# Positivity in $T$-Equivariant $K$-theory of partial flag varieties
associated to Kac-Moody groups

Joseph Compton and Shrawan Kumar

Abstract: We prove sign-alternation of the product structure constants in the
basis dual to the basis consisting of the structure sheaves of Schubert
varieties in the torus-equivariant Grothendieck group of coherent sheaves on
the partial flag varieties $G/P$ associated to an arbitrary symmetrizable Kac-
Moody group $G$, where $P$ is any parabolic subgroup of finite type. This
extends the previous work of Kumar from $G/B$ to $G/P$. When $G$ is of finite
type, i.e., it is a semisimple group, then it was proved by Anderson-Griffeth-
Miller.

##  1\. Introduction

This is a continuation of Kumar [Kum17], the notation of which we freely use.

Let $G$ be any symmetrizable Kac-Moody group over $\mathbb{C}$ completed along
the negative roots and $G^{\text{min}}\subset G$ the ‘minimal’ Kac-Moody group
as in [Kum02, §7.4]. Let $B$ be the standard (positive) Borel subgroup,
$B^{-}$ the standard negative Borel subgroup, $H=B\cap B^{-}$ the standard
maximal torus and $W$ the Weyl group. Let $\bar{X}=G/B$ be the ‘thick’ flag
variety (introduced by Kashiwara) which contains the standard flag variety
$X=G^{\text{min}}/B$. Let $T$ be the adjoint torus, i.e.,
$T:=H/Z(G^{\text{min}})$, where $Z(G^{\text{min}})$ denotes the center of
$G^{\text{min}}$ and let $R(T)$ denote the representation ring of $T$. For any
$w\in W$ (the Weyl group), we have the Schubert cell $C_{w}:=BwB/B\subset X$,
the Schubert variety $X_{w}:=\overline{C_{w}}\subset X$, the opposite Schubert
cell $C^{w}:=B^{-}wB/B\subset\bar{X}$, and the opposite Schubert variety
$X^{w}:=\overline{C^{w}}\subset\bar{X}$. When $G$ is a (finite-dimensional)
semisimple group, it is referred to as the finite case.
Let $P$ be any standard parabolic subgroup of $G$ of finite type (i.e., the
Levi component of $P$ is finite dimensional) and set $W^{P}$ to be the set of
minimal length coset representatives of $W/W_{P}$, where $W_{P}$ is the Weyl
group of $P$. Then, for any $w\in W^{P}$, we have the Schubert cell
$C_{w}^{P}:=BwP/P\subset{X_{P}}$, Schubert variety
$X_{w}^{P}:=\overline{C_{w}^{P}}\subset{X_{P}}$, the opposite Schubert cell
$C^{w}_{P}:=B^{-}wP/P\subset\bar{X}_{P}$, and the opposite Schubert variety
$X^{w}_{P}:=\overline{C^{w}_{P}}\subset\bar{X}_{P}$ where
$X_{P}=G^{\text{min}}/P$ and $\bar{X}_{P}=G/P$.
Let $K^{0}_{T}(\bar{X}_{P})$ be the Grothendieck group of $T$-equivariant
coherent $\mathcal{O}_{\bar{X}_{P}}$-modules of the nonquasi-compact scheme
$\bar{X}_{P}$ (cf. Definition 3.1). Let $\\{[\xi^{w}_{P}]\\}_{w\in W^{P}}$ be
the ‘basis’ of $K^{0}_{T}(\bar{X}_{P})$ given by
$\xi^{w}_{P}:=\mathcal{O}_{X^{w}_{P}}(-\partial X^{w}_{P})$ (where $\partial
X^{w}_{P}:=X^{w}_{P}\setminus C^{w}_{P}$) and express the product in
$K^{0}_{T}(\bar{X}_{P})$ in this ‘basis’:

$[\xi^{u}_{P}]\cdot[\xi^{v}_{P}]=\sum_{w\in
W^{P}}d_{u,v}^{w}(P)[\xi^{w}_{P}],\,\,\,\text{ for some
unique}\,d_{u,v}^{w}(P)\in R(T).$ (1)

The above sum, in general, is infinite.

The following result is our main theorem (cf. Theorem 5.5). This was first
conjectured by Graham-Kumar [GK08] in the finite case, proven in this case by
Anderson-Griffeth-Miller [AGM11], and then proven in the general Kac-Moody
case for $G/B$ by Kumar [Kum17].

######  Theorem 1.1.

For any $u,v,w\in W^{P}$,

$(-1)^{l(u)+l(v)+l(w)}d_{u,v}^{w}(P)\in\mathbb{Z}_{\geq
0}[(e^{-\alpha_{1}}-1),\cdots,(e^{-\alpha_{r}}-1)],$

where $\\{\alpha_{1},\cdots,\alpha_{r}\\}$ are the simple roots, i.e.,
$(-1)^{l(u)+l(v)+l(w)}d_{u,v}^{w}(P)$ is a polynomial in the variables
$x_{1}=e^{-\alpha_{1}}-1,\cdots,x_{r}=e^{-\alpha_{r}}-1$ with nonnegative
integral coefficients.

The proof of the above theorem generally follows the proof in [Kum17] for the
full flag variety $\bar{X}$. However, several of the geometric and
cohomological results in the case of the full flag variety had to be
generalized to the partial flag varieties $\bar{X}_{P}$. In particular, the
definition of the basis in terms of the dualizing sheaves had to be modified
in the parabolic case as the dualizing sheaf in this general case is not
suitable.

In another related work, Baldwin-Kumar [BK17] proved an analogue of the above
theorem for the basis consisting of the structure sheaves of the opposite
Schubert varieties.

We have added an appendix to determine the dualizing sheaf
$\omega_{X^{w}_{P}}$ of the Cohen-Macaulay scheme
$X^{w}_{P}\subset\bar{X}_{P}$ for any $w\in W^{P}$. Even though this result is
not used in the paper, we believe that it is interesting on its own.

Acknowledgements: The second author was partially supported by the NSF grant
DMS-1802328.

##  2\. Notation

We work over the field $\mathbb{C}$ of complex numbers. By a variety, we mean
an algebraic variety over $\mathbb{C}$ which is reduced, but not necessarily
irreducible. For a scheme $X$ and a closed subscheme $Y$,
$\mathcal{O}_{X}(-Y)$ denotes the ideal sheaf of $Y$ in $X$.
Let $G$ be any symmetrizable Kac-Moody group completed along the negative
roots (as opposed to along the positive roots as in [Kum02, Chapter 6]), and
let $G^{\text{min}}\subset G$ be the ‘minimal’ Kac-Moody group as in [Kum02,
§7.4]. Let $B$ be the standard (positive) Borel subgroup, $B^{-}$ the standard
negative Borel subgroup, $H=B\cap B^{-}$ the standard maximal torus and $W$
the Weyl group [Kum02, Chapter 6]. For any standard parabolic subgroup $P$ of
$G$ of finite type (i.e., the Levi component of $P$ is finite dimensional),
let

$\bar{X}_{P}=G/P$

be the ‘thick’ partial flag variety which contains the standard Kac-Moody
partial flag ind-variety

$X_{P}=G^{\text{min}}/P.$

If $G$ is not of finite type, then $\bar{X}_{P}$ is an infinite-dimensional
nonquasi-compact scheme (cf. [Kas89, $\S$4] for the case of $P=B$; the case of
general $P$ is similar) and $X_{P}$ is an ind-projective variety [Kum02,
§7.1]. The group $G^{\text{min}}$ (in particular, the maximal torus $H$) acts
on $X_{P}$ and $\bar{X}_{P}$. Let $T$ be the quotient $H/Z(G^{\text{min}})$,
where $Z(G^{\text{min}})$ is the center of $G^{\text{min}}$. Then the action
of $H$ on $\bar{X}_{P}$ (and $X_{P}$) descends to an action of $T$.

Let $W^{P}$ denote the set of minimal length coset representatives in the
quotient $W/W_{P}$, where $W_{P}\subset W$ is the Weyl group of $P$. Then, for
any $w\in W^{P}$, we have the Schubert cell

$C_{w}^{P}:=BwP/P\subset X_{P},$

the Schubert variety

$X_{w}^{P}:=\overline{C_{w}^{P}}=\bigsqcup_{\begin{subarray}{c}u\leq w\\\ u\in
W^{P}\end{subarray}}C_{u}^{P}\subset X_{P},$

the opposite Schubert cell

$C^{w}_{P}:=B^{-}wP/P\subset\bar{X}_{P},$

and the opposite Schubert variety

$X^{w}_{P}:=\overline{C^{w}_{P}}=\bigsqcup_{\begin{subarray}{c}u\geq w\\\ u\in
W^{P}\end{subarray}}C^{u}_{P}\subset\bar{X}_{P},$

all endowed with the reduced subscheme structures. When $P=B$, we drop the
qualification $P$, thus $\bar{X}=\bar{X}_{B},X_{w}=X_{w}^{B}$ etc. Then,
$X_{w}^{P}$ is a (finite-dimensional) irreducible projective subvariety of
$X_{P}$ and $X^{w}_{P}$ is a finite-codimensional irreducible subscheme of
$\bar{X}_{P}$. We denote by $Z_{w}$ the Bott-Samelson-Demazure-Hansen (BSDH)
variety as in [Kum02, §7.1], which is a $B$-equivariant desingularization of
$X_{w}^{P}$ [Kum02, Prop. 7.1.15]. Further, $X_{w}^{P}$ is normal and has
rational singularities (in particular, Cohen-Macaulay for short CM) [Kum02,
Theorem 8.2.2].
We also define the boundary of the Schubert variety by

$\partial X_{w}^{P}:=X_{w}^{P}\ \backslash\ C_{w}^{P}$

with the reduced subscheme structure. For any $u,w\in W^{P}$ with $u\leq w$,
we have the Richardson variety

$X_{w}^{u}(P):=X^{u}_{P}\cap X_{w}^{P}\subset X_{P}$

endowed with the reduced subvariety structure. By [KS14, Proposition 5.3]
together with [BK05, Lemma 1.1.8], $X_{P}$ is Frobenius split in any
characteristic $p>0$ compatibly splitting $\\{X_{w}^{P},X^{w}_{P}\cap
X_{P}\\}_{w\in W^{P}}$. In particular, any scheme theoretic intersection
$X^{P}_{w_{1}}\cap\dots\cap X^{P}_{w_{m}}\cap X_{P}^{v_{1}}\cap\dots\cap
X_{P}^{v_{n}}$ (for $m\geq 1$) is reduced.

We denote by $Z_{w}^{u}$ the $T$-equivariant desingularization of $X_{w}^{u}$
as in [Kum17, Theorem 6.8]. For $u,w\in W^{P}$, the canonical projection map
$X^{u}_{w}\to X^{u}_{w}(P)$ is a proper birational morphism, where $C^{u}\cap
C_{w}$ maps isomorphically onto a $T$-stable open subset of $C^{u}_{P}\cap
C^{P}_{w}$ (cf. [Deo87, Lemma 4.3]). Hence, $Z^{u}_{w}$ is a $T$-equivariant
desingularization of $X^{u}_{w}(P)$.

We denote the representation ring of $T$ by $R(T)$. Let
$\\{\alpha_{1},\cdots,\alpha_{r}\\}\subset\mathfrak{h}^{*}$ be the set of
simple roots,
$\\{\alpha_{1}^{\vee},\cdots,\alpha_{r}^{\vee}\\}\subset\mathfrak{h}$ the set
of simple coroots, and $\\{s_{1},\cdots,s_{r}\\}\subset W$ the corresponding
set of simple reflections, where $\mathfrak{h}:=\text{Lie}(H)$. Let
$\rho\in\mathfrak{h}^{*}$ be any integral weight satisfying

$\rho(\alpha_{i}^{\vee})=1,\ \text{for all }\ 1\leq i\leq r.$

If $G$ is a finite-dimensional semisimple group, $\rho$ is unique, but for a
general KM group $G$, it may not be unique. Denote by $\rho_{Y}:=\sum_{i\in
Y}\varpi_{i}$, where $\\{\varpi_{1},\cdots,\varpi_{r}\\}$ are fixed
fundamental weights and $Y\subset\\{1,\cdots,r\\}$ corresponds to the Levi
component of $P$, i.e., $\\{\alpha_{i}\\}_{i\in Y}$ are the simple roots of
the Levi component $L$ of $P$ containing $H$. (Observe that, in general,
$\varpi_{i}$ are not unique.) We then define the integral weight
$\hat{\rho}_{Y}\in\mathfrak{h}^{*}$ by

$\hat{\rho}_{Y}:=\rho-\rho_{Y}.$

For any integral weight $\lambda$, let $\mathbb{C}_{\lambda}$ denote the one-
dimensional representation of $H$ on $\mathbb{C}$ given by $h\cdot
z=\lambda(h)z$ for $h\in H$ and $z\in\mathbb{C}$. It uniquely extends to a
representation of $B$. We call $\lambda$ a $P$-weight if
$\dot{\lambda}(\alpha_{i}^{\vee})=0$ for all $i\in Y$, where $\dot{\lambda}$
is the derivative of $\lambda$. If $\lambda$ is a $P$-weight, this action
extends uniquely to an action of $P$ and we define the $G$-equivariant line
bundle $\mathcal{L}^{P}(\lambda)$ on $\bar{X}_{P}$ by

$\mathcal{L}^{P}(\lambda):=G\times^{P}\mathbb{C}_{-\lambda},$

where for any representation $V$ of $P$, $G\times^{P}V:=(G\times V)/P$ where
$P$ acts on $G\times V$ by $(g,v)\cdot p=(gp,p^{-1}v)$ for $g\in G,\ v\in V$
and $p\in P$. We also define the line bundle

$e^{\lambda}:=\bar{X}_{P}\times\mathbb{C}_{\lambda},$

which while trivial as a non-equivariant line bundle, is equivariantly non-
trivial with the diagonal action of $H$.

##  3\. Identification of the dual of the structure sheaf basis

######  Definition 3.1.

For a quasi-compact scheme $Y$, an $\mathcal{O}_{Y}$-module $\mathcal{S}$ is
called _coherent_ if it is finitely presented as an $\mathcal{O}_{Y}$-module
and any $\mathcal{O}_{Y}$-submodule of finite type admits a finite
presentation.

A subset $S\subset W^{P}$ is called an _ideal_ if $x\in S$ and $v\leq x$ imply
$v\in S$. An $\mathcal{O}_{\bar{X}_{P}}$-module $\mathcal{S}$ is called
_coherent_ if $\mathcal{S}|_{V^{S}}$ is a coherent
$\mathcal{O}_{V^{S}}$-module for any finite ideal $S\subset W^{P}$, where
$V^{S}$ is the quasi-compact open subset of $\bar{X}_{P}$ defined by

$V^{S}=\bigcup_{w\in S}B^{-}wP/P=\bigcup_{w\in S}wU^{-}P/P,$

where $U^{-}$ is the unipotent part of $B^{-}$. Observe that
$\\{wU^{-}P/P\\}_{w\in W^{P}}$ is an open cover of $\bar{X}_{P}$. Let
$K^{0}_{T}(\bar{X}_{P})$ denote the Grothendieck group of $T$-equivariant
coherent $\mathcal{O}_{\bar{X}_{P}}$-modules $\mathcal{S}$. Since the
coherence condition on $\mathcal{S}$ is imposed only for
$\mathcal{S}|_{V^{S}}$ for finite ideals $S$, $K^{0}_{T}(\bar{X}_{P})$ can be
thought of as the inverse limit of $K^{0}_{T}(V^{S})$ as $S$ varies over all
finite ideals of $W^{P}$ [KS09, §2].

Similarly, define
$K_{0}^{T}(X_{P}):=\text{Lim}_{n\to\infty}K_{0}^{T}(X^{P}_{n})$, where
$\\{X^{P}_{n}\\}_{n\geq 1}$ is the filtration of $X_{P}$ giving $X_{P}$ its
ind-projective variety structure (i.e., $X^{P}_{n}=\bigcup_{l(w)\leq n,w\in
W^{P}}BwP/P$) and $K_{0}^{T}(X^{P}_{n})$ is the Grothendieck group of
$T$-equivariant coherent sheaves on the projective variety $X_{n}^{P}$.

For any $w\in W^{P}$,

$[\mathcal{O}_{X_{w}^{P}}]\in K_{0}^{T}(X_{P}),$

where $[\mathcal{O}_{X_{w}^{P}}]$ denotes the class of
$\mathcal{O}_{X_{w}^{P}}$ in $K_{0}^{T}(X_{P})$.

######  Lemma 3.2.

$\\{[\mathcal{O}_{X_{w}^{P}}]\\}_{w\in W^{P}}$ forms a basis of
$K_{0}^{T}(X_{P})$ as an $R(T)$-module.

###### Proof.

Apply [CG97, §5.2.14 and Theorem 5.4.17]. ∎

For any $u\in W^{P}$, $\mathcal{O}_{X_{P}^{u}}$ is a coherent
$\mathcal{O}_{\bar{X}_{P}}$-module. We record the following lemma due to
Kashiwara-Shimozono [KS09, Proof of Lemma 8.1].

######  Lemma 3.3.

Any $T$-equivariant coherent sheaf $\mathcal{S}$ on $V^{u}$ admits a free
resolution in $\textup{Coh}_{T}(\mathcal{O}_{V^{u}})$:

$0\to S_{n}\otimes\mathcal{O}_{V^{u}}\to\cdots\to
S_{1}\otimes\mathcal{O}_{V^{u}}\to
S_{0}\otimes\mathcal{O}_{V^{u}}\to\mathcal{S}\to 0,$

where $S_{k}$ are finite-dimensional $T$-modules,
$V^{u}:=uU^{-}P/P\subset\bar{X}_{P}$, and
$\textup{Coh}_{T}(\mathcal{O}_{V^{u}})$ is the abelian category of
$T$-equivariant coherent $\mathcal{O}_{V^{u}}$-modules.

We define a pairing

$\langle{\ ,\ }\rangle:K^{0}_{T}(\bar{X}_{P})\otimes K_{0}^{T}(X_{P})\to
R(T),\hskip
28.45274pt\langle{[\mathcal{S}],[\mathcal{F}]}\rangle=\sum_{i}(-1)^{i}\chi_{T}(X^{P}_{n},\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{\bar{X}_{P}}}(\mathcal{S},\mathcal{F})),$
(2)

where $\mathcal{S}$ is a $T$-equivariant coherent sheaf on $\bar{X}_{P}$ and
$\mathcal{F}$ is a $T$-equivariant coherent sheaf on $X_{P}$ supported on
$X_{n}^{P}$ for some $n$, and where $\chi_{T}$ is the $T$-equivariant Euler-
Poincaré characteristic. This pairing is well defined by [Kum17, Proof of
Lemma 3.5].

######  Definition 3.4.

For any $u\in W^{P}$, set the $T$-equivariant sheaf on $\bar{X}_{P}$,

$\xi^{u}_{P}:=\mathcal{O}_{X^{u}_{P}}(-\partial X^{u}_{P}),\,\,\text{where
$\partial X^{u}_{P}:=X^{u}_{P}\setminus C^{u}_{P}$ with the reduced scheme
structure.}$

######  Proposition 3.5.

$\\{[\xi^{u}_{P}]\\}_{u\in W^{P}}$ forms an infinite basis of
$K^{0}_{T}(\bar{X}_{P})$ as an $R(T)$-module, where by infinite basis we mean
that $K^{0}_{T}(\bar{X}_{P})=\prod_{u\in W^{P}}\,R(T)[\xi^{u}_{P}]$.

###### Proof.

By the same proof as in [KS09, $\S$2] $\\{[\mathcal{O}_{X_{P}^{v}}]\\}_{v\in
W^{P}}$ is an infinite basis of $K^{0}_{T}(\bar{X}_{P})$. Moreover,
$\displaystyle[\xi^{u}_{P}]=[\mathcal{O}_{X^{u}_{P}}]+\sum_{\begin{subarray}{c}w^{\prime}>u\\\
w^{\prime}\in
W^{P}\end{subarray}}r_{w^{\prime}}[\mathcal{O}_{X^{w^{\prime}}_{P}}]$ for some
$r_{w^{\prime}}\in R(T)$, the lemma follows. ∎

######  Proposition 3.6.

For any finite union $Y=\bigcup_{i=1}^{k}X^{v_{i}}_{P}$ of opposite Schubert
varieties and any $w$ in $W^{P}$,

  1. (a)

$\mathscr{T}{\kern-3.0pt{or}}_{j}^{\mathcal{O}_{\bar{X}_{P}}}(\mathcal{O}_{Y},\mathcal{O}_{X_{w}^{P}})=0$
for all $j>0$.

  2. (b)

$H^{j}(X_{n}^{P},\mathcal{O}_{Y\cap X_{w}^{P}})=0$ for all $j>0$, where $n$ is
any positive integer such that $X_{n}^{P}\supset X_{w}^{P}$.

In particular, this applies to the case $Y=\partial X^{u}_{P}$.

###### Proof.

This follows by the same argument as [Kum17, Proof of Corollary 5.7]. Observe
that $X^{u}_{w}(P)$ is irreducible by Lemma 7.4. Moreover, the vanishing
$H^{j}(X_{n}^{P},\mathcal{O}_{X_{w}^{v}(P)})=0$, for all $j>0$, follows from
the corresponding result for $\bar{X}$ (cf. [KS14, Corollary 3.2]) and
applying the Leray spectral sequence for the bundle
$\pi:\bar{X}\to\bar{X}_{P}$, since
$\pi^{-1}(X^{v}_{w}(P))=X^{v}_{ww_{o}^{P}}$, where $w_{o}^{P}$ is the longest
element of $W_{P}$. ∎

######  Proposition 3.7.

For any $v,w\in W^{P}$,

$\mathscr{T}{\kern-3.0pt{or}}_{j}^{\mathcal{O}_{\bar{X}_{P}}}(\xi^{v}_{P},\mathcal{O}_{X_{w}^{P}})=0\hskip
14.22636pt\text{for all}\ j>0.$

###### Proof.

By Proposition 3.6(a), we have the vanishing

$\mathscr{T}{\kern-3.0pt{or}}_{j}^{\mathcal{O}_{\bar{X}_{P}}}(\mathcal{O}_{X^{v}_{P}},\mathcal{O}_{X_{w}^{P}})=0\hskip
14.22636pt\text{for all}\ j>0.$

The proposition follows from this together with Proposition 3.6 part (a)
applied to $Y=\partial X^{v}_{P}$ and the long exact sequence for
$\mathscr{T}{\kern-3.0pt{or}}$ associated to the short exact sequence

$0\to\xi^{v}_{P}\to\mathcal{O}_{X^{v}_{P}}\to\mathcal{O}_{\partial
X^{v}_{P}}\to 0.\qed$

######  Proposition 3.8.

For any $u,v\in W^{P}$,

$\langle{[\xi^{u}_{P}],[\mathcal{O}_{X_{v}^{P}}]}\rangle=\delta_{u,v}.$

###### Proof.

For any $u,w\in W^{P}$, the pairing is by definition

$\langle{[\xi^{u}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle=\sum_{i}(-1)^{i}\chi_{T}(X^{P}_{n},\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{\bar{X}_{P}}}(\xi^{u}_{P},\mathcal{O}_{X_{w}^{P}})),$

with $n$ taken such that $n\geq l(w)$. By Proposition 3.7, this becomes

$\langle{[\xi^{u}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle=\chi_{T}(X_{n}^{P},\xi^{u}_{P}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\mathcal{O}_{X_{w}^{P}}).$
(3)

From Proposition 3.6(a) and the definition
$\xi^{u}_{P}:=\mathcal{O}_{X^{u}_{P}}(-\partial X^{u}_{P})$, we have the sheaf
exact sequence

$0\to\xi^{u}_{P}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\mathcal{O}_{X_{w}^{P}}\to\mathcal{O}_{X^{u}_{P}}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\mathcal{O}_{X_{w}^{P}}\to\mathcal{O}_{\partial
X^{u}_{P}}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\mathcal{O}_{X_{w}^{P}}\to 0.$

Hence,

$\chi_{T}(X_{n}^{P},\xi^{u}_{P}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\mathcal{O}_{X_{w}^{P}})=\chi_{T}(X_{n}^{P},\mathcal{O}_{X^{u}_{w}(P)})-\chi_{T}(X_{n}^{P},\mathcal{O}_{(\partial
X^{u}_{P})\cap X_{w}^{P}}).$ (4)

From Lemma 7.4, the Richardson variety $X^{u}_{w}(P)$ is irreducible (when
nonempty, i.e., $u\leq w$) and hence $(\partial X^{u}_{P})\cap
X_{w}^{P}=\bigcup_{u<v\leq w}X^{v}_{w}(P)$ is connected (when nonempty) since
$wP\in X^{v}_{w}(P)$ for all $u<v\leq w$. If $u\not\leq w$, $X^{u}_{w}(P)$ is
empty, and hence Equations 3 and 4 imply
$\langle{[\xi^{u}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle=0$. Hence, we assume
$u\leq w$ so that $X^{u}_{w}(P)$ is nonempty. Since $X^{u}_{w}(P)$ has
rational singularities (Lemma 7.4)

$H^{i}(X_{n}^{P},\mathcal{O}_{X^{u}_{w}(P)})=0\hskip 14.22636pt\text{for all}\
i>0.$

Further, by Proposition 3.6 (b) applied to $\partial X^{u}_{P}$,

$H^{i}(X_{n}^{P},\mathcal{O}_{(\partial{X^{u}_{P}})\cap X_{w}^{P}})=0\hskip
14.22636pt\text{for all}\ i>0.$

Thus, for $u\leq w$,

$\chi_{T}(X_{n}^{P},\mathcal{O}_{X^{u}_{w}(P)})=1$

and for $u<w$,

$\chi_{T}(X_{n}^{P},\mathcal{O}_{(\partial X^{u}_{P})\cap X_{w}^{P}})=1.$

Therefore, when $u<w$,
$\langle{[\xi^{u}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle=0$. Finally, if
$u=w$, we have $\langle{[\xi^{w}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle=1$
since $\partial X^{u}_{P}\cap X^{P}_{w}$ is empty in this case. ∎

Let $\Delta:X_{P}\to X_{P}\times X_{P}$ be the diagonal map. Express the
coproduct in $K_{0}^{T}(X_{P})$:

$\Delta_{*}[\mathcal{O}_{X_{w}^{P}}]=\sum_{u,v\in
W^{P}}q_{u,v}^{w}(P)[\mathcal{O}_{X_{u}^{P}}]\otimes[\mathcal{O}_{X_{v}^{P}}],\,\,\,\text{for
$q^{w}_{u,v}\in R(T)$}.$

Also, express the product in $K_{T}^{0}(\bar{X}_{P})$

$[\xi^{u}_{P}]\cdot[\xi^{v}_{P}]=\sum_{w\in
W^{P}}d_{u,v}^{w}(P)[\xi^{w}_{P}],\,\,\,\text{for $d^{w}_{u,v}(P)\in R(T)$}.$
(5)

######  Proposition 3.9.

For any $u,v,w\in W^{P}$,

$q_{u,v}^{w}(P)=d_{u,v}^{w}(P).$

###### Proof.

Let $\bar{\Delta}:\bar{X}_{P}\to\bar{X}_{P}\times\bar{X}_{P}$ be the diagonal
map. Then, for any $w\in W^{P}$,

$\displaystyle\langle{\bar{\Delta}^{*}[\xi^{u}_{P}\boxtimes\xi^{v}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle$
$\displaystyle=\langle{[\xi^{u}_{P}\boxtimes\xi^{v}_{P}],\Delta_{*}[\mathcal{O}_{X_{w}^{P}}]}\rangle$
$\displaystyle=\langle{[\xi^{u}_{P}\boxtimes\xi^{v}_{P}],\sum_{u^{\prime},v^{\prime}\in
W^{P}}q_{u^{\prime},v^{\prime}}^{w}(P)[\mathcal{O}_{X_{u^{\prime}}^{P}}]\otimes[\mathcal{O}_{X_{v^{\prime}}^{P}}]}\rangle$
$\displaystyle=q_{u,v}^{w}(P),\hskip 28.45274pt\text{by Proposition
\ref{duality}}.$

On the other hand, since
$[\xi^{u}_{P}]\cdot[\xi^{v}_{P}]=\bar{\Delta}^{*}[\xi^{u}_{P}\boxtimes\xi^{v}_{P}],$
we also have

$\displaystyle\langle{\bar{\Delta}^{*}[\xi^{u}_{P}\boxtimes\xi^{v}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle$
$\displaystyle=\langle{[\xi^{u}_{P}]\cdot[\xi^{v}_{P}],[\mathcal{O}_{X_{w}^{P}}]}\rangle$
$\displaystyle=\left\langle\sum_{w^{\prime}\in
W^{P}}d_{u,v}^{w^{\prime}}(P)[\xi^{w^{\prime}}_{P}],[\mathcal{O}_{X_{w}^{P}}]\right\rangle$
$\displaystyle=d_{u,v}^{w}(P),\hskip 28.45274pt\text{by Proposition
\ref{duality}}\qed$

##  4\. The Mixing Space and Mixing Group

In this section, we introduce the mixing space $(X_{P})_{\mathbb{P}}$, which
is a bundle over a product of projective spaces with fiber $X_{P}$. This
allows the reduction from the $T$-equivariant $K$-theory to the non-
equivariant $K$-theory. We then introduce the mixing group $\Gamma$ whose
action is sufficient to allow for a transversality result used to prove part
of our main technical result.

Fix a large $N$ and let $\mathbb{P}:=(\mathbb{P}^{N})^{r}$, where $r=\dim T$.
For any $\mathbf{j}=(j_{1},\cdots,j_{r})\in[N]^{r}$, where
$[N]:=\\{0,1,\cdots,N\\}$, set

$\mathbb{P}^{\mathbf{j}}=\mathbb{P}^{N-{j_{1}}}\times\cdots\times\mathbb{P}^{N-{j_{r}}}\text{
and
}\mathbb{P}_{\mathbf{j}}=\mathbb{P}^{j_{1}}\times\cdots\times\mathbb{P}^{j_{r}}.$

We also define the boundary of $\mathbb{P}_{\mathbf{j}}$ by

$\partial\mathbb{P}_{\mathbf{j}}:=(\mathbb{P}^{j_{1}-1}\times\mathbb{P}^{j_{2}}\times\cdots\times\mathbb{P}^{j_{r}})\cup\cdots\cup(\mathbb{P}^{j_{1}}\times\mathbb{P}^{j_{2}}\times\cdots\times\mathbb{P}^{j_{r}-1}),$

where we interpret $\mathbb{P}^{-1}:=\varnothing$ as the empty set. Throughout
the paper, we fix an identification $T\cong(\mathbb{C}^{*})^{r}$ under
$t\mapsto(e^{\alpha_{1}}(t),\cdots,e^{\alpha_{r}}(t))$.

######  Definition 4.1.

Let $E(T)_{\mathbb{P}}:=(\mathbb{C}^{N+1}\setminus\\{0\\})^{r}$ be the total
space of the standard principal $T$-bundle $E(T)_{\mathbb{P}}\to\mathbb{P}$.
Let
$\pi_{X_{P}}:(X_{P})_{\mathbb{P}}:=E(T)_{\mathbb{P}}\times^{T}X_{P}\to\mathbb{P}$
be the fibration with fiber $X_{P}$ associated to the principal $T$-bundle
$E(T)_{\mathbb{P}}\to\mathbb{P}$, where we twist the action of $T$ on $X_{P}$
via

$t\odot x=t^{-1}x.$ (6)

For any $T$-subscheme $Y\subset X_{P}$, denote
$Y_{\mathbb{P}}:=E(T)_{\mathbb{P}}\times^{T}Y\subset(X_{P})_{\mathbb{P}}$.

The following proposition follows easily by using [CG97, $\S$5.2.14 and
Theorem 5.4.17] applied to the vector bundle
$(BwP/P)_{\mathbb{P}}\to\mathbb{P}$.

######  Proposition 4.2.

$K_{0}((X_{P})_{\mathbb{P}}):=\textup{Lim}_{n\to\infty}K_{0}((X^{P}_{n})_{\mathbb{P}})$
is a free module over the ring $K_{0}(\mathbb{P})=K^{0}(\mathbb{P})$ with
basis $\\{[\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}]\\}_{w\in W^{P}}$.
Therefore, $K_{0}((X_{P})_{\mathbb{P}})$ has a $\mathbb{Z}$-basis

$\\{\pi_{X_{P}}^{*}([\mathcal{O}_{\mathbb{P}^{\mathbf{j}}}])\cdot[\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}]\\}_{\mathbf{j}\in[N]^{r},\
w\in W^{P}},$

where we view $[\mathcal{O}_{\mathbb{P}^{\mathbf{j}}}]$ as an element of
$K_{0}(\mathbb{P})=K^{0}(\mathbb{P})$.

Let $Y_{P}:=X_{P}\times X_{P}$. The diagonal map $\Delta:X_{P}\to Y_{P}$ gives
rise to the embedding

$\tilde{\Delta}:(X_{P})_{\mathbb{P}}\to(Y_{P})_{\mathbb{P}}=E(T)_{\mathbb{P}}\times^{T}Y_{P}\cong(X_{P})_{\mathbb{P}}\times_{\mathbb{P}}(X_{P})_{\mathbb{P}}.$

Therefore, we have (denoting the projection
$(Y_{P})_{\mathbb{P}}\to\mathbb{P}$ by $\pi_{Y_{P}}$)

$\tilde{\Delta}_{*}[\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}]=\sum_{\begin{subarray}{c}u,v\in
W^{P}\\\
\mathbf{j}\in[N]^{r}\end{subarray}}c_{u,v}^{w}(\mathbf{j})\pi^{*}_{Y_{P}}([\mathcal{O}_{\mathbb{P}^{\mathbf{j}}}])\cdot[\mathcal{O}_{(X_{u}^{P}\times
X_{v}^{P})_{\mathbb{P}}}]\in K_{0}((Y_{P})_{\mathbb{P}})$ (7)

for some $c_{u,v}^{w}(\mathbf{j})\in\mathbb{Z}$.

Let $\bar{Y}_{P}=\bar{X}_{P}\times\bar{X}_{P}$ and let
$K^{0}((\bar{Y}_{P})_{\mathbb{P}})$ denote the Grothendieck group associated
to the semi-group of coherent
$\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}$-modules. Define, for $u,v\in
W^{P}$,

$\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}:=\mathcal{O}_{(X^{u}_{P}\times
X^{v}_{P})_{\mathbb{P}}}(-\partial((X^{u}_{P}\times
X^{v}_{P})_{\mathbb{P}}))\in K^{0}((\bar{Y}_{P})_{\mathbb{P}}),$

where $\partial((X^{u}_{P}\times X^{v}_{P})_{\mathbb{P}}):=((\partial
X^{u}_{P}\times X^{v}_{P})\cup(X^{u}_{P}\times\partial
X^{v}_{P}))_{\mathbb{P}}$.

######  Lemma 4.3.

With the notation as above,

$c_{u,v}^{w}(\mathbf{j})=\langle{\pi_{\bar{Y}_{P}}^{*}[\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})]\cdot[\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}],\tilde{\Delta}_{*}[\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}]}\rangle,$
(8)

where $\pi_{\bar{Y}_{P}}:(\bar{Y}_{P})_{\mathbb{P}}\to\mathbb{P}$ is the
projection and the pairing $\langle{\ ,\
}\rangle:K^{0}((\bar{Y}_{P})_{\mathbb{P}})\otimes
K_{0}((Y_{P})_{\mathbb{P}})\to\mathbb{Z}$ is defined similar to (2) above.
Explicitly,

$\langle{[\mathcal{S}],[\mathcal{F}]}\rangle=\sum_{i}(-1)^{i}\chi((\bar{Y}_{P})_{\mathbb{P}},\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\mathcal{S},\mathcal{F})),$
(9)

where $\chi$ denotes the (non-equivariant) Euler-Poincaré characteristic.

###### Proof.

We compute

$\displaystyle\langle{\pi_{\bar{Y}_{P}}^{*}[\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})]\cdot[\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}],\tilde{\Delta}_{*}[\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}]}\rangle$
$\displaystyle\hskip
8.53581pt=\langle{\pi_{\bar{Y}_{P}}^{*}[\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})]\cdot[\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}],\sum_{\begin{subarray}{c}u^{\prime},v^{\prime}\in
W^{P}\\\
\mathbf{j}^{\prime}\in[N]^{r}\end{subarray}}c_{u^{\prime},v^{\prime}}^{w}(\mathbf{j}^{\prime})\pi^{*}_{Y_{P}}([\mathcal{O}_{\mathbb{P}^{\mathbf{j}^{\prime}}}])\cdot[\mathcal{O}_{(X_{u^{\prime}}^{P}\times
X_{v^{\prime}}^{P})_{\mathbb{P}}}]}\rangle$ $\displaystyle\hskip
8.53581pt=c_{u,v}^{w}(\mathbf{j})\hskip 14.22636pt\text{by
\lx@cref{creftypecap~refnum}{duality} and \cite[cite]{[\@@bibref{}{Ku}{}{},
identity (20)]}}.\qed$

######  Definition 4.4.

Let $T$ act on $B$ via

$t\cdot b=t^{-1}bt$

where $t\in T$ and $b\in B$. Then there is a natural action of $\Delta T$ on
$B\times B$. Let $(B^{2})_{\mathbb{P}}$ be the ind-group scheme over
$\mathbb{P}$:

$(B^{2})_{\mathbb{P}}:=E(T)_{\mathbb{P}}\times^{T}(B\times B)\to\mathbb{P}$

and let $\Gamma_{0}$ denote the group of global sections of
$(B^{2})_{\mathbb{P}}$ under pointwise multiplication. Thus, $\Gamma_{0}$ can
be identified with the set of regular maps $f:E(T)_{\mathbb{P}}\to B\times B$
such that $f(e\cdot t)=t^{-1}\cdot f(e)$ for all $e\in E(T)_{\mathbb{P}}$ and
$t\in T$. Now, $GL(N+1)^{r}$ acts canonically on $(B^{2})_{\mathbb{P}}$ in a
way compatible with its action on $\mathbb{P}$ and acts on $\Gamma_{0}$ via
its pull-back. We define the mixing group
$\Gamma=\Gamma_{B^{2}}:=\Gamma_{0}\rtimes GL(N+1)^{r}$:

$1\to\Gamma_{0}\to\Gamma\to GL(N+1)^{r}\to 1.$

By the comments following [Kum17, Lemmas 4.7 and 4.8], we have the following
two lemmas.

######  Lemma 4.5.

$\Gamma$ is connected.

######  Lemma 4.6.

Given any $\bar{e}\in\mathbb{P}$ and any $(b_{1},b_{2})$ in the fiber of
$(B^{2})_{\mathbb{P}}$ over $\bar{e}$, there exists a section
$\gamma\in\Gamma_{0}$ such that $\gamma(\bar{e})=(b_{1},b_{2})$.

Define the action of $\Gamma$ on $(Y_{P})_{\mathbb{P}}$ by

$(\gamma,g)\cdot[e,(y,y^{\prime})]=[ge,\gamma(ge)\cdot(y,y^{\prime})]$

for $\gamma\in\Gamma_{0}$, $g\in GL(N+1)^{r}$, $e\in E(T)_{\mathbb{P}}$, and
$(y,y^{\prime})\in{Y}_{P}$, where the action of $\Gamma_{0}$ is via the
standard action of $B^{2}$ on ${Y}_{P}$. From Lemma 4.6, it follows that the
orbits of the $\Gamma$-action on $(Y_{P})_{\mathbb{P}}$ are precisely
$\\{(C_{u}^{P}\times C_{v}^{P})_{\mathbb{P}}\\}_{u,v\in W^{P}}$.

######  Proposition 4.7.

For any coherent sheaf $\mathcal{S}$ on $\mathbb{P}$, and any $u,v\in W^{P}$,

$\pi^{*}[\mathcal{S}]\cdot[\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}]=[\pi^{*}(\mathcal{S})\otimes_{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}})]\in
K^{0}((\bar{Y}_{P})_{\mathbb{P}}),$

where we abbreviate $\pi_{\bar{Y}_{P}}$ by $\pi$ and
$\pi^{*}(\mathcal{S})=\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}\otimes_{\mathcal{O}_{\mathbb{P}}}\mathcal{S}$.
In particular,

$\pi^{*}[\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})]\cdot[\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}]=[\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes_{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}})].$

###### Proof.

This follows from the same argument as [Kum17, Propositon 4.9]. ∎

##  5\. Statement of Main Results

The following is our main technical result. The proof of its two parts are
given in Sections 6 and 10 respectively.

######  Theorem 5.1.

For general $\gamma\in\Gamma$, $\mathbf{j}\in[N]^{r}$, and any $u,v,w\in
W^{P}$,

  1. (A)

$\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}),\gamma_{*}\tilde{\Delta}_{*}(\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}))=0$
for all $i>0$, where $\gamma\in\Gamma$ is viewed as an automorphism of the
scheme $(Y_{P})_{\mathbb{P}}$.

  2. (B)

For $c_{u,v}^{w}(\mathbf{j})\neq 0$, where $c_{u,v}^{w}(\mathbf{j})$ is
defined by (7) (see also (8)),

$H^{p}((\bar{Y}_{P})_{\mathbb{P}},\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}})\otimes\gamma_{*}\tilde{\Delta}_{*}(\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}))=0$

for all $p\neq|\mathbf{j}|+l(w)-(l(u)+l(v))$, where
$|\mathbf{j}|=\sum_{i=1}^{r}j_{i}$.

Since $\Gamma$ is connected (cf. Lemma 4.5), we have the following result as
an immediate corollary of Lemma 4.3, Proposition 4.7, and Theorem 5.1.

######  Corollary 5.2.

$(-1)^{|\mathbf{j}|+l(w)-(l(u)+l(v))}c_{u,v}^{w}(\mathbf{j})\in\mathbb{Z}_{\geq
0}$.

Recall the definition of the structure constants $d_{u,v}^{w}(P)\in R(T)$ (cf.
(5)) for the product in $K^{0}_{T}(\bar{X}_{P})$.

######  Lemma 5.3.

For any $u,v,w\in W^{P}$,
$d^{w}_{u,v}(P)\in\mathbb{Z}[(e^{-\alpha_{1}}-1),\dots,(e^{-\alpha_{r}}-1)]$.

###### Proof.

By [GK08, Proposition 3.5], which is valid in the symmetrizable Kac-Moody case
by the same proof, for any $u,v,w\in W^{P}$,

$d^{w}_{u,v}(P)=\sum_{u^{\prime}\in uW_{P},\\\ v^{\prime}\in
vW_{P}}\,d^{w}_{u^{\prime},v^{\prime}}(B).$ (10)

Now,
$d^{w}_{u^{\prime},v^{\prime}}(B)\in\mathbb{Z}[(e^{-\alpha_{1}}-1),\dots,(e^{-\alpha_{r}}-1)]$
by [Kum17, Lemma 4.12]. Hence, the lemma follows from the identity (10). ∎

The following lemma follows easily from the identity (7), Proposition 3.9,
[GK08, Lemma 6.2] (which is valid in the Kac-Moody case as well) (see also
[AGM11, §3]).

######  Lemma 5.4.

For any $u,v,w\in W^{P}$, we can choose a large enough $N$ (depending on
$u,v,w$) and write (cf. Lemma 5.3)

$d_{u,v}^{w}(P)=\sum_{\mathbf{j}}d_{u,v}^{w}(\mathbf{j})(e^{-\alpha_{1}}-1)^{j_{1}}\cdots(e^{-\alpha_{r}}-1)^{j_{r}}$

for some unique $d_{u,v}^{w}(\mathbf{j})\in\mathbb{Z}$, where
$\mathbf{j}=(j_{1},\cdots,j_{r})$. Then

$d_{u,v}^{w}(\mathbf{j})=(-1)^{|\mathbf{j}|}c_{u,v}^{w}(\mathbf{j}).$ (11)

The following main theorem of this paper is an immediate consequence of
Corollary 5.2 and Lemma 5.4, which was proved in the $G/B$ case by Kumar
[Kum17].

######  Theorem 5.5.

For any $u,v,w\in W^{P}$, and any symmetrizable Kac-Moody group $G$ and
parabolic subgroup $P$ of finite type, the structure constants in
$K^{0}_{T}((\bar{X}_{P})_{\mathbb{P}})$ satisfy

$(-1)^{l(u)+l(v)+l(w)}d_{u,v}^{w}(P)\in\mathbb{Z}_{\geq
0}[(e^{-\alpha_{1}}-1),\cdots,(e^{-\alpha_{r}}-1)].$ (12)

##  6\. Proof of part (a) of Theorem 5.1

###### Proof of Theorem 5.1(A).

Note that since the assertion is local in $\mathbb{P}$, we assume
$(\bar{Y}_{P})_{\mathbb{P}}\simeq\mathbb{P}\times\bar{Y}_{P}$. Then,

$\displaystyle\pi^{*}\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})\simeq\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})\boxtimes\mathcal{O}_{\bar{Y}_{P}},$
(13)
$\displaystyle\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}\cong\mathcal{O}_{\mathbb{P}}\boxtimes(\xi^{u}_{P}\boxtimes\xi^{v}_{P}),$
(14) $\displaystyle\mathcal{O}_{(X_{w}^{P}\times
X_{w}^{P})_{\mathbb{P}}}\simeq\mathcal{O}_{\mathbb{P}}\boxtimes(\mathcal{O}_{X_{w}^{P}}\boxtimes\mathcal{O}_{X_{w}^{P}}).$
(15)

We first show that for any $\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}$-module
$\mathcal{S}$ (where $(Y_{w}^{P})_{\mathbb{P}}:=(X_{w}^{P}\times
X_{w}^{P})_{\mathbb{P}}$),

$\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}),\mathcal{S})\\\
\simeq\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}}(\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}\otimes_{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}\left(\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}})\right),\mathcal{S}).$
(16)

To prove (16), we observe the following: Let $R,S$ be commutative rings with a
ring homomorphism $R\to S$, $M$ an $R$-module and $N$ an $S$-module. Then
$N\otimes_{S}(S\otimes_{R}M)\simeq N\otimes_{R}M$. This gives the following
isomorphism, provided $\textup{Tor}_{j}^{R}(S,M)=0$ for all $j>0$:

$\textup{Tor}^{R}_{i}(M,N)\simeq\textup{Tor}^{S}_{i}(S\otimes_{R}M,N).$ (17)

Then (16) follows with $R=\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}},\
S=\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}},\
M=\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}),$
and $N=\mathcal{S}$, provided that
$\mathscr{T}{\kern-3.0pt{or}}_{j}^{R}(S,M)=0$ for all $j>0$. But this follows
from the Kunneth formula, and the isomorphisms (13) - (15) together with
Proposition 3.7.

By Proposition 3.7 and isomorphism (17) applied to
$R=\mathcal{O}_{\bar{X}_{P}},\ S=\mathcal{O}_{X_{w}^{P}},\ M=\xi^{u}_{P},$ and
$N=\mathcal{O}_{X_{x}^{P}}$ (for $x\leq w,x\in W^{P}$), we have

$\mathscr{T}{\kern-3.0pt{or}}_{j}^{\mathcal{O}_{X_{w}^{P}}}(\mathcal{O}_{X_{w}^{P}}\otimes_{\mathcal{O}_{\bar{X}_{P}}}\xi^{u}_{P},\mathcal{O}_{X_{x}^{P}})=0\hskip
28.45274pt\text{for all}\ x\leq w,\ j>0.$ (18)

By Lemma 4.6, the closures of the $\Gamma$-orbits in
$(Y_{w}^{P})_{\mathbb{P}}$ are precisely $(X_{x}^{P}\times
X_{y}^{P})_{\mathbb{P}}$ for $x,y\leq w$ and $x,y\in W^{P}$. Setting
$\mathcal{F}:=\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}\otimes_{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\pi^{*}\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}})\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}))$,
the identities eqs. 13, 14 and 15 and (18) imply that $\mathcal{F}$ is
homologically transverse to the closures of the $\Gamma$-orbits in
$(Y_{w}^{P})_{\mathbb{P}}$. Then applying [AGM11, Theorem 2.3] (with their
$G=\Gamma,X=(Y^{P}_{w})_{\mathbb{P}},\mathcal{E}=\tilde{\Delta}_{*}\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}$
and their $\mathcal{F}=\mathcal{F}$), we have the following:

$\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}}(\mathcal{O}_{(Y_{w}^{P})_{\mathbb{P}}}\otimes_{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}})),\gamma_{*}\tilde{\Delta}_{*}\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}})=0\hskip
28.45274pt\text{for all}\ i>0.$ (19)

(We note here that although $\Gamma$ is infinite-dimensional, its action on
$(Y_{w}^{P})_{\mathbb{P}}$ factors through a finite-dimensional quotient group
$\bar{\Gamma}$ of $\Gamma$.) Finally, observe that
$\gamma(\tilde{\Delta}(X_{w}^{P})_{\mathbb{P}})\subset(Y_{w}^{P})_{\mathbb{P}}$,
and thus eq. 16 and eq. 19 imply

$\mathscr{T}{\kern-3.0pt{or}}_{i}^{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\pi^{*}(\mathcal{O}_{\mathbb{P}_{\mathbf{j}}}(-\partial\mathbb{P}_{\mathbf{j}}))\otimes(\widetilde{\xi^{u}_{P}\boxtimes\xi^{v}_{P}}),\gamma_{*}\tilde{\Delta}_{*}(\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}}))=0\hskip
28.45274pt\text{for all}\ i>0.\qed$

This proves Theorem 5.1(A).

##  7\. Study of the Richardson Varieties

For any $u,v\leq w$ in $W^{P}$ set $X^{u,v}_{w}:=X^{u}_{w}\times X^{v}_{w}$
where $X^{u}_{w}:=X^{u}\cap X_{w}$. Similarly, we set
$X^{u,v}_{w}(P):=X^{u}_{w}(P)\times X^{v}_{w}(P)$ where
$X^{u}_{w}(P):=X^{u}_{P}\cap X_{w}^{P}$. We also write
$X^{2}_{w}(P):=X_{w}^{P}\times X_{w}^{P}$ and from now on we denote by
$Y_{\mathbf{j}}$ (for any $T$-stable subscheme $Y\subset\bar{X}_{P}$) the
inverse image of $\mathbb{P}_{\mathbf{j}}$ under the standard quotient map
$E(T)_{\mathbb{P}}\times^{T}Y\to\mathbb{P}$.

The action of $B$ on $X_{w}^{P}$ factors through the action of a finite
dimensional quotient group $\bar{B}=B_{w}$ containing the maximal torus $H$.
Similarly, the action of $\Gamma$ on $(X_{w}^{2}(P))_{\mathbb{P}}$ descends to
an action of a finite dimensional quotient group $\bar{\Gamma}=\Gamma_{w}$:

$\Gamma\twoheadrightarrow\bar{\Gamma}=\Gamma_{w}\twoheadrightarrow
GL(N+1)^{r}.$

Further, we can (and do) take $\bar{\Gamma}=\bar{\Gamma}_{0}\rtimes
GL(N+1)^{r}$, where $\bar{\Gamma}_{0}$ is the group of global sections of the
bundle
$(\bar{B}^{2})_{\mathbb{P}}:=E(T)_{\mathbb{P}}\times^{T}\bar{B}^{2}\to\mathbb{P}$.

######  Lemma 7.1.

For any $\mathbf{j}=(j_{1},\cdots,j_{r})\in[N]^{r}$ and $u,v\leq w\in W^{P}$,
the map

$m:\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}\to(X^{2}_{w}(P))_{\mathbb{P}}$

is flat, where $m(\gamma,x)=\gamma\cdot\pi_{2}(x)$ and
$\pi_{2}:(X_{w}^{u,v}(P))_{\mathbf{j}}\to(X^{2}_{w}(P))_{\mathbb{P}}$ is the
map induced from the inclusion $p:X^{u}_{w}(P)\times X^{v}_{w}(P)\to
X_{w}^{2}(P)$. Similarly, its restriction
$\hat{m}:\bar{\Gamma}\times\partial((X^{u,v}_{w}(P))_{\mathbf{j}})\to(X^{2}_{w}(P))_{\mathbb{P}}$
is also flat, where

$\partial((X^{u,v}_{w}(P))_{\mathbf{j}}):=((\partial
X^{u,v}_{P})\cap(X^{2}_{w}(P))_{\mathbf{j}})\cup(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}},$

$(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$ denotes the inverse image
of $\partial\mathbb{P}_{\mathbf{j}}$ under the quotient map
$E(T)_{\mathbb{P}}\times^{T}(X^{u,v}_{w}(P))\to\mathbb{P}$ and $\partial
X^{u,v}_{P}:=(\partial X^{u}_{P}\times X^{v}_{P})\cup(X^{u}_{P}\times\partial
X^{v}_{P})$.

###### Proof.

Consider the following diagram:

${\bar{\Gamma}_{0}\times
X^{u,v}_{w}(P)}$${\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}}$${\text{GL}(N+1)^{r}\times\mathbb{P}_{\mathbf{j}}}$${X^{2}_{w}(P)}$${(X^{2}_{w}(P))_{\mathbb{P}}}$${\mathbb{P}}$$\scriptstyle{m^{\prime}}$$\scriptstyle{m}$$\scriptstyle{m^{\prime\prime}}$

Here the two right horizontal maps are locally trivial fibrations with fibers
in the leftmost spaces. From this and the fact that $m^{\prime\prime}$ is
smooth (see the proof of [Kum17, Lemma 6.11]), to show that $m$ is flat it is
enough to show that $m^{\prime}$ is a flat morphism. From Lemma 4.6, it
suffices to show that $(\bar{B}^{2})\times X^{u,v}_{w}(P)\to X^{2}_{w}(P)$ is
flat. This follows from the proof of [Kum17, Lemma 6.10] for $\bar{X}_{P}$.

Observe also that, by the same proof as that of [Kum17, Lemma 6.10], the map
$(\bar{B}^{2})\times((\partial X^{u,v}_{P})\cap X^{2}_{w}(P))\to X^{2}_{w}(P)$
is flat. Hence to show that $\hat{m}$ is flat, we first observe that the
restrictions to the components $\Gamma_{1}:=\bar{\Gamma}\times((\partial
X^{u,v}_{P})\cap X^{2}_{w}(P))_{\mathbf{j}}$,
$\Gamma_{2}:=\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$,
and $\Gamma_{1}\cap\Gamma_{2}$ are all flat maps (following the same argument
as the first part of this proof). Therefore $\hat{m}$ is flat on
$\Gamma_{1}\cup\Gamma_{2}$, since for any affine scheme $Y=Y_{1}\cup Y_{2}$
with closed subschemes $Y_{1},Y_{2}$ and a morphism $f:Y\to X$ of schemes,
there is an exact sequence of $k[X]$-modules:

$0\to k[Y]\to k[Y_{1}]\oplus k[Y_{2}]\to k[Y_{1}\cap Y_{2}]\to 0.\qed$

As in Section 2, for $v\leq w\in W^{P}$, let

$\pi^{v}_{w}:Z^{v}_{w}\to X^{v}_{w}(P)$

be the $T$-equivariant desingularization of $X^{v}_{w}(P)$. For $u,v,w\in
W^{P}$, let $Z^{u,v}_{w}:=Z^{u}_{w}\times Z^{v}_{w}$ under the diagonal action
of $T$ and $\left(Z^{u,v}_{w}\right)_{\mathbf{j}}$ is the inverse image of
$\mathbb{P}_{\mathbf{j}}$ under the map
$E(T)_{\mathbb{P}}\displaystyle\mathop{\times}^{T}Z^{u,v}_{w}\to\mathbb{P}$.

We record the following from [Kum17, Lemma 6.11]:

######  Lemma 7.2.

For any $\mathbf{j}\in[N]^{r}$ and $u,v\leq w\in W^{P}$, the map

$\tilde{m}:\bar{\Gamma}\times(Z^{u,v}_{w})_{\mathbf{j}}\to(Z^{2}_{w})_{\mathbb{P}}$

is a smooth morphism, where $\tilde{m}$ is defined similarly to the map $m$ in
Lemma 7.1.

######  Lemma 7.3.

For any $u\in W^{P}$, $X^{u}_{P}$ is normal and CM.

###### Proof.

The standard projection map $p:G/B\to G/P$ is a locally trivial fibration with
fibers isomorphic to the finite dimensional flag variety $P/B$ (since, by
assumption, $P$ is of finite type). By [KS09, Propositions 3.2 and 3.4] the
lemma follows for $X^{u}$. Hence, the lemma follows for $X^{u}_{P}$ since
$X^{u}\to X^{u}_{P}$ is a locally trivial fibration with fiber $P/B$. Thus,
locally $X^{u}$ is isomorphic with $X^{u}_{P}\times\mathbb{A}^{n}$, for
$n=\dim P/B$. Alternatively, see [sta, Lemma 10.164.3] for normality and [sta,
Lemma 75.26.5] for CM property. ∎

######  Lemma 7.4.

For any symmetrizable Kac-Moody group $G$ and any $u\leq w$ in $W^{P}$, the
Richardson variety $X^{u}_{w}(P)\subset\bar{X}_{P}$ is irreducible, normal,
and CM with rational singularities. Moreover, $C_{w}^{P}\cap C^{u}_{P}$ is an
open dense subset of $X^{u}_{w}(P)$.

###### Proof.

Observe that under the locally trivial fibration $\pi:\bar{X}\to\bar{X}_{P}$
with fiber $P/B$, $\pi^{-1}(X^{u}_{w}(P))=X^{u}_{\hat{w}}$, where $\hat{w}$ is
the longest element in the coset $wW_{P}$. Hence, the irreducibility,
normality and CM property of $X^{u}_{w}(P)$ follows, as in the above lemma
7.3, from the corresponding properties of $X^{u}_{\hat{w}}$ (cf. [Kum17,
Propositon 6.6]). Similarly, the rational singularities of $X^{u}_{w}(P)$
follows from the corresponding result for $X^{u}_{\hat{w}}$ (cf. [KS14,
Theorem 3.1]).

Finally, $C_{w}^{P}\cap C^{u}_{P}$ is clearly an open subset of $X^{u}_{w}(P)$
and therefore dense since it is nonempty by [Kum02, Lemma 7.3.10]. ∎

######  Lemma 7.5.

$\mathcal{O}_{X^{v}_{P}}(-\partial X^{v}_{P})$ is a CM
$\mathcal{O}_{X^{v}_{P}}$-module and $\mathcal{O}_{\partial X^{v}_{P}}$ is a
CM ring.

###### Proof.

From the locally trivial fibration $G/B\to G/P$ with fiber $P/B$, we see that
locally

$\bar{X}\simeq\bar{X}_{P}\times\mathbb{A}^{n},\,\,X^{v}\simeq
X^{v}_{P}\times\mathbb{A}^{n},\,\,\text{and}\,\,\partial X^{v}\simeq\partial
X^{v}_{P}\times\mathbb{A}^{n}.$ (20)

Thus, locally as
$\mathcal{O}_{X^{v}}\simeq\mathcal{O}_{X^{v}_{P}}\otimes\mathcal{O}_{\mathbb{A}^{n}}$-modules,
we get

$\mathcal{O}_{X^{v}}(-\partial
X^{v})\simeq\left(\mathcal{O}_{X^{v}_{P}}(-\partial
X^{v}_{P})\right)\otimes\mathcal{O}_{\mathbb{A}^{n}}.$ (21)

Since $\mathcal{O}_{X^{v}}(-\partial X^{v})$ is a CM
$\mathcal{O}_{X^{v}}$-module (cf. [Kum17, Corollary 10.5]), from the above
decomposition (21), we get that $\mathcal{O}_{X^{v}_{P}}(-\partial X^{v}_{P})$
is a CM $\mathcal{O}_{X^{v}_{P}}$-module.

Now, $\mathcal{O}_{\partial X^{v}}$ being a CM ring (cf. [Kum17, Corollary
10.5]), so is $\mathcal{O}_{\partial X^{v}_{P}}$ by using the decompositions
(20). ∎

Remark Following the same proof as that of Lemma 7.4, replacing $X^{u}_{P}$ by
$\partial X^{u}_{P}$ and observing that $\partial X^{u}\cap X_{\hat{w}}$ is CM
(cf. [Kum17, Remark 6.7]), we also have that $(\partial X^{u}_{P})\cap
X_{w}^{P}$ is CM.

##  8\. The schemes $\mathcal{Z}_{P}$ and $\widetilde{\mathcal{Z}}$

Let $u,v\leq w\in W^{P}$. As in Section 7, denote by
$Z^{u,v}_{w}:=Z^{u}_{w}\times Z^{v}_{w}$, where $Z^{u}_{w}$ is the
$T$-equivariant desingularization of $X^{u}_{w}(P)$ as in Section 2. We also
let $Z^{2}_{w}:=Z_{w}\times Z_{w}$, where $Z_{w}$ is a BSDH variety as in
[Kum02, §7.1.3]. For any $\mathbf{j}\in[N]^{r}$, let
$(X^{u,v}_{w}(P))_{\mathbf{j}}$ and $(Z^{u,v}_{w})_{\mathbf{j}}$ denote the
inverse image of $\mathbb{P}_{\mathbf{j}}$ through the maps
$E(T)_{\mathbb{P}}\times^{T}X^{u,v}_{w}(P)\to\mathbb{P}$ and
$E(T)_{\mathbb{P}}\times^{T}Z^{u,v}_{w}\to\mathbb{P}$ respectively.

We define the scheme $\widetilde{\mathcal{Z}}$ to be the fiber product
$(\bar{\Gamma}\times(Z^{u,v}_{w})_{\mathbf{j}})\times_{(Z^{2}_{w})_{\mathbb{P}}}\widetilde{\Delta}((Z_{w})_{\mathbb{P}})$
and $\mathcal{Z}_{P}$ to be the fiber product
$(\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}})\times_{(X^{2}_{w}(P))_{\mathbb{P}}}\widetilde{\Delta}((X^{P}_{w})_{\mathbb{P}})$
as in the commutative diagram:

${\widetilde{\mathcal{Z}}}$${\widetilde{\Delta}((Z_{w})_{\mathbb{P}})}$${\bar{\Gamma}\times(Z^{u,v}_{w})_{\mathbf{j}}}$${(Z^{2}_{w})_{\mathbb{P}}}$${\bar{\Gamma}}$${\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}}$${(X^{2}_{w}(P))_{\mathbb{P}}}$${\mathcal{Z}_{P}}$${\widetilde{\Delta}((X_{w}^{P})_{\mathbb{P}})}$$\scriptstyle{\tilde{\mu}}$(smooth)$\scriptstyle{\tilde{i}}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{f}$${\square}$$\scriptstyle{\tilde{m}}$(smooth)$\scriptstyle{\theta}$$\scriptstyle{\beta}$$\scriptstyle{m}$(flat)${\square}$$\scriptstyle{\pi}$$\scriptstyle{i}$$\scriptstyle{\mu}$(flat)

The map $\theta$ is induced by the product of the desingularizations
$\pi^{u}_{w}:Z^{u}_{w}\to X^{u}_{w}(P)$ and $\pi^{v}_{w}:Z^{v}_{w}\to
X^{v}_{w}(P)$. In particular, $\theta$ is a birational map for $u,v\leq w\in
W^{P}$. The map $\beta$ is induced from the BSDH desingularization $Z_{w}\to
X_{w}^{P}$. The maps $\pi$ and $\widetilde{\pi}$ are obtained from the
projections to the $\bar{\Gamma}$-factor via the maps $i$ and $\tilde{i}$
respectively. The map $f$ is the restriction of $\theta$ to
$\widetilde{\mathcal{Z}}$ (via $\tilde{i}$) with image inside
$\mathcal{Z}_{P}$.

######  Proposition 8.1.

The schemes $\mathcal{Z}_{P}$ and $\widetilde{\mathcal{Z}}$ are irreducible
and the map $f:\widetilde{\mathcal{Z}}\to\mathcal{Z}_{P}$ is a proper
birational map. Thus $\widetilde{\mathcal{Z}}$ is a desingularization of
$\mathcal{Z}_{P}$. Moreover, $\mathcal{Z}_{P}$ is CM with

$\dim(\mathcal{Z}_{P})=|\mathbf{j}|+l(w)-(l(u)+l(v))+\dim(\bar{\Gamma}),$ (22)

where $|\mathbf{j}|=\sum_{i=1}^{r}j_{i}$ for
$\mathbf{j}=(j_{1},\cdots,j_{r})$.

###### Proof.

We first note that the top Cartesian square is precisely the same as that of
[Kum17, §7], and hence (as in [Kum17, Equation 59])

$\dim(\widetilde{\mathcal{Z}})=\dim(\bar{\Gamma})+|\mathbf{j}|+l(w)-l(u)-l(v)$
(23)

and $\widetilde{\mathcal{Z}}$ is irreducible from the proof of [Kum17,
Proposition 7.4].

We next show that $\mathcal{Z}_{P}$ is of pure dimension. Since $m$ is a flat
morphism, Im $(m)$ is an open subset of $(X^{2}_{w}(P))_{\mathbb{P}}$ ([Har77,
Chap. III, Exercise 9.1]). Moreover, clearly Im
$(m)\supset(C^{2}_{w}(P))_{\mathbb{P}}$, thus Im $(m)$ intersects
$\tilde{\Delta}((X^{2}_{w}(P)_{\mathbb{P}})$. Applying [Har77, Chap. III,
Corollary 9.6] first to the morphism
$m:\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}\to$ Im $(m)$ and then its
restriction $\mu$ to $\mathcal{Z}_{P}$, we see that $\mathcal{Z}_{P}$ is pure
dimensional.

We next show that $\mathcal{Z}_{P}$ is irreducible. Since
$\widetilde{\mathcal{Z}}$ is irreducible, so is its open subset
$\widetilde{\mathcal{Z}}\cap\widetilde{m}^{-1}((C^{2}_{w}(P))_{\mathbb{P}})$.
Now
$\widetilde{\mathcal{Z}}\cap\widetilde{m}^{-1}((C^{2}_{w}(P))_{\mathbb{P}})$
maps surjectively onto the open subset $\mathcal{Z}_{P}\cap
m^{-1}((C^{2}_{w}(P))_{\mathbb{P}})$ of $\mathcal{Z}_{P}$ under $f$ and hence
$\mathcal{Z}_{1}:=\overline{\mathcal{Z}_{P}\cap
m^{-1}((C^{2}_{w}(P))_{\mathbb{P}})}$ is an irreducible component of
$\mathcal{Z}_{P}$. Let $\mathcal{Z}_{2}$ be another irreducible component of
$\mathcal{Z}_{P}$ so that
$\mu(\mathcal{Z}_{2})\subset\tilde{\Delta}((X^{P}_{w}\setminus
C^{P}_{w})_{\mathbb{P}})$. But $\dim(\tilde{\Delta}((X^{P}_{w}\setminus
C^{P}_{w})_{\mathbb{P}}))<\dim(\tilde{\Delta}((X^{P}_{w})_{\mathbb{P}}))$ and
all of the fibers of $\mu|_{\mathcal{Z}_{2}}$ have dimension no more than any
fiber of $\mu$, so $\dim(\mathcal{Z}_{2})<\dim(\mathcal{Z}_{1})$, which is a
contradiction since $\mathcal{Z}_{P}$ is of pure dimension. Thus
$\mathcal{Z}_{P}=\mathcal{Z}_{1}$ is irreducible.

The map $f$ is clearly proper and an isomorphism when restricted to the open
subset

$\widetilde{\mathcal{Z}}\cap(\bar{\Gamma}\times((C^{u}_{P}\cap
C_{w}^{P})\times(C^{v}_{P}\cap C_{w}^{P}))_{\mathbf{j}}),$

where we identify the inverse image $(\pi^{u}_{w})^{-1}(C^{u}_{P}\cap
C_{w}^{P})$ inside $Z^{u}_{w}$ with $C^{u}_{P}\cap C_{w}^{P}$. Hence
$\dim(\mathcal{Z}_{P})=\dim(\widetilde{\mathcal{Z}})$ and (22) holds by the
equation (23). Therefore,

$\text{codim}_{\mathcal{Z}_{P}}(\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}})=\text{codim}_{\tilde{\Delta}((X^{P}_{w})_{\mathbb{P}})}((X^{2}_{w}(P))_{\mathbb{P}})=l(w).$

The fact that $\mathcal{Z}_{P}$ is CM follows from the fact that $m$ is flat
as well as [Kum17, Lemma 7.2] and Lemma 7.4. ∎

######  Lemma 8.2.

The scheme $\mathcal{Z}_{P}$ is normal, irreducible, and CM.

###### Proof.

The only thing that remains to be shown is that $\mathcal{Z}_{P}$ is normal.
Consider the map

$\mu_{o}:G\times^{U^{-}}X^{u}_{P}(S_{u}^{\prime})\to\bar{X}_{P},\hskip
28.45274pt[g,x]\mapsto g\cdot x,$

where $X^{u}_{P}(S_{u}^{\prime}):=X^{u}_{P}\cap V^{S_{u}^{\prime}}$,
$V^{S_{u}^{\prime}}:=\bigcup_{w\in S_{u}^{\prime}}B^{-}wx_{o}$, $x_{o}$ is the
base point $1.P$ of $\bar{X}_{P}$, and $S_{u}^{\prime}:=\\{v\in W^{P}\mid
l(v)\leq l(u)+1\\}$. Since $\mu_{o}$ is $G$-equivariant, it is a locally
trivial fibration with fibers $F^{u}:=\bigcup_{u\leq v;\ l(v)\leq
l(u)+1}Pv^{-1}U^{-}/U^{-}$. Since $X^{u}_{P}$ is normal by Lemma 7.3 and any
$B^{-}$ orbit in $X^{u}_{P}(S_{u}^{\prime})$ has codimension $\leq 1$ in
$X^{u}_{P}$, $X^{u}_{P}(S_{u}^{\prime})$ is smooth and so is $F^{u}$. (Here
the smoothness of $F^{u}$ means that there exists a closed normal subgroup
$B_{1}$ of $B$ of finite codimension such that $B_{1}$ acts freely and
properly on $F^{u}$ and the quotient $B_{1}\setminus F^{u}$ is a smooth scheme
of finite type over $\mathbb{C}$.) Therefore $\mu_{o}$ is a smooth morphism.
Hence so is the restriction of $\mu_{o}$ to the open subset $B\times
X^{u}_{P}(S_{u}^{\prime})$, and further so is the restriction to the inverse
image of $X_{w}^{P}$, $\mu_{o}(w):B\times(X^{u}_{P}(S_{u}^{\prime})\cap
X_{w}^{P})\to X_{w}^{P}$. Clearly $\mu_{o}(w)$ factors through a smooth
morphism $\bar{\mu}_{o}(w):\bar{B}\times(X^{u}_{P}(S_{u}^{\prime})\cap
X_{w}^{P})\to X_{w}^{P}$, where $\bar{B}$ is a finite-dimensional quotient of
$B$. Hence,
$\bar{\mu}_{o}(w)^{-1}(\mathring{X}_{w}^{P})=\bar{B}\times(X^{u}_{P}(S_{u}^{\prime})\cap(\mathring{X}_{w}^{P}))$
is a smooth variety, where
$\mathring{X}_{w}^{P}:=X_{w}^{P}\setminus\Sigma_{w}^{P}$ and $\Sigma_{w}^{P}$
is the singular locus of $X_{w}^{P}$.

Following the same argument as in the proof of Lemma 7.1 and restricting the
middle vertical map to the open subset
$\bar{m}:\bar{\Gamma}\times((X^{u}_{P}(S_{u}^{\prime})\times
X^{v}_{P}(S_{v}^{\prime}))\cap
X^{2}_{w}(P))_{\mathbf{j}}\to(X^{2}_{w}(P))_{\mathbb{P}}$ and the left
vertical map accordingly, we have that $\bar{m}$ is a smooth morphism (with
open image $Y$). Hence, the restriction of $\bar{m}$ to
$\hat{m}:\bar{m}^{-1}(\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}}))\to\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$
is also smooth. (Observe that $Y$ does intersect
$\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$ since $Y$ being a
$\bar{\Gamma}$-stable open subset of $(X^{2}_{w}(P))_{\mathbb{P}}$,
$(C^{2}_{w}(P))_{\mathbb{P}}\subset Y$.) Therefore the open set
$\hat{m}^{-1}(\tilde{\Delta}((\mathring{X}_{w}^{P})_{\mathbb{P}}))$ in
$\mathcal{Z}_{P}=m^{-1}(\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}}))$ is a smooth
variety. Denote the complement of
$\bar{\Gamma}\times((X^{u}_{P}(S_{u}^{\prime})\times
X^{v}_{P}(S_{v}^{\prime}))\cap X^{2}_{w}(P))_{\mathbf{j}}$ in
$\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}$ by $F$ and denote
$\hat{m}^{-1}(\tilde{\Delta}((\Sigma_{w}^{P})_{\mathbb{P}}))$ by $F^{\prime}$.
Then $F^{\prime}$ has codimension $\geq 2$ in
$\bar{m}^{-1}(\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}}))$, and hence in
$\mathcal{Z}_{P}$. Also, $F$ is of codimension $\geq 2$ in
$\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}$. If $F$ is nonempty, then
the restriction of $m$ to $F$ is flat (following the proof of Lemma 7.1) with
image an open subset of $(X^{2}_{w}(P))_{\mathbb{P}}$ intersecting
$\tilde{\Delta}((X^{P}_{w})_{\mathbb{P}})$. Therefore the codimension of
$F\cap\mathcal{Z}_{P}$ in $\mathcal{Z}_{P}$ is $\geq 2$ and hence so is the
complement of the smooth locus of $\mathcal{Z}_{P}$ in $\mathcal{Z}_{P}$.
Finally, since $\mathcal{Z}_{P}$ is CM, it is normal by Serre’s criterion
[Har77, Ch. II, Theorem 8.22(A)]. ∎

######  Proposition 8.3.

The scheme $\mathcal{Z}_{P}$ has rational singularities.

###### Proof.

Since $\mu$ is flat and $\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$ has
rational singularities [Kum02, Theorem 8.2.2(c)], it is enough to show (by
[Elk78, Théorème 5]) that the fibers of $\mu$ are disjoint unions of
irreducible varieties with rational singularities.

Let $x\in\tilde{\Delta}((C_{w^{\prime}}^{P})_{\mathbb{P}})$, where
$w^{\prime}\leq w\in W^{P}$. Then from [Kum17, Thoerem 7.6] and Lemma 4.6,

$\text{Stab}(x)\setminus\mu^{-1}(x)\simeq((X^{u}_{P}\cap
C_{w^{\prime}}^{P})\times(X^{v}_{P}\cap C^{P}_{w^{\prime}}))_{\mathbf{j}},$
(24)

where the stabilizer Stab$(x)$ is taken with respect to the $\bar{\Gamma}$
action on $(X^{2}_{w}(P))_{\mathbb{P}}$. From [Ser58, Proposition 3, §2.5],
the map $\bar{\Gamma}\to\text{Stab}(x)\setminus\bar{\Gamma}$ is locally
trivial in the étale topology.

We have the pull-back diagram

${\mu^{-1}(x)}$${\subseteq}$${\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}}$${\text{Stab}(x)\setminus\mu^{-1}(x)}$${\subseteq}$${(\text{Stab}(x)\setminus\bar{\Gamma})\times(X^{u,v}_{w}(P))_{\mathbf{j}}\,,}$

where the left vertical map is a locally trivial fibration in the étale
topology since so is the right side map. Therefore, since $\text{Stab}(x)$ is
smooth and $\text{Stab}(x)\setminus\mu^{-1}(x)$ has rational singularities (by
the equation (24) and Lemma 7.4), $\mu^{-1}(x)$ is a disjoint union of
irreducible varieties with rational singularities by [KM08, Corollary 5.11]. ∎

######  Proposition 8.4.

The scheme $\partial\mathcal{Z}_{P}$ is of pure codimension 1 in
$\mathcal{Z}_{P}$ and is CM, where we define the closed subscheme
$\partial\mathcal{Z}_{P}$ of $\mathcal{Z}_{P}$ as

$\partial\mathcal{Z}_{P}:=(\bar{\Gamma}\times\partial((X^{u,v}_{w}(P))_{\mathbf{j}}))\times_{(X^{2}_{w}(P))_{\mathbb{P}}}\widetilde{\Delta}((X_{w}^{P})_{\mathbb{P}}),$

where $\partial((X^{u,v}_{w}(P))_{\mathbf{j}})$ is as in Lemma 7.1.

###### Proof.

First recall that
$\bar{\Gamma}\times\partial((X^{u,v}_{w}(P))_{\mathbf{j}})\xrightarrow{\hat{m}}(X^{2}_{w}(P))_{\mathbb{P}}$
is a flat morphism (cf. Lemma 7.1) and observe that
$\partial((X^{u,v}_{w}(P))_{\mathbf{j}})$ is pure of codimension 1 in
$(X^{u,v}_{w}(P))_{\mathbf{j}}$. We consider two cases:
$\partial\mathbb{P}_{\mathbf{j}}\neq\emptyset$ or
$\partial\mathbb{P}_{\mathbf{j}}=\emptyset$. In the first case, $\text{Im
}\hat{m}=\text{Im }m$. In the latter case, we have

$\text{Im }\hat{m}\supset\left(\left(\left(\bigcup_{u\to
u^{\prime}\leq\theta\leq
w}C_{\theta}^{P}\right)\times\left(\bigcup_{v\leq\theta^{\prime}\leq
w}C_{\theta^{\prime}}^{P}\right)\right)\cup\left(\left(\bigcup_{u\leq\theta\leq
w}C_{\theta}^{P}\right)\times\left(\bigcup_{v\to
v^{\prime}\leq\theta^{\prime}\leq
w}C_{\theta^{\prime}}^{P}\right)\right)\right)_{\mathbb{P}}.$

In either case we have that, if nonempty, $\text{Im }\hat{m}$ intersects
$\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$. Moreover, since $\hat{m}$ is flat,
$\text{Im }\hat{m}$ is open in $(X^{2}_{w}(P))_{\mathbb{P}}$. Thus, by [Har77,
Ch. III, Corollary 9.6], each fiber of $\hat{m}$ (if nonempty) is pure of
dimension

$\dim(\bar{\Gamma})+\dim((X^{u,v}_{w}(P))_{\mathbf{j}})-\dim((X^{2}_{w}(P))_{\mathbb{P}})-1.$

Applying [Har77, Ch. III, Corollary 9.6] again to the restriction of $\hat{m}$
to $\partial\mathcal{Z}_{P}$ via $\mu$, we have that $\partial\mathcal{Z}_{P}$
is pure of dimension

$\dim(\bar{\Gamma})+\dim((X^{u,v}_{w}(P))_{\mathbf{j}})-\dim((X^{2}_{w}(P))_{\mathbb{P}})-1+\dim(\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})).$

Hence, from (22) it is clear that $\partial\mathcal{Z}_{P}$ is pure of
codimension 1 in $\mathcal{Z}_{P}$. From Lemma 7.4 and the remark following
Lemma 7.5, both of $((\partial X^{u}_{P})\cap X_{w}^{P})\times X^{v}_{w}(P)$
and $X^{u}_{w}(P)\times((\partial X^{v}_{P})\cap X_{w}^{P})$ are CM. Hence so
is their intersection, which is of pure codimension 1 in each of them. Thus
their union is CM (cf. [Kum02, Theorem A.36]) and hence so is $((\partial
X^{u,v}_{P})\cap X^{2}_{w}(P))_{\mathbf{j}}$. We also have that
$(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$ and the intersection

$((\partial X^{u,v}_{P})\cap
X^{2}_{w}(P))_{\mathbf{j}}\cap(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}=((\partial
X^{u,v}_{P}\cap X^{2}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$

are CM since $\partial\mathbb{P}_{\mathbf{j}}$ is. Since $((\partial
X^{u,v}_{P})\cap X^{2}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$ is CM of
pure codimension 1 in each of
$(X^{u,v}_{w}(P))_{\partial\mathbb{P}_{\mathbf{j}}}$ and $((\partial
X^{u,v}_{P})\cap X^{2}_{w}(P))_{\mathbf{j}}$, their union
$\partial((X^{u,v}_{w}(P))_{\mathbf{j}})$ is CM. Finally, applying [Kum17,
Lemma 7.2] to $\hat{m}$ shows $\partial\mathcal{Z}_{P}$ is CM. ∎

######  Corollary 8.5.

Assume that $c_{u,v}^{w}(\mathbf{j})\neq 0$, where $c_{u,v}^{w}(\mathbf{j})$
is defined by the equation (7) (also see Lemma 4.3). Then for general
$\gamma\in\bar{\Gamma}$, the fiber
$N_{\gamma}:=\pi^{-1}(\gamma)\subset\mathcal{Z}_{P}$ is CM of pure dimension,
where $\pi:\mathcal{Z}_{P}\to\bar{\Gamma}$ is defined at the beginning of this
section. In fact, for any $\gamma\in\bar{\Gamma}$ such that $N_{\gamma}$ is
pure of dimension

$\dim(N_{\gamma})=\dim(\mathcal{Z}_{P})-\dim(\bar{\Gamma})=|\mathbf{j}|+l(w)-l(u)-l(v),$
(25)

$N_{\gamma}$ is CM (and this condition holds for general $\gamma$).

Similarly, if $|\mathbf{j}|+l(w)-l(u)-l(v)>0$, then for general
$\gamma\in\bar{\Gamma}$, the fiber
$M_{\gamma}:=\pi_{1}^{-1}(\gamma)\subset\partial\mathcal{Z}_{P}$ is CM of pure
codimension 1 in $N_{\gamma}$, where $\pi_{1}$ is the restriction of $\pi$ to
$\partial\mathcal{Z}_{P}$. If $|\mathbf{j}|+l(w)-l(u)-l(v)=0$, then for
general $\gamma\in\bar{\Gamma}$, $M_{\gamma}$ is empty.

In particular, for general $\gamma\in\bar{\Gamma}$,

$\mathscr{E}{\kern-1.0pt{xt}}^{i}_{\mathcal{O}_{N_{\gamma}}}(\mathcal{O}_{N_{\gamma}}(-M_{\gamma}),\omega_{N_{\gamma}})=0\hskip
5.69054pt\text{ for all }i>0.$ (26)

###### Proof.

This follows the same argument as the proof of [Kum17, Corollary 7.9]. ∎

##  9\. Study of
$R^{p}f_{*}(\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}))$

Throughout this section we assume $c_{u,v}^{w}(\mathbf{j})\neq 0$, where
$c_{u,v}^{w}(\mathbf{j})$ is defined by the identity (7). We also follow the
notation of the commutative diagram at the beginning of Section 8.

######  Lemma 9.1.

For $u\in W^{P}$, the line bundle
$\mathcal{L}^{P}(\hat{\rho}_{Y})|_{X^{u}_{P}}$ has a section with zero set
precisely equal to $\partial X^{u}_{P}$. In particular,

$\mathcal{L}^{P}(\hat{\rho}_{Y})|_{X^{u}_{w}(P)}\sim\sum_{i}b_{i}X_{i}\hskip
11.38109pt\text{ for some }b_{i}>0,$

where the $X_{i}$ are the irreducible components of $(\partial X^{u}_{P})\cap
X_{w}^{P}$.

###### Proof.

Let $L(\hat{\rho}_{Y})$ denote the integrable highest weight
$G^{\text{min}}$-module with highest weight $\hat{\rho}_{Y}$ and let
$L(\hat{\rho}_{Y})^{\vee}$ be its restricted dual, i.e., the direct sum of the
dual of its weight spaces. Also, let $\mathcal{L}$ be the tautological line
bundle over $\mathbb{P}(L(\hat{\rho}_{Y}))$. Consider the linear map
$\beta:L(\hat{\rho}_{Y})^{\vee}\to
H^{0}(\mathbb{P}(L(\hat{\rho}_{Y})),\mathcal{L}^{*})$, where
$\beta(f)(x)=(x,f|_{x})$ for $f\in L(\hat{\rho}_{Y})^{\vee}$ and
$x\in\mathbb{P}(L(\hat{\rho}_{Y}))$. Further, let
$i^{*}:H^{0}(\mathbb{P}(L(\hat{\rho}_{Y})),\mathcal{L}^{*})\to
H^{0}(\bar{X}_{P},\mathcal{L}^{P}(\hat{\rho}_{Y}))$ be induced from
$i:\bar{X}_{P}\to\mathbb{P}(L(\hat{\rho}_{Y}))$ taking
$gP\mapsto[ge_{\hat{\rho}_{Y}}]$, where $e_{\hat{\rho}_{Y}}$ is a highest
weight vector in $L(\hat{\rho}_{Y})$. Set
$\chi:=i^{*}\circ\beta:L(\hat{\rho}_{Y})^{\vee}\xrightarrow{\sim}H^{0}(\bar{X}_{P},\mathcal{L}^{P}(\hat{\rho}_{Y}))$,
which is the Borel-Weil isomorphism (see [Kum02, §8.1.21]) given by
$\chi(f)(gP)=[g,f(ge_{\hat{\rho}_{Y}})]$.

Let $e_{u\hat{\rho}_{Y}}$ be the extremal weight vector of $L(\hat{\rho}_{Y})$
with weight $u\hat{\rho}_{Y}$ and $e_{u\hat{\rho}_{Y}}^{*}$ the linear form
which takes value 1 on $e_{u\hat{\rho}_{Y}}$ and 0 on any weight vector of
weight different from $u\hat{\rho}_{Y}$. It is easy to see (from [Kum02, Lemma
8.3.3 and Proposition 1.4.2(a)]) that the section
$\chi(e_{u\hat{\rho}_{Y}}^{*})|_{X^{u}_{P}}$ has zero set exactly $\partial
X^{u}_{P}$. This proves the lemma. ∎

Recall that a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on an irreducible
projective variety $X$ is said to be _nef_ (resp. _big_) if $D$ has
nonnegative intersection with every irreducible curve in $X$ (resp.
$\dim(H^{0}(X,\mathcal{O}_{X}(mD)))>cm^{\dim(X)}$ for some $c>0$ and $m\gg
1$). Note that if $D$ is ample, then it is _nef_ and _big_ (cf. [KM08,
Proposition 2.61]).

For a proper morphism $\pi:X\to Y$ between schemes and a $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisor $D$ on $X$, $D$ is said to be $\pi$-_nef_ (resp.
$\pi$-_big_) if $D$ has nonnegative intersection with every irreducible curve
in $X$ contracted by $\pi$ (resp. $\pi_{*}\mathcal{O}_{X}(mD)>cm^{n}$ for some
$c>0$ and $m\gg 1$, where $n$ is the dimension of a general fiber of $\pi$).

######  Proposition 9.2.

There exists a nef and big line bundle $\mathcal{M}$ on
$(Z^{u,v}_{w})_{\mathbf{j}}$ with a section with support
$\partial((Z^{u,v}_{w})_{\mathbf{j}})$, where
$\partial((Z^{u,v}_{w})_{\mathbf{j}})$ is defined to be the inverse image of
$\partial((X^{u,v}_{w}(P))_{\mathbf{j}})$ under the canonical map
$(Z^{u,v}_{w})_{\mathbf{j}}\to(X^{u,v}_{w}(P))_{\mathbf{j}}$ induced by the
$T$-equivariant map $\pi^{u,v}_{w}:Z^{u,v}_{w}\to X^{u,v}_{w}(P)$. Moreover,
$\mathcal{M}$ can be chosen to be the pull-back of an ample line bundle
$\mathcal{M}^{\prime}$ on $(X^{u,v}_{w}(P))_{\mathbf{j}}$.

###### Proof.

Let $\mathcal{H}$ be an ample line bundle on $\mathbb{P}_{\mathbf{j}}$ with a
section with support precisely $\partial\mathbb{P}_{\mathbf{j}}$. Let
$\mathcal{L}^{P}_{Z^{u,v}_{w}}(\hat{\rho}_{Y}\boxtimes\hat{\rho}_{Y})$ be the
pull-back of the line bundle
$\mathcal{L}^{P}(\hat{\rho}_{Y})\boxtimes\mathcal{L}^{P}(\hat{\rho}_{Y})$ on
$\bar{X}_{P}\times\bar{X}_{P}$ via the morphism
$Z^{u,v}_{w}\to\bar{X}_{P}\times\bar{X}_{P}$. The line bundle
$e^{u\hat{\rho}_{Y}+v\hat{\rho}_{Y}}\mathcal{L}^{P}_{Z^{u,v}_{w}}(\hat{\rho}_{Y}\boxtimes\hat{\rho}_{Y})$
is $T$-equivariant, hence we have the line bundle

$\tilde{\mathcal{L}}^{P}_{Z^{u,v}_{w}}(-\hat{\rho}_{Y}\boxtimes-\hat{\rho}_{Y}):=E(T)_{\mathbf{j}}\times^{T}e^{u\hat{\rho}_{Y}+v\hat{\rho}_{Y}}\mathcal{L}^{P}_{Z^{u,v}_{w}}(\hat{\rho}_{Y}\boxtimes\hat{\rho}_{Y})\to(Z^{u,v}_{w})_{\mathbf{j}}.$

Set
$\mathcal{M}:=\tilde{\mathcal{L}}^{P}_{Z^{u,v}_{w}}(-\hat{\rho}_{Y}\boxtimes-\hat{\rho}_{Y})\otimes\pi^{*}(\mathcal{H}^{N})$,
where $\pi$ is the standard quotient map
$E(T)_{\mathbf{j}}\times^{T}Z^{u,v}_{w}\to\mathbb{P}_{\mathbf{j}}$ and $N\gg
0$. Take
$\theta:(Z^{u,v}_{w})_{\mathbf{j}}\to\tilde{\mathcal{L}}^{P}_{Z^{u,v}_{w}}(-\hat{\rho}_{Y}\boxtimes-\hat{\rho}_{Y})$
to be the section given by

$[e,z]\mapsto[e,1_{u\hat{\rho}_{Y}+v\hat{\rho}_{Y}}\otimes(\bar{\chi}(e^{*}_{u\hat{\rho}_{Y}})\boxtimes\bar{\chi}(e^{*}_{v\hat{\rho}_{Y}}))(z)],$

where $e\in E(T)_{\mathbf{j}},\ z\in Z^{u,v}_{w},\
1_{u\hat{\rho}_{Y}+v\hat{\rho}_{Y}}$ is the constant section of the trivial
line bundle over $Z^{u,v}_{w}$ with $H$ action given by
$u\hat{\rho}_{Y}+v\hat{\rho}_{Y}$, and $\bar{\chi}\boxtimes\bar{\chi}$ is the
pull-back of the Borel-Weil isomorphism as in the proof of Lemma 9.1. Let
$\sigma$ be any section of $\mathcal{H}^{N}$ with support precisely
$\partial\mathbb{P}_{\mathbf{j}}$ and $\hat{\sigma}$ its pull-back to
$(Z^{u,v}_{w})_{\mathbf{j}}$. It is easy to see that the support of
$\theta\otimes\hat{\sigma}$ is precisely
$\partial((Z^{u,v}_{w})_{\mathbf{j}})$.

Moreover, the line bundle $\mathcal{M}$ is the pull-back of the line bundle
$\mathcal{M}^{\prime}:=\tilde{\mathcal{L}}^{\prime}(-\hat{\rho}_{Y}\boxtimes-\hat{\rho}_{Y})\otimes\pi_{1}^{*}(\mathcal{H}^{N})$
on $E(T)_{\mathbf{j}}\times^{T}X^{u,v}_{w}(P)$ via the morphism

$E(T)_{\mathbf{j}}\times^{T}Z^{u,v}_{w}\to
E(T)_{\mathbf{j}}\times^{T}X^{u,v}_{w}(P),$

where $\pi_{1}$ is the standard quotient map
$E(T)_{\mathbf{j}}\times^{T}X^{u,v}_{w}(P)\to\mathbb{P}_{\mathbf{j}}$ and

$\tilde{\mathcal{L}}^{\prime}(-\hat{\rho}_{Y}\boxtimes-\hat{\rho}_{Y}):=E(T)_{\mathbf{j}}\times^{T}\left(e^{u\hat{\rho}_{Y}+v\hat{\rho}_{Y}}(\mathcal{L}^{P}(\hat{\rho}_{Y})\boxtimes\mathcal{L}^{P}(\hat{\rho}_{Y}))|_{X^{u,v}_{w}(P)}\right).$

From [KM08, Proposition 1.45 and Theorems 1.37 and 1.42],
$\mathcal{M}^{\prime}$ is ample on $(X^{u,v}_{w}(P))_{\mathbf{j}}$ for large
enough $N>0$. Since $\pi$ is a birational morphism and $\mathcal{M}^{\prime}$
is ample, $\mathcal{M}$ is nef and big by [Deb01, §1.29]. ∎

######  Proposition 9.3.

For $\tilde{\pi}:\widetilde{\mathcal{Z}}\to\bar{\Gamma}$,

$R^{p}\tilde{\pi}_{*}(\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}))=0\hskip
14.22636pt\text{for all}\,\,p>0,$

where $\partial\widetilde{\mathcal{Z}}:=f^{-1}(\partial\mathcal{Z}_{P})$ and
$\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}):=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(\mathcal{O}_{\widetilde{\mathcal{Z}}}(-\partial\widetilde{\mathcal{Z}}),\omega_{\widetilde{\mathcal{Z}}}).$
Here we take $\partial\mathcal{Z}_{P}$ with the reduced scheme structure.

(Observe that $f$ being a desingularization of a normal scheme
$\mathcal{Z}_{P}$ and $\partial\mathcal{Z}_{P}$ being reduced,
$\partial\widetilde{\mathcal{Z}}$ is also a reduced scheme.)

###### Proof.

As guaranteed by Proposition 9.2, let $\mathcal{M}$ be a nef and big line
bundle on $(Z^{u,v}_{w})_{\mathbf{j}}$ with divisor $\sum_{i=1}^{d}b_{i}Z_{i}$
(where $b_{i}>0$ for all $i$) supported precisely in
$\partial((Z^{u,v}_{w})_{\mathbf{j}})$. Further, $\mathcal{M}$ can be taken to
be the pull-back of an ample line bundle $\mathcal{M}^{\prime}$ on
$(X^{u,v}_{w}(P))_{\mathbf{j}}$. Let $N$ be an integer so that $N>b_{i}$ for
all $i$.

By the proof of Proposition 9.2, the line bundle
$\epsilon\boxtimes\mathcal{M}^{\prime}$ over
$\bar{\Gamma}\times(X^{u,v}_{w}(P))_{\mathbf{j}}$ restricted to
$\mathcal{Z}_{P}$ via $i$ has a section with support
$\partial\mathcal{Z}_{P}$, where $\partial\mathcal{Z}_{P}$ is defined in
Proposition 8.4 and $\epsilon$ is the trivial line bundle over $\bar{\Gamma}$.
Hence, the pull-back of the line bundle
$f^{*}(i^{*}(\epsilon\boxtimes\mathcal{M}^{\prime}))$ over
$\widetilde{\mathcal{Z}}$ has a section with support
$f^{-1}(\partial\mathcal{Z}_{P})=\partial\widetilde{\mathcal{Z}}$. Since
$\partial\widetilde{\mathcal{Z}}$ is the zero set of a line bundle on
$\widetilde{\mathcal{Z}}$, $\partial\widetilde{\mathcal{Z}}$ is a pure scheme
of codimension 1 in $\widetilde{\mathcal{Z}}$. Let $\mathcal{L}$ be the line
bundle on the smooth scheme $\widetilde{\mathcal{Z}}$ associated to the
reduced divisor $\partial\widetilde{\mathcal{Z}}$ and let $D$ be the divisor
$\sum_{i}(N-b_{i})\widetilde{Z}_{i}$ on $\widetilde{\mathcal{Z}}$, where

$\widetilde{\mathcal{Z}}_{i}:=(\bar{\Gamma}\times
Z_{i})\times_{(Z^{2}_{w})_{\mathbb{P}}}\tilde{\Delta}((Z_{w})_{\mathbb{P}}).$

Then, as proved in [Kum17, Proof of Proposition 8.4], each $\widetilde{Z}_{i}$
is a smooth irreducible divisor of $\widetilde{\mathcal{Z}}$, and for any
collection $\widetilde{Z}_{i_{1}},\cdots,\widetilde{Z}_{i_{q}}$, $1\leq
i_{1}<\cdots<i_{q}\leq d$, the intersection
$\bigcap_{j=1}^{q}\widetilde{Z}_{i_{j}}$ (if non-empty) is smooth and of pure
codimension $q$ in $\widetilde{\mathcal{Z}}$. (Observe that
$\widetilde{\mathcal{Z}}$ and $\widetilde{\mathcal{Z}}_{i}$ here coincide with
the same in [Kum17, Proposition 8.4].) Hence, the $\widetilde{Z}_{i}$’s are
all distinct and $\partial\widetilde{\mathcal{Z}}=\sum\widetilde{Z}_{i}$ is a
simple normal crossing divisor. Clearly,

$\mathcal{L}^{N}(-D)=\mathcal{O}_{\widetilde{\mathcal{Z}}}\left(\sum_{i}b_{i}\widetilde{Z}_{i}\right)\simeq\tilde{i}^{*}\left(\mathcal{O}_{\bar{\Gamma}\times(Z^{u,v}_{w})_{\mathbf{j}}}\left(\sum_{i}b_{i}(\bar{\Gamma}\times
Z_{i})\right)\right).$

By [Deb01, §1.6], since $\sum_{i}b_{i}Z_{i}$ is a nef divisor (by assumption),
and since $\tilde{i}$ is injective, $\mathcal{L}^{N}(-D)$ is
$\tilde{\pi}$-nef.

We next show that $\mathcal{L}^{N}(-D)$ is $\tilde{\pi}$-big. Since
$\mathcal{M}$ was chosen to be the pull-back of an ample line bundle
$\mathcal{M}^{\prime}$ on $(X^{u,v}_{w}(P))_{\mathbf{j}}$, we have that
$\mathcal{L}^{N}(-D)$ is the pull-back of the line bundle
$\mathcal{S}:=i^{*}(\epsilon\boxtimes\mathcal{M}^{\prime})$ on
$\mathcal{Z}_{P}$ via $f$. But $\mathcal{M}^{\prime}$ being ample implies
$\mathcal{S}$ is $\pi$-big. Further, since $f$ is birational, the fibers of
$\tilde{\pi}$ for general $\gamma$ have the same dimension as the fibers of
$\pi$ (use [Sha13, Chap. I, $\S$6.3, Theorem 1.25]). Hence
$\mathcal{L}^{N}(-D)$ is $\tilde{\pi}$-big.

Since $f$ is proper and birational it is surjective. We also have that $\pi$
is surjective (proof of [Kum17, Corollary 7.9]), hence so is $\tilde{\pi}$.
Now apply [Kum17, Theorem 8.3]. ∎

######  Theorem 9.4.

For the morphism $f:\widetilde{\mathcal{Z}}\to\mathcal{Z}_{P}$,

  1. (a)

$R^{p}f_{*}(\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}))=0$
for all $p>0$, and

  2. (b)

$f_{*}(\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}))=\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P})$,
where
$\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P}):=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{{\mathcal{Z}_{P}}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),\omega_{\mathcal{Z}_{P}}).$

###### Proof.

For (a), note that $f$ is surjective since it is proper and birational. As in
the previous proposition, $\mathcal{L}^{N}(-D)$ is $\tilde{\pi}$-nef and
$\tilde{\pi}$-big. Now the fibers of $\tilde{\pi}$ contain the fibers of $f$,
so that $\mathcal{L}^{N}(-D)$ is also $f$-nef. Since $f$ is birational, we
also have that $\mathcal{L}^{N}(-D)$ is $f$-big. Then (a) follows from [Kum17,
Theorem 8.3].

(b) Since $\partial\widetilde{\mathcal{Z}}=f^{-1}(\partial\mathcal{Z}_{P})$ is
the scheme-theoretic inverse image with the reduced scheme structure on
$\partial\mathcal{Z}_{P}$, the morphism

$f^{*}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}))\to\mathcal{O}_{\widetilde{\mathcal{Z}}}(-\partial\widetilde{\mathcal{Z}})$

is surjective (cf. [sta, Tag 01HJ, Lemma 25.4.7]) with kernel supported on a
proper closed subset of $\widetilde{\mathcal{Z}}$ (since $f$ is a
desingularization). Thus the kernel is a torsion sheaf and so the dual map

$\mathcal{O}_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}})\to\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(f^{*}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P})),\mathcal{O}_{\widetilde{\mathcal{Z}}})$
(27)

is an isomorphism, where
$\mathcal{O}_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}):=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(\mathcal{O}_{\widetilde{\mathcal{Z}}}(-\partial\widetilde{\mathcal{Z}}),\mathcal{O}_{\widetilde{\mathcal{Z}}})$.

Finally,

$\displaystyle
f_{*}(\omega_{\widetilde{\mathcal{Z}}}(\partial\widetilde{\mathcal{Z}}))$
$\displaystyle=f_{*}(\omega_{\widetilde{\mathcal{Z}}}\otimes\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(\mathcal{O}_{\widetilde{\mathcal{Z}}}(-\partial\widetilde{\mathcal{Z}}),\mathcal{O}_{\widetilde{\mathcal{Z}}}))$
$\displaystyle=f_{*}(\omega_{\widetilde{\mathcal{Z}}}\otimes\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(f^{*}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P})),\mathcal{O}_{\widetilde{\mathcal{Z}}}))\hskip
14.22636pt\text{by \eqref{dualhom}}$
$\displaystyle=f_{*}\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\widetilde{\mathcal{Z}}}}(f^{*}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P})),\omega_{\widetilde{\mathcal{Z}}})$
$\displaystyle=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{{\mathcal{Z}_{P}}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),f_{*}\omega_{\widetilde{\mathcal{Z}}})\hskip
14.22636pt\text{by adjunction \cite[cite]{[\@@bibref{}{H}{}{}, Ch. II, \S
5]}}$
$\displaystyle=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{{\mathcal{Z}_{P}}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),\omega_{\mathcal{Z}_{P}})\hskip
14.22636pt\text{by \lx@cref{creftypecap~refnum}{ratsings} and
\cite[cite]{[\@@bibref{}{KM}{}{}, Theorem 5.10]} }$
$\displaystyle=\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P}).\qed$

The following is an immediate consequence of Proposition 9.3, Theorem 9.4 and
the Grothendieck spectral sequence [Jan03, Part I, Proposition 4.1] applied to
$\tilde{\pi}=\pi\circ f$.

######  Corollary 9.5.

The morphism $\pi:\mathcal{Z}_{P}\to\bar{\Gamma}$ from the diagram satisfies

$R^{p}\pi_{*}(\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P}))=0\hskip
14.22636pt\text{for all}\ p>0.$

##  10\. Proof of Part (b) of Theorem 5.1

From Proposition 3.6, a similar argument to the proof of part (A) of Theorem
5.1 gives the vanishing

$\mathscr{T}{\kern-3.0pt{or}}_{1}^{\mathcal{O}_{(\bar{Y}_{P})_{\mathbb{P}}}}(\gamma_{*}\tilde{\Delta}_{*}\mathcal{O}_{(X_{w}^{P})_{\mathbb{P}}},\mathcal{O}_{\partial((X^{u,v}_{P})_{\mathbf{j}})})=0\hskip
14.22636pt\text{for general}\ \gamma\in\bar{\Gamma}.$

This together with the definition that
$\xi^{u}_{P}:=\mathcal{O}_{X^{u}_{P}}(-\partial X^{u}_{P})$ implies that part
(B) of Theorem 5.1 is equivalent to the following

######  Theorem 10.1.

Assume $c_{u,v}^{w}(\mathbf{j})\neq 0$. For general $\gamma\in\bar{\Gamma}$,

$H^{p}(\bar{N}_{\gamma},\mathcal{O}_{\bar{N}_{\gamma}}(-\bar{M}_{\gamma}))=0\hskip
14.22636pt\text{for all}\ p\neq|\mathbf{j}|+l(w)-l(u)-l(v),$

where $\bar{M}_{\gamma}:=M_{\gamma^{-1}}$ is the subscheme
$(\partial((X^{u,v}_{P})_{\mathbf{j}}))\cap\gamma\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$
and $\mathcal{O}_{\bar{N}_{\gamma}}(-\bar{M}_{\gamma})$ is the ideal sheaf of
$\bar{M}_{\gamma}$ in
$\bar{N}_{\gamma}:=(X^{u,v}_{P})_{\mathbf{j}}\cap\gamma\tilde{\Delta}((X_{w}^{P})_{\mathbb{P}})$.

###### Proof.

Since $\mathcal{Z}_{P}$ and $\partial\mathcal{Z}_{P}$ are CM and
$\partial\mathcal{Z}_{P}$ is of codimension 1 in $\mathcal{Z}_{P}$ (cf.
Propositions 8.1 and 8.4), we have

$\mathscr{E}{\kern-1.0pt{xt}}^{i}_{\mathcal{O}_{\mathcal{Z}_{P}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),\omega_{\mathcal{Z}_{P}})=0\hskip
14.22636pt\text{for all}\ i\geq 1.$

Similarly, by the identity (26), we also have for general
$\gamma\in\bar{\Gamma}$,

$\mathscr{E}{\kern-1.0pt{xt}}^{i}_{\mathcal{O}_{\bar{N}_{\gamma}}}(\mathcal{O}_{\bar{N}_{\gamma}}(-\bar{M}_{\gamma}),\omega_{\bar{N}_{\gamma}})=0\hskip
14.22636pt\text{for all}\ i\geq 1.$

By the Serre duality [Har77, Ch. III, Theorem 7.6] applied to
$\bar{N}_{\gamma}$ and the local-to-global Ext spectral sequence [God58, Ch.
II, Théorèm 7.3.3], the theorem is equivalent to the vanishing

$H^{p}(\bar{N}_{\gamma},\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\bar{N}_{\gamma}}}(\mathcal{O}_{\bar{N}_{\gamma}}(-\bar{M}_{\gamma}),\omega_{\bar{N}_{\gamma}}))=0\hskip
14.22636pt\text{for all}\ p>0,$ (28)

since $\bar{N}_{\gamma}$ is CM (for general $\gamma\in\bar{\Gamma}$) and
$\dim(\bar{N}_{\gamma})=|\mathbf{j}|+l(w)-l(u)-l(v)$ (cf. Corollary 8.5).

Let
$\omega_{\bar{N}_{\gamma}}(\bar{M}_{\gamma}):=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\bar{N}_{\gamma}}}(\mathcal{O}_{\bar{N}_{\gamma}}(-\bar{M}_{\gamma}),\omega_{\bar{N}_{\gamma}})$.
Then for general $\gamma\in\bar{\Gamma}$,

$\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P})|_{\pi^{-1}(\gamma^{-1})}\simeq\omega_{\pi^{-1}(\gamma^{-1})}(\partial\mathcal{Z}_{P}\cap\pi^{-1}(\gamma^{-1}))=\omega_{\bar{N}_{\gamma}}(\bar{M}_{\gamma}).$
(29)

To prove this, observe that by [Sha13, Ch. I, §6.3, Theorem 1.25] and [Har77,
Ch. III, Exercise 10.9], there exists an open nonempty subset
$\bar{\Gamma}_{o}\subset\bar{\Gamma}$ such that
$\pi:\pi^{-1}(\bar{\Gamma}_{o})\to\bar{\Gamma}_{o}$ is a flat morphism. Since
$\bar{\Gamma}_{o}$ is smooth and $\mathcal{Z}_{P}$ and
$\partial\mathcal{Z}_{P}$ are CM, and since the assertion is local in
$\bar{\Gamma}$, it is enough to observe (see [ILL+07, Corollary 11.35]) that
for a nonzero function $\theta$ on $\bar{\Gamma}_{o}$, there is an isomorphism
of $\mathcal{O}_{\mathcal{Z}_{P}^{\theta}}$-modules

$\mathcal{S}/\theta\cdot\mathcal{S}\simeq\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\mathcal{Z}_{P}^{\theta}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P})/\theta\cdot\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),\omega_{\mathcal{Z}_{P}^{\theta}}),$

where $\mathcal{Z}_{P}^{\theta}$ is the zero scheme of $\theta$ in
$\mathcal{Z}_{P}$ and
$\mathcal{S}:=\mathscr{H}{\kern-3.0pt{om}}_{\mathcal{O}_{\mathcal{Z}_{P}}}(\mathcal{O}_{\mathcal{Z}_{P}}(-\partial\mathcal{Z}_{P}),\omega_{\mathcal{Z}_{P}})$.
Taking $\theta$ in a local coordinate system, we can continue and get (29).

Now $R^{p}\pi_{*}(\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P}))=0$ for
$p>0$ (cf. Corollary 9.5) implies that for general $\gamma\in\bar{\Gamma}$,
(28) holds. This follows from the fact that since $\mathcal{Z}_{P}$ and
$\partial\mathcal{Z}_{P}$ are CM, $\bar{\Gamma}_{o}$ is smooth, and
$\pi:\pi^{-1}(\bar{\Gamma}_{o})\to\bar{\Gamma}_{o}$ is flat, we have
$\omega_{\mathcal{Z}_{P}}(\partial\mathcal{Z}_{P})$ is flat over
$\bar{\Gamma}_{o}$ (see the proof of [Kum17, Theorem 9.1]). Therefore (28)
follows from the semicontinuity theorem ([Har77, Ch. III, Theorem 12.8 and
Corollary 12.9]). Hence the theorem is proven and therefore so is part (B) of
Theorem 5.1. Thus, Corollary 5.2 is proved. ∎

##  11\. Appendix

The aim of this appendix is to determine the dualizing sheaf
$\omega_{X^{w}_{P}}$ of the CM scheme $X^{w}_{P}\subset\bar{X}_{P}$ for any
$w\in W^{P}$ (in the thick Kac-Moody flag variety). Even though this result is
not used in the paper, we believe that it is interesting on its own for
potentially its future use. When $P=B$, this result was proved by Kashiwara
(cf. [Kum17, Theorem 10.4]). Moreover, in the finite case (i.e., when $G$ is a
semisimple group), this result for any $P$ is obtained in [KRW20, Theorem
3.3].

From [Jan03, Part II, §4.2], the dualizing sheaf of $\bar{X}_{P}$ in the
finite case is given by

$\omega_{\bar{X}_{P}}=\mathcal{L}^{P}(-2\rho+2\rho^{Y}),$

where $\rho^{Y}$ denotes the half sum of positive roots coming from the Levi
component of $P$.

Given any Cohen-Macaulay subscheme $Y$ of a smooth variety $X$, its dualizing
sheaf $\omega_{Y}$ is given by

$\omega_{Y}\simeq\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{X}}^{\text{codim}\
Y}(\mathcal{O}_{Y},\mathcal{O}_{X})\otimes\omega_{X}.$

In particular, by Lemma 7.3, following the analogy with the schemes of finite
type, we define the dualizing sheaf of $X^{w}_{P}$ by

$\omega_{X^{w}_{P}}:=\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{\bar{X}_{P}}}^{l(w)}(\mathcal{O}_{X^{w}_{P}},\mathcal{O}_{\bar{X}_{P}})\otimes\mathcal{L}^{P}(-2\rho+2\rho^{Y}).$

Recall the definitions of $\rho_{Y}=\sum_{i\in Y}\varpi_{i}$ and
$\hat{\rho}_{Y}=\rho-\rho_{Y}$ from Section 2. For any $w\in W^{P}$, let

$\displaystyle\hat{\xi}^{w}_{P}:$
$\displaystyle=\mathbb{C}_{-\rho+w\rho_{Y}}\otimes\omega_{X^{w}_{P}}\otimes\mathcal{L}^{P}(\rho+\rho_{Y}-2\rho^{Y})$
$\displaystyle=\mathbb{C}_{-\rho+w\rho_{Y}}\otimes\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{\bar{X}_{P}}}^{l(w)}(\mathcal{O}_{X^{w}_{P}},\mathcal{O}_{\bar{X}_{P}})\otimes\mathcal{L}^{P}(-\hat{\rho}_{Y}).$

######  Lemma 11.1.

Restricted to the open cell $C^{w}_{P}\subset X^{w}_{P}$, we have a
$B^{-}$-equivariant isomorphism

$(\hat{\xi}^{w}_{P})|_{C^{w}_{P}}\simeq(\mathcal{O}_{X^{w}_{P}})|_{C^{w}_{P}}.$

###### Proof.

Because of the $B^{-}$-equivariance of $\hat{\xi}^{w}_{P}$, it suffices to
show that $i^{*}_{w}\hat{\xi}^{w}_{P}$ is trivial as a $T$-module, where
$i_{w}:\\{pt\\}\to\bar{X}_{P}$ is the map sending the point to the fixed point
$w$. First, note that for any character $\lambda$ of $P$,
$i^{*}_{w}\mathcal{L}^{P}(\lambda)=\mathbb{C}_{-w\lambda}$. Further,

$i^{*}_{w}\left(\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{\bar{X}_{P}}}^{l(w)}\left(\mathcal{O}_{X^{w}_{P}},\mathcal{O}_{\bar{X}_{P}}\right)\right)\simeq\det\left(\frac{T_{w}(\bar{X}_{P})}{T_{w}(X^{w}_{P})}\right)\simeq\mathbb{C}_{\rho-w\rho}.$

The last equality follows from the computation of the tangent spaces:

$\displaystyle T_{w}(X^{w}_{P})$
$\displaystyle=T_{w}(B^{-}wP/P)=T_{w}\left(w(w^{-1}B^{-}w\cap
U^{-}_{P})P/P\right)=\bigoplus_{\beta\in\Delta^{-}\cap
w(\Delta^{-}\setminus\Delta^{-}_{Y})}\mathfrak{g}_{\beta}$ $\displaystyle
T_{w}(\bar{X}_{P})$ $\displaystyle=\bigoplus_{\beta\in
w(\Delta^{-}\setminus\Delta_{Y}^{-})}\mathfrak{g}_{\beta},$

where $\Delta^{-}$ is the set of negative roots, $\Delta^{-}_{Y}$ the set of
negative roots of the Levi component of $P$, and $U^{-}_{P}$ is the unipotent
radical of the opposite parabolic of $P$. Then by [Kum02, Corollary 1.3.22],

$\det\left(\frac{T_{w}(\bar{X}_{P})}{T_{w}(X^{w}_{P})}\right)=\det\left(\bigoplus_{\beta\in\Delta^{+}\cap
w(\Delta^{-}\setminus\Delta^{-}_{Y})}\mathfrak{g}_{\beta}\right)=\det\left(\bigoplus_{\beta\in\Delta^{+}\cap
w\Delta^{-}}\mathfrak{g}_{\beta}\right)=\mathbb{C}_{\rho-w\rho},\,\,\,\text{since
$w\in W^{P}$}.$

Therefore the conclusion of the lemma follows since the weight of
$i^{*}_{w}\hat{\xi}^{w}_{P}$ is given by

$-\rho+w\rho_{Y}+(\rho-w\rho)-w(-\hat{\rho}_{Y})=w\rho_{Y}-w\rho+w\rho-w\rho_{Y}=0.\qed$

Let $\displaystyle V^{w}_{P}:=C^{w}_{P}\cup\bigcup_{v\leftarrow w}C^{v}_{P}$,
where the notation $v\leftarrow w$ indicates that $l(w)=l(v)-1$ and $v>w$.
Then, $V^{w}_{P}$ is a smooth open subset of $X^{w}_{P}$ and
$\hat{\xi}^{w}_{P}|_{V^{w}_{P}}$ is an invertible $B^{-}$-equivariant
$\mathcal{O}_{V^{w}_{P}}$-module. By Lemma 11.1,

$\hat{\xi}^{w}_{P}|_{V^{w}_{P}}\simeq\mathcal{O}_{X^{w}_{P}}\left(-\sum_{v\leftarrow
w}m_{w,v}^{P}X^{v}_{P}\right)|_{V^{w}_{P}},\hskip 14.22636pt\text{for some}\
m_{w,v}^{P}\in\mathbb{Z}.$

######  Lemma 11.2.

The coefficients $m_{w,v}^{P}$ are given by the formula

$m_{w,v}^{P}=1-\langle{w\rho_{Y},\beta^{\vee}}\rangle,$

where $\beta$ is the positive root so that $s_{\beta}w=v$.

###### Proof.

Let $v\in W^{P}$ be such that $v\leftarrow w$. We first compute

$\displaystyle
i^{*}_{v}\left(\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{\bar{X}_{P}}}^{l(w)}(\mathcal{O}_{X^{w}_{P}},\mathcal{O}_{\bar{X}_{P}})\right)$
$\displaystyle\simeq\det\left(\frac{T_{v}(\bar{X}_{P})}{T_{v}(X^{w}_{P})}\right)$
$\displaystyle\simeq\det\left(\frac{T_{v}(\bar{X}_{P})}{T_{v}(X^{v}_{P})}\right)\otimes\det\left(\frac{T_{v}(X^{w}_{P})}{T_{v}(X^{v}_{P})}\right)^{*}$
$\displaystyle\simeq\mathbb{C}_{\rho-v\rho-\beta},\,\,\,\text{by
\cite[cite]{[\@@bibref{}{Ku}{}{}, Lemma 10.3]}}.$

Hence,

$\displaystyle i^{*}_{v}\hat{\xi}^{w}_{P}$
$\displaystyle\simeq\mathbb{C}_{-\rho+w\rho_{Y}}\otimes\mathbb{C}_{\rho-v\rho-\beta}\otimes\mathbb{C}_{v\hat{\rho}_{Y}}$
$\displaystyle=\mathbb{C}_{w\rho_{Y}-\beta-v\rho_{Y}}$
$\displaystyle=\mathbb{C}_{(\langle{w\rho_{Y},\beta^{\vee}}\rangle-1)\beta}.$
(30)

On the other hand

$\displaystyle i^{*}_{v}\left(\mathcal{O}_{X^{w}_{P}}\left(-\sum_{u\leftarrow
w}m_{w,u}^{P}X^{u}_{P}\right)\right)$
$\displaystyle=\det\left(\frac{T_{v}(X^{w}_{P})}{T_{v}(X^{v}_{P})}\right)^{\otimes-
m_{w,v}^{P}}$ $\displaystyle=\mathbb{C}_{\beta}^{\otimes-m_{w,v}^{P}}$
$\displaystyle=\mathbb{C}_{-m_{w,v}^{P}\beta}.$ (31)

Equating the equations (11) and (11), we have the desired result. ∎

######  Theorem 11.3.

For any $w\in W^{P}$, we have a $B^{-}$-equivariant isomorphism

$\hat{\xi}^{w}_{P}\simeq\mathcal{O}_{X^{w}_{P}}\left(-\sum_{v\leftarrow
w}m_{w,v}^{P}X^{v}_{P}\right),$

where $m_{w,v}^{P}$ are as in Lemma 11.2. Therefore, the dualizing sheaf of
$X^{w}_{P}$ is $T$-equivariantly isomorphic to

$\mathbb{C}_{\rho-w\rho_{Y}}\otimes\mathcal{O}_{X^{w}_{P}}\left(-\sum_{v\leftarrow
w}m_{w,v}^{P}X^{v}_{P}\right)\otimes\mathcal{L}^{P}(2\rho^{Y}-\rho-\rho_{Y}).$

###### Proof.

Let $j:V^{w}_{P}\hookrightarrow X^{w}_{P}$ be the inclusion map and set
$D:=\displaystyle\sum_{v\leftarrow w}m_{w,v}^{P}X^{v}_{P}$. We have the
following commutative diagram with exact rows

${0}$${\mathcal{O}_{X^{w}_{P}}(-D)}$${\mathcal{O}_{X^{w}_{P}}}$${\mathcal{O}_{D}}$${0}$${0}$${j_{*}j^{-1}(\mathcal{O}_{X^{w}_{P}}(-D))}$${j_{*}j^{-1}(\mathcal{O}_{X^{w}_{P}})}$${j_{*}j^{-1}\mathcal{O}_{D}.}$$\scriptstyle{\simeq}$

Here the middle vertical arrow is an isomorphism since $X^{w}_{P}$ is normal
(cf. Lemma 7.3) and $X^{w}_{P}\setminus V^{w}_{P}$ is of codimension at least
2 in $X^{w}_{P}$. The right vertical map is injective since
$\overline{\text{supp}(D)\cap V^{w}_{P}}=\text{supp}(D)$. This implies that
the leftmost vertical map is an isomorphism. We also have that
$\hat{\xi}^{w}_{P}\simeq j_{*}j^{-1}(\hat{\xi}^{w}_{P})$ since
$\hat{\xi}^{w}_{P}$ is a CM $\mathcal{O}_{X^{w}_{P}}$-module. Moreover, by
Lemmas 7.3 and 11.2,

$\hat{\xi}^{w}_{P}\simeq j_{*}j^{-1}(\hat{\xi}^{w}_{P})\simeq
j_{*}j^{-1}\mathcal{O}_{X^{w}_{P}}(-D)\simeq\mathcal{O}_{X^{w}_{P}}(-D).\qed$

For $P=B$, clearly $\rho_{Y}=0$ and the coefficients
$m_{w,v}^{P}=1-\langle{w\rho_{Y},\beta^{\vee}}\rangle=1$. This gives the
following theorem of Kashiwara ([Kum17, Theorem 10.4])

######  Theorem 11.4.

For any $w\in W$, we have a $B^{-}$-equivariant isomorphism

$\xi^{w}\simeq\mathcal{O}_{X^{w}}(-\partial X^{w}),$

where
$\xi^{w}:=\mathbb{C}_{-\rho}\otimes\mathcal{L}(-\rho)\otimes\mathscr{E}{\kern-1.0pt{xt}}_{\mathcal{O}_{\bar{X}}}^{l(w)}(\mathcal{O}_{X^{w}},\mathcal{O}_{\bar{X}}).$

######  Example 11.5.

(a) For the affine Kac-Moody group $G=\widehat{G_{o}}$, where $G_{o}$ is a
simple simply-connected complex algebraic group, taking the standard maximal
parabolic subgroup $P$, we see that if we take $w=e,v=s_{0}$, then
$m^{P}_{w,v}=1.$

(b) Let us take $G=\widehat{SL_{2}}$ and $P$ the standard maximal parabolic
subgroup (as above). Then, for $w=s_{0},v=s_{1}s_{0}$, $m^{P}_{w,v}=0.$ If we
take $w=s_{1}s_{0},v=s_{0}s_{1}s_{0}$, then $m^{P}_{w,v}=-1.$

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J. C. and S.K.: Department of Mathematics, University of North Carolina,
Chapel Hill, NC 27599-3250, USA (emails<EMAIL_ADDRESS><EMAIL_ADDRESS>