})^{-1}(e_{i})\right)\operatorname{PKGW}_{\mu_{2}}(\vec{\gamma}^{(2)},e_{j}).$

The rest of the appendix is devoted to the proof of Proposition A.3. While it
should not be difficult to prove the result using the machineries developed by
Li [32, 33], we take a more direct approach, namely we work with the moduli of
stable maps to the degeneration family instead of working with the stack of
expanded degenerations and its associated moduli of relative stable maps.

Define

$C:=\\{(t,[x:y:z])\in\mathbb{A}^{1}\times\mathbb{P}^{2}|~{}xy=tz^{2}\\}.$

Then the projection onto $\mathbb{A}^{1}$ defines a flat family
$C\rightarrow\mathbb{A}^{1}$. Denote by $C_{t}$ the fiber over
$t\in\mathbb{A}^{1}$. If $t\neq 0$, $C_{t}$ is isomorphic to $\mathbb{P}^{1}$,
and if $t=0$, it is isomorphic to the nodal rational curve $C_{1}\cup C_{2}$
where $C_{1},C_{2}\simeq\mathbb{P}^{1}$. In [13], we constructed a locally
trivial $G/P$-fibration

$p:P_{\mu_{1},\mu_{2}}\rightarrow C$

such that

$P_{\mu_{1},\mu_{2}}|_{C_{t\neq 0}}\simeq P_{\mu_{1}+\mu_{2}}(G/P)~{}\text{
and }~{}P_{\mu_{1},\mu_{2}}|_{C_{i}}\simeq P_{\mu_{i}}(G/P).$

By the construction, $P_{\mu_{1},\mu_{2}}$ admits a $T$-action such that $p$
is $T$-equivariant where $T$ acts $C$ trivially. Moreover, the above
isomorphisms are $T$-equivariant. Let $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
Like $P_{\mu}(G/P)$, there is a line bundle on $P_{\mu_{1},\mu_{2}}$ which
restricts to $L_{\rho}=G\times_{P}\mathbb{C}_{-\rho}$ on each fiber of $p$. By
abuse of notation, we denote this line bundle by $\mathcal{L}_{\rho}$.

Fix some sections
$s_{1}^{(1)},\ldots,s_{k_{1}}^{(1)},s_{1}^{(2)},\ldots,s_{k_{2}}^{(2)}$ of the
family $C\rightarrow\mathbb{A}^{1}$ such that
$s_{1}^{(i)}(0),\ldots,s_{k_{i}}^{(i)}(0)$ lie in $C_{i}\subset C_{t=0}$ away
from the intersection $C_{1}\cap C_{2}$. For each $s=1,\ldots,k_{1}$, the
divisor
$D_{s}^{(1)}:=P_{\mu_{1},\mu_{2}}\times_{(p,s_{s}^{(1)})}\mathbb{A}^{1}$ of
$P_{\mu_{1},\mu_{2}}$ is canonically isomorphic to $\mathbb{A}^{1}\times G/P$.
Denote by $\iota_{s}^{(1)}:D_{s}^{(1)}\hookrightarrow P_{\mu_{1},\mu_{2}}$ the
inclusion and $j_{s}^{(1)}:D_{s}^{(1)}\rightarrow G/P$ the projection. For
each $t=1,\ldots,k_{2}$, we define $D_{t}^{(2)}$, $\iota_{t}^{(2)}$ and
$j_{t}^{(2)}$ in a similar way.

From now on, fix $\beta\in\pi_{2}(G/P)$. We have to show

$\operatorname{PKGW}^{\beta}_{\mu_{1}+\mu_{2}}(\vec{\gamma}^{(1)},\vec{\gamma}^{(2)})=\sum_{r=0}^{\infty}\sum_{i,j\in
I}\sum_{\vec{\beta}}(-1)^{r}g^{ij}\operatorname{PKGW}^{\beta_{1}}_{\mu_{1}}(\vec{\gamma}^{(1)},I^{\beta_{11},\ldots,\beta_{1r}}(e_{i}))\operatorname{PKGW}^{\beta_{2}}_{\mu_{2}}(\vec{\gamma}^{(2)},e_{j}).$
(A.1)

Here,

  * •

the third sum runs over all tuples
$\vec{\beta}=(\beta_{1};\beta_{11},\ldots,\beta_{1r};\beta_{2})$ of elements
of $\pi_{2}(G/P)$ such that $\beta_{1i}\neq 0$ for all $i=1,\ldots,r$ and
$\beta_{1}+\sum_{i=1}^{r}\beta_{1i}+\beta_{2}=\beta$; and

  * •

$I^{\beta_{11},\ldots,\beta_{1r}}$ is a linear operator on $K_{T}(G/P)$
defined by

$I^{\beta_{11},\ldots,\beta_{1r}}:=(\operatorname{ev}_{2}^{\beta_{11}})_{*}\circ(\operatorname{ev}_{1}^{\beta_{11}})^{*}\circ\cdots\circ(\operatorname{ev}_{2}^{\beta_{1r}})_{*}\circ(\operatorname{ev}_{1}^{\beta_{1r}})^{*}$

where
$\operatorname{ev}_{1}^{\beta^{\prime}},\operatorname{ev}_{2}^{\beta^{\prime}}:\overline{\mathcal{M}}_{0,2}(G/P,\beta^{\prime})\rightarrow
G/P$ are the evaluation maps.

Denote by $A$ the semigroup of effective curve classes in
$P_{\mu_{1},\mu_{2}}$. Let $a\in A$ and $k\in\mathbb{N}$. Consider the Artin
stack $\mathfrak{M}_{0,k,A,a}$ defined in [15]. Roughly speaking, it
parametrizes genus zero $k$-pointed prestable curves each of whose irreducible
components is assigned an element of $A$ such that the sum of these elements
is equal to $a$ and this assignment satisfies a stability condition. There is
a natural morphism
$\overline{\mathcal{M}}_{0,k}(P_{\mu_{1},\mu_{2}},a)\rightarrow\mathfrak{M}_{0,k,A,a}$
defined by forgetting the target but remembering the degrees of the
restrictions of any stable maps to the irreducible components of the domain
curves. It is shown in loc. cit. that $\mathfrak{M}_{0,k,A,a}$ is étale over
the usual Artin stack $\mathfrak{M}_{0,k}$ of prestable curves. It follows
that $\mathfrak{M}_{0,k,A,a}$ is smooth and the standard virtual structure
sheaf
$\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{0,k}(P_{\mu_{1},\mu_{2}},a)}$ is
equal to the one constructed using the relative perfect obstruction theory
associated to the above morphism.

######  Definition A.4.

  1. (1)

Define a Deligne-Mumford stack

$\overline{\mathcal{M}}:=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,k_{1}+k_{2}}(P_{\mu_{1},\mu_{2}},\widetilde{\beta})$

where $\widetilde{\beta}$ runs over all classes in
$\pi_{2}(P_{\mu_{1},\mu_{2}})$ satisfying

    1. (i)

$p_{*}\widetilde{\beta}=[C_{1}]+[C_{2}]$, the fiber class of the flat family
$C\rightarrow\mathbb{A}^{1}$; and

    2. (ii)

$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.

  2. (2)

Define an Artin stack

$\mathfrak{M}_{A}:=\bigcup_{\widetilde{\beta}}\mathfrak{M}_{0,k_{1}+k_{2},A,\widetilde{\beta}}$

where $\widetilde{\beta}$ runs over the same set in (1) above.

There is a proper morphism
$\pi:\overline{\mathcal{M}}\rightarrow\mathbb{A}^{1}$ sending each stable map
$u$ to the unique $t\in\mathbb{A}^{1}$ such that $p\circ u$ lands in $C_{t}$.
There is also a natural morphism
$\nu:\overline{\mathcal{M}}\rightarrow\mathfrak{M}_{A}$ which is the union of
the morphisms
$\overline{\mathcal{M}}_{0,k_{1}+k_{2}}(P_{\mu_{1},\mu_{2}},\widetilde{\beta})\rightarrow\mathfrak{M}_{0,k_{1}+k_{2},A,\widetilde{\beta}}$
mentioned above.

######  Definition A.5.

Let $r$ be a non-negative integer.

  1. (1)

Define $\Omega^{r}$ to be the set of modular graphs with $A$-structure [7]
satisfying

    1. (a)

the number of vertices is $r+2$;

    2. (b)

the genus associated to every vertex is zero;

    3. (c)

the underlying graph without tails is a linear graph;

    4. (d)

the degrees associated to the two end vertices of the graph in (c) are classes
which are projected via $p$ to $[C_{1}]$ and $[C_{2}]$ respectively;

    5. (e)

the sum of degrees satisfies conditions (i) and (ii) in the definition of
$\overline{\mathcal{M}}$ above; and

    6. (f)

the number of tails is $k_{1}+k_{2}$.

  2. (2)

Define $\widetilde{\Omega}^{r}$ to be the set of modular graphs with
$A$-structure satisfying (a) to (e) above and that the number of tails is
zero.

It follows from (d) and (e) that for any $\sigma\in\Omega^{r}$ or
$\widetilde{\Omega}^{r}$, the degrees associated to the intermediate vertices
are fiber classes with respect to the fibration
$p:P_{\mu_{1},\mu_{2}}\rightarrow C$.

Define an injective map
$\widetilde{\Omega}^{r}\hookrightarrow\Omega^{r}:\sigma\mapsto\overline{\sigma}$
as follows. For each $\sigma\in\widetilde{\Omega}^{r}$, define
$\overline{\sigma}$ to be the modular graph obtained from $\sigma$ by
attaching the first $k_{1}$ tails to the end vertex corresponding to $C_{1}$
and the rest to the other end vertex. See (d) in the definition of
$\Omega^{r}$. We will identify $\widetilde{\Omega}^{r}$ with the image of this
injective map.

For any $\sigma\in\Omega^{r}$, we have a similarly-defined Deligne-Mumford
stack $\overline{\mathcal{M}}(\sigma)$ (resp. Artin stack
$\mathfrak{M}_{A}(\sigma)$) parametrizing stable maps (resp. prestable curves)
which are at least singular as described by the graph $\sigma$. If
$\sigma\in\Omega^{0}$, then $\mathfrak{M}_{A}(\sigma)$ is a Weil divisor of
$\mathfrak{M}_{A}$. Since $\mathfrak{M}_{A}$ is smooth, we have the associated
Cartier divisor $\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$
(for simplicity, the section is hidden from the notation). For each
$\ell\in\mathbb{N}$, define $\mathfrak{M}^{\leqslant\ell}_{0,0}$ to be the
moduli stack of genus zero prestable curves with at most $\ell$ nodes. We have
a smooth divisor $\mathcal{D}$ of $\mathfrak{M}^{\leqslant 1}_{0,0}$ which is
the complement of the open substack $\mathfrak{M}^{\leqslant 0}_{0,0}$.

######  Lemma A.6.

There exists a commutative diagram of stacks

${\overline{\mathcal{M}}}$${\mathfrak{M}_{A}}$${\mathbb{A}^{1}}$${\mathfrak{M}^{\leqslant
1}_{0,0}}$$\scriptstyle{\pi}$$\scriptstyle{\nu}$$\scriptstyle{g}$$\scriptstyle{f}$
(A.2)

such that $f^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$ and
$g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$
as Cartier divisors.

###### Proof.

We postpone the proof until the end. ∎

Observe that
$\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)\subset\mathfrak{M}_{A}$
is a normal crossing divisor and the intersection of these divisors
$\mathfrak{M}_{A}(\sigma)$ over any finite subset of $\Omega^{0}$ is of the
form $\mathfrak{M}_{A}(\sigma^{\prime})$ for some modular graph
$\sigma^{\prime}$ with $A$-structure. It is easy to see that in our case these
modular graphs are precisely the elements of $\Omega^{r}$, $r\in\mathbb{N}$.
Therefore, by the inclusion-exclusion principle [31],

$\left[\mathcal{O}_{\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)}\right]=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\Omega^{r}}\left[\mathcal{O}_{\mathfrak{M}_{A}(\sigma)}\right].$
(A.3)

By Lemma A.6, (A.3) and some standard arguments in virtual pullbacks [37], we
have

$\text{LHS of
}\eqref{appeq1}=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\Omega^{r}}\chi_{\overline{\mathcal{M}}(\sigma)}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\sigma)}\otimes
e_{1}\otimes e_{2}\right)$ (A.4)

where
$e_{1}:=\bigotimes_{s=1}^{k_{1}}\operatorname{ev}_{s}^{*}(\iota_{s}^{(1)})_{*}(j_{s}^{(1)})^{*}\gamma_{s}^{(1)}$
and
$e_{2}:=\bigotimes_{t=1}^{k_{2}}\operatorname{ev}_{k_{1}+t}^{*}(\iota_{t}^{(2)})_{*}(j_{t}^{(2)})^{*}\gamma_{t}^{(2)}$.

Let $\sigma\in\Omega^{r}$. Recall the subset
$\widetilde{\Omega}^{r}\subseteq\Omega^{r}$ defined above. Suppose
$\sigma\not\in\widetilde{\Omega}^{r}$. It is easy to see that one of the
evaluation maps on $\overline{\mathcal{M}}(\sigma)$ factors through the open
immersion $P_{\mu_{1},\mu_{2}}\setminus D_{s}^{(1)}\hookrightarrow
P_{\mu_{1},\mu_{2}}$ or $P_{\mu_{1},\mu_{2}}\setminus
D_{t}^{(2)}\hookrightarrow P_{\mu_{1},\mu_{2}}$ for some $s=1,\ldots,k_{1}$ or
$t=1,\ldots,k_{2}$. It follows that the summand in (A.4) is zero for $\sigma$.
Therefore, we have

$\text{RHS of
}\eqref{appeq4}=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\widetilde{\Omega}^{r}}\chi_{\overline{\mathcal{M}}(\overline{\sigma})}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\overline{\sigma})}\otimes
e_{1}\otimes e_{2}\right).$ (A.5)

Now let $\sigma\in\widetilde{\Omega}^{r}$. Denote the degrees of $\sigma$ by
$a_{1}^{\sigma},a_{11}^{\sigma},\ldots,a_{1r}^{\sigma},a_{2}^{\sigma}$, in the
order compatible with the linear graph, starting with the end vertex
corresponding to $C_{1}$. Define $P_{\mu_{i}}:=P_{\mu_{1},\mu_{2}}|_{C_{i}}$,
$F:=P_{\mu_{1},\mu_{2}}|_{C_{1}\cap C_{2}}$ and
$\iota_{F}^{(i)}:F\hookrightarrow P_{\mu_{i}}$ to be the inclusion. For each
$i=1,2$, define

$\overline{\mathcal{M}}_{i}:=\overline{\mathcal{M}}_{0,k_{i}+1}(P_{\mu_{i}},a_{i}^{\sigma})\times_{(\operatorname{ev}_{k_{i}+1},\iota_{F}^{(i)})}F$

and $\operatorname{ev}_{(i)}:\overline{\mathcal{M}}_{i}\rightarrow F$ to be
the morphism induced by $\operatorname{ev}_{k_{i}+1}$. Then Proposition A.3
follows if we can show

$\displaystyle\chi_{\overline{\mathcal{M}}(\overline{\sigma})}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\overline{\sigma})}\otimes
e_{1}\otimes e_{2}\right)$ $\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\chi_{\overline{\mathcal{M}}_{1}}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{1}}\otimes
e_{1}\otimes\operatorname{ev}_{(1)}^{*}\left(I^{a_{11}^{\sigma},\ldots,a_{1r}^{\sigma}}(e_{i})\right)\right)\chi_{\overline{\mathcal{M}}_{2}}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{2}}\otimes
e_{2}\otimes\operatorname{ev}_{(2)}^{*}e_{j}\right).$ (A.6)

The last equation follows from a similar argument which is used to prove the
“cutting edges” axiom in [6]. More precisely, we prove

######  Lemma A.7.

There exists a Cartesian diagram

${\overline{\mathcal{M}}(\overline{\sigma})}$${\overline{\mathcal{M}}_{1}\times\displaystyle\left(\prod_{i=1}^{r}\overline{\mathcal{M}}_{0,2}(F,a^{\sigma}_{1i})\right)\times\overline{\mathcal{M}}_{2}}$${F^{r+1}}$${F^{2r+2}}$$\scriptstyle{\Delta_{F}^{r+1}}$
(A.7)

such that the obstruction theories on the two stacks in the upper row are
compatible over $\Delta_{F}^{r+1}$.

###### Proof.

Denote by
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
the universal curves of $\overline{\mathcal{M}}(\overline{\sigma})$
corresponding to different vertices of $\overline{\sigma}$, and by
$u_{1},u_{11},\ldots,u_{1r},u_{2}$ the universal stable maps. Observe that
over any $\mathbb{C}$-point of $\overline{\mathcal{M}}(\overline{\sigma})$,
$u_{1}$, $u_{1i}$ and $u_{2}$ factor through scheme-theoretically the
inclusions of $P_{\mu_{1}}$, $F$ and $P_{\mu_{2}}$ into $P_{\mu_{1},\mu_{2}}$
respectively. Using the fact that $C_{1}$ and $C_{2}$ are $(-1)$-curves in $C$
and the theorem of cohomology and base change, these stable maps (over
$\overline{\mathcal{M}}(\overline{\sigma})$) in fact land in these subschemes.
This defines the horizontal arrow in (A.7). The vertical arrows are defined to
be the evaluation maps at those marked points which do not correspond to any
of the insertions $\vec{\gamma}^{(1)}$ and $\vec{\gamma}^{(2)}$. It is
straightforward to verify that (A.7) is Cartesian.

It remains to verify the compatibility of obstruction theories. Let
$\mathfrak{C}$ denote the universal curve of
$\overline{\mathcal{M}}(\overline{\sigma})$ obtained by gluing
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
according to the combinatorics of $\overline{\sigma}$. Let
$u:\mathfrak{C}\rightarrow P_{\mu_{1},\mu_{2}}$ be the universal stable map,
$\eta:\mathfrak{C}_{1}\amalg\cdots\amalg\mathfrak{C}_{2}\rightarrow\mathfrak{C}$
the gluing map, and
$z_{0},\ldots,z_{r}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}$
the sections “lying between” $\mathfrak{C}_{1}$ and $\mathfrak{C}_{11}$,
$\mathfrak{C}_{11}$ and $\mathfrak{C}_{12}$, etc. For each $i=1,2$, we remove
the component $\mathfrak{C}_{3-i}$ from $\mathfrak{C}$, that is, we glue all
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
except $\mathfrak{C}_{3-i}$. Denote by $\widetilde{\mathfrak{C}}_{i}$ the
resulting curve and by
$\eta_{i}:\widetilde{\mathfrak{C}}_{i}\hookrightarrow\mathfrak{C}$ the
inclusion.

Define $\mathcal{F}_{i}$ to be the kernel of the natural epimorphism

$u_{i}^{*}T_{P_{\mu_{i}}}\rightarrow(x_{i})_{*}(u_{i}\circ
x_{i})^{*}N_{F/P_{\mu_{i}}}\rightarrow 0$

where
$x_{1}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}_{1}$
(resp.
$x_{2}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}_{2}$)
is the section which is used to glue $\mathfrak{C}_{1}$ and
$\mathfrak{C}_{11}$ (resp. $\mathfrak{C}_{1r}$ and $\mathfrak{C}_{2}$).
(Recall $u_{i}$ lands in $P_{\mu_{i}}$.) Notice that $\mathcal{F}_{i}$ is the
sheaf giving rise to the deformation and obstruction spaces for
$\overline{\mathcal{M}}_{i}$ relative to
$\mathfrak{M}_{A}(\overline{\sigma})$. It is straightforward to verify that
there exists a coherent sheaf $\mathcal{E}$ on $\mathfrak{C}$ which fits into
the following two short exact sequences simultaneously:

$0\rightarrow\mathcal{E}\rightarrow\eta_{*}\left(\mathcal{F}_{1}\oplus\bigoplus_{i=1}^{r}u_{1i}^{*}T_{F}\oplus\mathcal{F}_{2}\right)\rightarrow\bigoplus_{j=0}^{r}(z_{j})_{*}(u\circ
z_{j})^{*}T_{F}\rightarrow 0$ (A.8) $0\rightarrow\mathcal{E}\rightarrow
u^{*}T_{P_{\mu_{1},\mu_{2}}}\rightarrow\bigoplus_{i=1}^{2}(\eta_{i})_{*}(u\circ\eta_{i})^{*}N_{P_{\mu_{i}}/P_{\mu_{1},\mu_{2}}}\rightarrow
0.$ (A.9)

Since $(u\circ\eta_{i})^{*}N_{P_{\mu_{i}}/P_{\mu_{1},\mu_{2}}}\simeq(p\circ
u\circ\eta_{i})^{*}\mathcal{O}_{C_{i}}(-1)$, the derived pushforward functor
sends the cokernel in (A.9) to the zero object in the derived category
$D(\overline{\mathcal{M}}(\overline{\sigma}))$ and so defines an isomorphism
between the objects corresponding to the other two coherent sheaves in the
same short exact sequence. Using this isomorphism and applying the argument in
[6] to (A.8), we obtain a homomorphism of distinguished triangles

${h^{*}E_{\overline{\mathcal{M}}_{right}}}$${E_{\overline{\mathcal{M}}_{left}}}$${v^{*}L_{\Delta^{r+1}_{F}}}$${h^{*}E_{\overline{\mathcal{M}}_{right}}[1]}$${h^{*}L_{\overline{\mathcal{M}}_{right}/\mathfrak{M}_{A}(\overline{\sigma})}}$${L_{\overline{\mathcal{M}}_{left}/\mathfrak{M}_{A}(\overline{\sigma})}}$${L_{\overline{\mathcal{M}}_{left}/\overline{\mathcal{M}}_{right}}}$${h^{*}L_{\overline{\mathcal{M}}_{right}/\mathfrak{M}_{A}(\overline{\sigma})}[1]}$

where $\overline{\mathcal{M}}_{left}$ (resp. $\overline{\mathcal{M}}_{right}$)
is the stack at the top left (resp. right) corner in diagram (A.7), and
$h:\overline{\mathcal{M}}_{left}\rightarrow\overline{\mathcal{M}}_{right}$ and
$v:\overline{\mathcal{M}}_{left}\rightarrow F^{r+1}$ are the morphisms in the
same diagram. This is the compatibility we need to prove. ∎

Equation (A.6) now follows from the functoriality of virtual structure sheaves
[31]. The proof is Proposition A.3 is complete.

####  Proof of Lemma A.6.

By making use of the “forgetting tails” morphisms for $\overline{\mathcal{M}}$
and $\mathfrak{M}_{A}$, we may assume $k_{1}=k_{2}=0$. Notice that the latter
morphism exists by [15, Proposition 2.1.1]. Define $f$ to be the morphism
representing the flat family $C\rightarrow\mathbb{A}^{1}$ defined at the
beginning.

We define $g$ as follows. Denote by $B$ the semigroup of effective curve
classes in $C$. The projection $p:P_{\mu_{1},\mu_{2}}\rightarrow C$ induces a
homomorphism $\phi:A\rightarrow B$ of semigroups. There is a morphism
$\mathfrak{M}_{A}\rightarrow\mathfrak{M}_{0,0,B,[C_{1}]+[C_{2}]}$ sending each
curve $S$ over $\operatorname{Spec}\mathbb{C}$ to the curve obtained by
replacing the degree $a\in A$ assigned to each irreducible component with
$\phi(a)\in B$ and then stabilizing the resulting curve in the sense of [15],
i.e. contracting those irreducible components labelled by $0\in B$ and having
less than three special points. Although the construction of this morphism is
standard, we provide the details in the next paragraph.

Let $T$ be a scheme and $S\rightarrow T$ a family of prestable curves
representing a $T$-point of $\mathfrak{M}_{0,0,A,a}$. By definition, we have a
constructible function $\deg:S\rightarrow A$ which records the degree assigned
to each irreducible component of the fiber over any $\mathbb{C}$-point of $T$.
We have to stabilize the family $S\rightarrow T$ with respect to the degree
function $\phi\circ\deg:S\rightarrow B$. As in the usual case, this type of
stabilization is universal so that we can construct it étale locally. Étale
locally, we can make $S$ stable in the usual sense by inserting sufficiently
many sections. Then we remove these sections one by one. Each time we remove a
section, we stabilize the family by contracting those components labelled by
$0\in B$ and having two special points. Such a stabilization does exist, by
[15, Proposition 2.1.1] which says that
$\mathfrak{C}_{0,N,B,b}\simeq\mathfrak{M}_{0,N+1,B,b}$ where
$\mathfrak{C}_{0,N,B,b}$ is the universal curve of $\mathfrak{M}_{0,N,B,b}$.
The family obtained after removing all these sections will be the desired
stabilization.

Our $g$ is then defined to be the composite

$\mathfrak{M}_{A}\rightarrow\mathfrak{M}_{0,0,B,[C_{1}]+[C_{2}]}\rightarrow\mathfrak{M}^{\leqslant
1}_{0,0}$

where the first morphism is the above morphism and the second is defined by
forgetting all the degrees without stabilizing. It is easy to see that $g$
does land in $\mathfrak{M}^{\leqslant 1}_{0,0}$.

It remains to verify the stated properties.

The commutativity of diagram (A.2).

Let $T$ be a scheme and $u:S\rightarrow P_{\mu_{1},\mu_{2}}$ a stable map over
$T$ representing a $T$-point of $\overline{\mathcal{M}}$. By [7, Lemma 2.2],
$u$ lies over a unique morphism $v:T\rightarrow\mathbb{A}^{1}$. By definition,
$f\circ\pi$ sends this $T$-point to the $T$-point of $\mathfrak{M}^{\leqslant
1}_{0,0}$ represented by the flat family $C\times_{\mathbb{A}^{1}}T$. Consider
the morphism $w:S\rightarrow C\times_{\mathbb{A}^{1}}T$ induced by $p\circ u$.
Observe that, over every $\mathbb{C}$-point of $T$, $w$ contracts precisely
those irreducible components which are mapped into the fibers of $p$. It
follows that $C\times_{\mathbb{A}^{1}}T$ is the stabilization of $p\circ u$
which is equal to the $T$-point with respect to the composite $g\circ\nu$.

The equality $f^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$.

This is straightforward. For example, one can make use of the quotient stack
presentation of $\mathfrak{M}^{\leqslant 1}_{0,0}$ given in [16]. More
precisely, $\mathfrak{M}^{\leqslant 1}_{0,0}$ is isomorphic to $[V/GL_{3}]$
where $V\subset\operatorname{Sym}^{2}\mathbb{C}^{3}$ is the space of symmetric
2-tensors on $\mathbb{C}^{3}$ with rank at least two, and the divisor
$\mathcal{D}$ corresponds to the space of those tensors whose rank is
precisely two.

The equality $g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$.

Observe that, in a smooth atlas, the equality
$g^{-1}(\mathcal{D})=\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)$
holds set-theoretically. It follows that
$g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))^{\otimes
m_{\sigma}}$ as Cartier divisors for some $m_{\sigma}\in\mathbb{Z}_{\geqslant
1}$. To determine $m_{\sigma}$, consider the morphism
$h_{\sigma}:\mathbb{A}^{1}\rightarrow\mathfrak{M}_{A}$ which represents the
flat family $C\rightarrow\mathbb{A}^{1}$ where $C_{1}$ and $C_{2}$ are
labelled by the degrees given by $\sigma$. By the previous paragraph, we have
$h_{\sigma}^{*}g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$. On the other
hand, it is easy to see that for any $\sigma^{\prime}\in\Omega^{0}$ not equal
to $\sigma$,
$h_{\sigma}^{*}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma^{\prime}))$
is the trivial Cartier divisor. This gives $m_{\sigma}=1$. $\square$

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