# Root components for tensor product of affine Kac-Moody Lie algebra modules

Samuel Jeralds  and  Shrawan Kumar  S. Jeralds and S. Kumar: Department of
Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Abstract

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra and let $\lambda,\mu$ be
two dominant integral weights for $\mathfrak{g}$. We prove that under some
mild restriction, for any positive root $\beta$, $V(\lambda)\otimes V(\mu)$
contains $V(\lambda+\mu-\beta)$ as a component, where $V(\lambda)$ denotes the
integrable highest weight (irreducible) $\mathfrak{g}$-module with highest
weight $\lambda$. This extends the corresponding result by Kumar from the case
of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie
algebras. One crucial ingredient in the proof is the action of Virasoro
algebra via the Goddard-Kent-Olive construction on the tensor product
$V(\lambda)\otimes V(\mu)$. Then, we prove the corresponding geometric results
including the higher cohomology vanishing on the $\mathcal{G}$-Schubert
varieties in the product partial flag variety
$\mathcal{G}/\mathcal{P}\times\mathcal{G}/\mathcal{P}$ with coefficients in
certain sheaves coming from the ideal sheaves of $\mathcal{G}$-sub Schubert
varieties. This allows us to prove the surjectivity of the Gaussian map.

##  2\. Introduction

Let $\mathfrak{[}g]$ be a symmetrizable Kac–Moody Lie algebra, and fix two
dominant integral weights $\lambda$, $\mu\in\mathscr{P}^{+}$. To these, we can
associate the integrable, highest weight (irreducible) representations
$V(\lambda)$ and $V(\mu)$. Then, the content of the tensor decomposition
problem is to express the product $V(\lambda)\otimes V(\mu)$ as a direct sum
of irreducible components; that is, to find the decomposition

$V(\lambda)\otimes V(\mu)=\bigoplus_{\nu\in\mathscr{P}^{+}}V(\nu)^{\oplus
m_{\lambda,\mu}^{\nu}},$

where $m_{\lambda,\mu}^{\nu}\in\mathbb{Z}_{\geq 0}$ is the multiplicity of
$V(\nu)$ in $V(\lambda)\otimes V(\mu)$. While this is a classical problem with
a straightforward statement, determining the multiplicities
$m_{\lambda,\mu}^{\nu}$ exactly–or even determining when
$m_{\lambda,\mu}^{\nu}>0$–is a challenging endeavor. Various algebraic,
geometric, and combinatorial methods have been developed to understand the
tensor decomposition problem; see [Ku4] for a survey in the case of (finite
dimensional) semisimple Lie algebras.

While having a complete description for the components of a tensor product is
desirable, many significant results in the literature demonstrate the
existence of ”families” of components; that is, components that are uniformly
described and exist for tensor product decompositions regardless of
$\mathfrak{[}g]$. One such example is given by the root components
$V(\lambda+\mu-\beta)$ for a positive root $\beta$.

We show the existence of root components for affine Lie algebras generalizing
the corresponding result in the finite case (i.e., when $\mathfrak{[}g]$ is a
semisimple Lie algebra) as in [Ku1]. Recall that by a Wahl triple (introduced
in [Ku1] though not christened as Wahl triple there) we mean a triple
$(\lambda,\mu,\beta)\in({\mathscr{P}^{+}})^{2}\times\Phi^{+}$ such that

  * (P1)

$\lambda+\mu-\beta\in\mathscr{P}^{+}$, and

  * (P2)

If $\lambda(\alpha_{i}^{\vee})=0$ or $\mu(\alpha_{i}^{\vee})=0$, then
$\beta-\alpha_{i}\not\in\Phi\sqcup\\{0\\}$,

where $\Phi$ (resp. $\Phi^{+}$) is the set of all the roots (resp., positive
roots).

The main representation theoretic result of the paper is the following theorem
(cf. Theorem 10.1).

Theorem I. For any affine Kac-Moody Lie algebra $\mathfrak{[}g]$ and Wahl
triple $(\lambda,\mu,\beta)\in({\mathscr{P}^{+}})^{2}\times\Phi^{+}$,

$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu).$

We construct the proof in three parts. First, notice that the conditions (P1)
and (P2) are invariant under adding $\delta$; that is, if
$(\lambda,\mu,\beta)$ is a Wahl triple, then so is
$(\lambda,\mu,\beta+k\delta)$ for any $k\in\mathbb{Z}_{\geq 0}$. This allows
us to make use of the Goddard-Kent-Olive construction of the Virasoro algebra
action on the tensor product $V(\lambda)\otimes V(\mu)$ and we explore its
action on the subspaces $W^{\lambda+\mu-\beta}$ (as in Definition 3.3). This
reduces the problem to certain ‘maximal root components’ via Proposition 4.9.
In the second part, closely following the construction of root components for
simple Lie algebras as in [Ku1], we show the existence of the bulk of the
maximal root components. Finally, in the third part, we construct the
remaining maximal root components that are excluded from the previous methods
explicitly using familiar, but ad hoc, constructions from the general tensor
decomposition problem using the PRV components.

We next prove the corresponding geometric results. Let $\mathcal{G}$ be the
‘maximal’ Kac-Moody group associated to the symmetrizable Kac-Moody Lie
algebra $\mathfrak{g}$ and let $\mathcal{P}$ be a standard parabolic subgroup
of $\mathcal{G}$ corresponding to a subset $S$ of the set of simple roots
$\\{\alpha_{1},\cdots,\alpha_{\ell}\\}$, i.e., $S$ is the set of simple roots
of the Levi group of $\mathcal{P}$. In the sequel, we abbreviate $S$ as the
subset of $\\{1,\cdots,\ell\\}$. In particular, for $S=\emptyset$, we have the
standard Borel subgroup $\mathcal{B}$ (corresponding to the Borel subalgebra
$\mathfrak{b}$). Let $W$ be the Weyl group of $\mathfrak{g}$ and let
$W_{\mathcal{P}}^{\prime}$ be the set of smallest length coset representatives
in $W/W_{\mathcal{P}}$, where $W_{\mathcal{P}}$ is the subgroup of $W$
generated by the simple reflections $\\{s_{k}\\}_{k\in S}$. For any pro-
algebraic $\mathcal{P}$-module $M$, by $\mathscr{L}(M)$ we mean the
corresponding homogeneous vector bundle on
$\mathcal{X}_{\mathcal{P}}:=\mathcal{G}/\mathcal{P}$ associated to the
principal $\mathcal{P}$-bundle: $\mathcal{G}\to\mathcal{G}/\mathcal{P}$ by the
representation $M$ of $\mathcal{P}$.

For any integral weight $\lambda\in\mathfrak{h}^{*}$ (where $\mathfrak{h}$ is
the Cartan subalgebra of $\mathfrak{g}$ with the corresponding standard
maximal torus $\mathcal{H}$ of $\mathcal{G}$), such that
$\lambda(\alpha_{k}^{\vee})=0$ for all $k\in S$, the one dimensional
$\mathcal{H}$-module $\mathbb{C}_{\lambda}$ (given by the character $\lambda$)
admits a unique $\mathcal{P}$-module structure (extending the
$\mathcal{H}$-module structure); in particular, we have the line bundle
$\mathscr{L}(\mathbb{C}_{\lambda})$ on $\mathcal{X}_{\mathcal{P}}$. We
abbreviate the line bundle $\mathscr{L}(\mathbb{C}_{-\lambda})$ by
$\mathscr{L}(\lambda)$ and its restriction to the Schubert variety
$X_{w}^{\mathcal{P}}:=\overline{\mathcal{B}w\mathcal{P}/\mathcal{P}}$ by
$\mathscr{L}_{w}(\lambda)$ (for any $w\in W_{\mathcal{P}}^{\prime}$). Given
two line bundles $\mathscr{L}(\lambda)$ and $\mathscr{L}(\mu)$, we can form
their external tensor product to get the line bundle
$\mathscr{L}(\lambda\boxtimes\mu)$ on
$\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$. A dominant
integral weight $\mu$ is called $S$-regular if $\mu(\alpha_{k}^{\vee})=0$ if
and only if $k\in S$. The set of such weights is denoted by
${\mathscr{P}^{+}_{S}}^{o}$.

Then, we prove that, for any $\mu\in{\mathscr{P}^{+}_{S}}^{o}$ and $w\in
W_{\mathcal{P}}^{\prime}$ such that $X_{w}^{\mathcal{P}}$ is
$\mathcal{P}$-stable under the left multiplication:

$H^{p}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{k}\otimes\mathscr{L}_{w}(\mu))=H^{p}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e}^{k})\otimes\mathscr{L}_{w}(\mu))=0,\,\,\text{for
all $p>0,k=1,2$},$

where $\mathscr{I}_{e}$ is the ideal sheaf of $X_{w}^{\mathcal{P}}$ at the
base point $e$ and $\mathscr{O}_{w}$ denotes the structure sheaf of
$X_{w}^{\mathcal{P}}$. We further explicitly determine
$H^{0}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))$
and
$H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu))$
(cf. Proposition 11.1 and Corollary 11.2).

For any $w\in W_{\mathcal{P}}^{\prime}$ such that the Schubert variety
$X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable, define the
$\mathcal{G}$-Schubert variety:

$\hat{\mathcal{X}}_{w}^{\mathcal{P}}:=\mathcal{G}\times^{\mathcal{P}}X_{w}^{\mathcal{P}}.$

Consider the isomorphism:

$\delta:\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}\simeq\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\,\,\,[g,x]\mapsto(g\mathcal{P},gx),\,\,\text{for
$g\in\mathcal{G}$ and $x\in\mathcal{X}_{\mathcal{P}}$}.$

We have the canonical embedding

$\hat{\mathcal{X}}_{w}^{\mathcal{P}}\hookrightarrow\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}.$

In particular, we can restrict the line bundle
$\mathscr{L}(\lambda\boxtimes\mu)$ via the above isomorphism $\delta$ to
$\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ to get the line bundle denoted
$\mathscr{L}_{w}(\lambda\boxtimes\mu)$. Then, for
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o},k=1,2,w\in W_{\mathcal{P}}^{\prime}$
as above and $p\geq 0$, we determine (cf. Lemma 12.1):

$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}^{k}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\,\,\text{and}\,\,H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}^{k}_{e})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)),$

in terms of the cohomology of the partial flag variety
$\mathcal{G}/\mathcal{P}$ with coefficients in explicit homogeneous vector
bundles, where $\hat{\mathscr{I}}_{e}$ denotes the ideal sheaf of
$\widehat{\mathcal{X}}_{e}^{\mathcal{P}}$ in
$\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ and $\hat{\mathscr{O}}_{w}$ is the
structure sheaf of $\hat{\mathcal{X}}_{w}^{\mathcal{P}}$. In fact, we show
that

$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,\,\,\text{for
all $p>0$}.$

Let $\Phi^{+}\subset\mathfrak{h}^{*}$ be the set of positive roots,
$\Phi^{+}_{S}:=\Phi^{+}\cap(\oplus_{k\in S}\,\mathbb{Z}_{\geq 0}\alpha_{k})$
and $\Phi^{+}(S):=\Phi^{+}\setminus\Phi^{+}_{S}.$

We next study the vanishing of the first cohomology
$H^{1}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))$
and prove the following crucial result (cf. Proposition 12.4):

Proposition I. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra. Then,
for any $\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ (where $S$ is an arbitrary
subset of the simple roots of $\mathfrak{g}$) and any $w\in
W_{\mathcal{P}}^{\prime}$ such that the Schubert variety $X_{w}^{\mathcal{P}}$
is $\mathcal{P}$-stable, consider the following two conditions:

(a)
$H^{1}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0.$

(b) For all the real roots $\beta\in\Phi^{+}(S)$, satisfying $S\subset\\{0\leq
i\leq\ell:\beta-\alpha_{i}\not\in\Phi^{+}\sqcup\\{0\\}\\}$ and
$\lambda+\mu-\beta\in\mathscr{P}^{+}$, there exists a
$f_{\beta}\in\operatorname{Hom}_{\mathfrak{b}}\left(\mathbb{C}_{\lambda+\mu-\beta}\otimes
V(\lambda)^{\vee},V(\mu)\right)$ such that

$X_{\beta}(f_{\beta}(\mathbb{C}_{\lambda+\mu-\beta}\otimes
v_{\lambda}^{*}))\neq 0,\,\,\text{for $X_{\beta}\neq
0\in\mathfrak{g}_{\beta}$},$

where $V(\lambda)^{\vee}$ is the restricted dual of $V(\lambda)$, and
$v_{\lambda}^{*}\neq 0\in[V(\lambda)^{\vee}]_{-\lambda}$.

Then, the condition (b) implies the condition (a).

Further, we show that under the assumptions of the above proposition, the
condition (b) of the proposition is satisfied in all the cases except possibly
$\mathring{\mathfrak{g}}$ of type $F_{4}$ or $G_{2}$ (cf. Proposition 12.6 for
a more precise result). The proof of Proposition 12.6 relies on explicit
constructions of root components obtained in the earlier sections including
that of the GKO operator and the following lemma (cf. Lemma 12.5):

Lemma II. Let $(\lambda,\mu,\beta)$ be a Wahl triple for a real root $\beta$
and let $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ be a
$\delta$-maximal root component. Observe that $\beta\in\mathring{\Phi}^{+}$ or
$\beta=\delta-\gamma$ for $\gamma\in\mathring{\Phi}^{+}$. Then, for
$\beta\in\mathring{\Phi}^{+}$, the validity of condition (b) of Proposition
12.4 for $V(\lambda+\mu-\beta)$ implies its validity for
$V(\lambda+\mu-\beta-k\delta)$ for any $k\geq 0$.

Moreover, if $\beta=\delta-\gamma$ for $\gamma\in\mathring{\Phi}^{+}$, then we
have an identity connecting the condition (b) of Proposition 12.4 for
$V(\lambda+\mu-\delta+\gamma)$ with that of $V(\lambda+\mu-\beta)$ (see the
identity (12)).

Combining the above Proposition I and Proposition 12.6, we obtain the
following main geometric result of the paper (cf. Thoerem 12.7):

Theorem III. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra and let
$w\in W_{\mathcal{P}}^{\prime}$ be such that the Schubert variety
$X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable. Then, for any
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ (where $S$ is an arbitrary subset of
the simple roots) such that the condition (b) of Proposition 12.4 is satisfied
for all the Wahl triples $(\lambda,\mu,\beta)$ for any real root
$\beta\in\Phi^{+}$,

$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,\,\,\,\text{for
all $p>0$}.$

In particular, the canonical Gaussian map

$H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\to
H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{I}}_{e}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))$

is surjective.

In particular, the theorem holds for any simply-laced
$\mathring{\mathfrak{g}}$ and $\mathring{\mathfrak{g}}$ of types
$B_{\ell},C_{\ell}$. Moreover, it also holds for $\mathring{\mathfrak{g}}$ of
type $F_{4}$ in the case $\mathcal{P}$ is the Borel subgroup $\mathcal{B}$.

As a fairly straight forward corollary of the above theorem, taking inverse
limits, we get the following (cf. Corollary 12.8):

Corollary IV. Under the notation and assumptions of the above theorem, the
canonical Gaussian map

$H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\mathscr{I}_{D}\otimes\mathscr{L}(\lambda\boxtimes\mu))\to
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\mathscr{I}_{D}/\tilde{\mathscr{I}}_{D}^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))$

is surjective, where $\mathscr{I}_{D}$ is the ideal sheaf of the diagonal
$D\subset\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$ and
$\tilde{\mathscr{I}}_{D}^{2}$ is defined as
$\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w)^{2}.$

##  3\. Affine Lie algebras (Notation and Preliminaries)

In this section, we recall the definition of affine Kac-Moody Lie algebras
$\mathfrak{[}g]$ and their root and weight lattices. For a more extensive
treatment of $\mathfrak{[}g]$ and its properties, see Chapters 6 and 7 of
[Kac].

Let $\mathring{\mathfrak{[}g]}$ be a finite-dimensional simple Lie algebra
over $\mathbb{C}$ with a fixed Borel subalgebra $\mathring{\mathfrak{[}b]}$
and Cartan subalgebra
$\mathring{\mathfrak{[}h]}\subset\mathring{\mathfrak{[}b]}$. We denote the
rank of $\mathring{\mathfrak{[}g]}$ (which is, by definition, the dimension of
$\mathring{\mathfrak{[}h]}$) by $\ell$. Let
$\mathring{\Phi}\subset\mathring{\mathfrak{[}h]}^{*}$ be the set of roots and
$\mathring{\Phi}^{+}$ (resp. $\mathring{\Phi}^{-}$) be the subset of positive
(resp. negative) roots. Then, the associated affine Kac-Moody Lie algebra is
given, as a vector space, by

$\mathfrak{[}g]=\mathring{\mathfrak{[}g]}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}d,$

where $K$ is the central element and $d$ the derivation or scaling element.
The Lie bracket in $\mathfrak{[}g]$ is given by

$\displaystyle[x\otimes t^{m}+zK+\zeta d,x^{\prime}\otimes
t^{n}+z^{\prime}K+\zeta^{\prime}d]=$ $\displaystyle[x,x^{\prime}]\otimes
t^{m+n}+n\zeta x^{\prime}\otimes t^{n}-m\zeta^{\prime}x\otimes t^{m}$ (1)
$\displaystyle+m\delta_{n,-m}(x|x^{\prime})_{\mathring{\mathfrak{[}g]}}K,$

for $x,x^{\prime}\in\mathring{\mathfrak{[}g]}$, $n,m\in\mathbb{Z}$,
$z,z^{\prime},\zeta,\zeta^{\prime}\in\mathbb{C}$, where $\delta_{m,-n}$ is the
Kronecker delta and $(\cdot|\cdot)_{\mathring{\mathfrak{[}g]}}$ is the
invariant form on $\mathring{\mathfrak{[}g]}$ normalized so that
$(\theta|\theta)_{\mathring{\mathfrak{[}g]}}=2$, where
$\theta\in\mathring{\Phi}$ is the highest root of $\mathring{\mathfrak{[}g]}$.
Through the paper we will always take this normalized form on
$\mathring{\mathfrak{[}g]}$.

The Cartan subalgebra of $\mathfrak{[}g]$ is given by
$\mathfrak{[}h]:=\mathring{\mathfrak{[}h]}\oplus(\mathbb{C}K+\mathbb{C}d)$. We
thus have

$\mathfrak{[}h]^{\ast}:=\mathring{\mathfrak{[}h]}^{\ast}\oplus(\mathbb{C}\delta+\mathbb{C}\Lambda_{0}),$

where $\delta$ is defined by $\delta(d)=1$,
$\delta|_{\mathring{\mathfrak{[}h]}\oplus\mathbb{C}K}=0$ and
$\Lambda_{0}(K)=1$,
$\Lambda_{0}|_{\mathring{\mathfrak{[}h]}\oplus\mathbb{C}d}=0$ and
$\mathring{\mathfrak{[}h]}^{\ast}$ is the dual of $\mathring{\mathfrak{[}h]}$
taking $d$ and $K$ to zero. Setting $\alpha_{0}:=\delta-\theta$,
$\alpha_{0}^{\vee}:=K-\theta^{\vee}$ for $\theta$ as above, and
$\\{\alpha_{i}\\}_{i=1}^{\ell}$, $\\{\alpha_{i}^{\vee}\\}_{i=1}^{\ell}$ the
set of simple roots and coroots of $\mathring{\mathfrak{[}g]}$, we have that
$\\{\alpha_{i}\\}_{i=0}^{\ell}$, $\\{\alpha_{i}^{\vee}\\}_{i=0}^{\ell}$ are
the simple roots and coroots of $\mathfrak{[}g]$.

Let $\Phi$ be the set of roots of $\mathfrak{[}g]$ and $\Phi^{+}$ (resp.
$\Phi^{-}$) be the subset of positive (resp. negative) roots. Then, we can
separate $\Phi=\Phi_{Re}\sqcup\Phi_{Im}$, a disjoint union of real and
imaginary roots. These are given precisely by

$\displaystyle\Phi_{Re}$
$\displaystyle=\\{\beta+k\delta:\beta\in\mathring{\Phi},\ k\in\mathbb{Z}\\},$
$\displaystyle\Phi_{Im}$ $\displaystyle=\\{k\delta:k\in\mathbb{Z}\backslash
0\\}.$

Finally, let $\\{\Lambda_{i}\\}_{i=0}^{\ell}$ be the set of fundamental
weights of $\mathfrak{[}g]$ defined by
$\Lambda_{i}(\alpha_{j}^{\vee})=\delta_{ij}$. Then, the dominant integral
weights of $\mathfrak{[}g]$ are given by
$\mathscr{P}^{+}=\displaystyle\left(\bigoplus_{i=0}^{\ell}\mathbb{Z}_{\geq
0}\Lambda_{i}\right)\oplus\mathbb{C}\delta.$ The integrable highest weight
(irreducible) $\mathfrak{[}g]$-modules are parameterized by $\mathscr{P}^{+}$.
For $\lambda\in\mathscr{P}^{+}$, let $V(\lambda)$ be the corresponding
integrable highest weight $\mathfrak{[}g]$-module.

Fix dominant weights $\lambda$, $\mu\in\mathscr{P}^{+}$, and consider the
tensor product $V(\lambda)\otimes V(\mu)$ of representations. Note that, as
$\delta(\alpha_{i}^{\vee})=0$ for all $i$, the representation $V(k\delta)$ is
one-dimensional for all $k\in\mathbb{C}$, and $V(\lambda)\otimes
V(k\delta)\cong V(\lambda+k\delta)$. Therefore, throughout the remainder of
this paper, we can assume without loss of generality that
$\lambda(d)=\mu(d)=0$ up to an appropriate twist by $V(k\delta)$.

######  Definition 3.1.

For $\lambda\in\mathscr{P}^{+}$, the integer $\lambda(K)$ is called the level
of $\lambda$. Further, for any $v\in V(\lambda)$ (not necessarily a highest
weight vector), we have $K.v=\lambda(K)v$, since $K$ is a central element of
$\mathfrak{[}g]$. So, we say that $V(\lambda)$ has level $\lambda(K)$.

If $\lambda$ is of level $l$ and $\mu$ is of level $m$, then each irreducible
component of $V(\lambda)\otimes V(\mu)$ is of level $l+m$. Indeed, the Cartan
component $V(\lambda+\mu)$ clearly has level $l+m$, and if $V(\nu)\subset
V(\lambda)\otimes V(\mu)$, then necessarily
$\lambda+\mu-\nu\in\oplus_{i=0}^{\ell}\mathbb{Z}_{\geq 0}\alpha_{i}$, and
$\alpha_{i}(K)=0$ for all $i$.

We also recall the following definition taken from [BrKu] of $\delta$-maximal
components of the tensor product.

######  Definition 3.2.

($\delta$-Maximal Components) A component $V(\nu)\subset V(\lambda)\otimes
V(\mu)$ is called $\delta$-maximal if $V(\nu+k\delta)\not\subset
V(\lambda)\otimes V(\mu)$ for any $k>0$.

Let $\mathfrak{[}g]^{\prime}:=[\mathfrak{[}g],\mathfrak{[}g]]$ be the derived
subalgebra of $\mathfrak{[}g]$. By the Lie bracket given in (3), we have
$\mathfrak{[}g]^{\prime}=\mathring{\mathfrak{[}g]}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}K$.
In particular, the $\mathfrak{[}g]^{\prime}$ action on $V(\lambda)$ cannot
detect the weight $\delta$. Nevertheless, it is known that the restriction of
$V(\lambda)$ to a $\mathfrak{[}g]^{\prime}$-module remains irreducible (cf.
Lemma 2.1.4 of [Ku3]). This allows us to make the following definition.

######  Definition 3.3.

Let $V(\nu)\subset V(\lambda)\otimes V(\mu)$ be a $\delta$-maximal component.
We denote by $W^{\nu}$ the subspace

$W^{\nu}:=\sum_{k\geq 0}V(\nu-k\delta)^{\oplus m_{k}}\subset V(\lambda)\otimes
V(\mu),$

where $m_{k}:=m_{\lambda,\mu}^{\nu-k\delta}$ is the multiplicity of
$V(\nu-k\delta)$ in $V(\lambda)\otimes V(\mu)$.

That is, $W^{\nu}$ is the $\nu$-isotypic component of $V(\lambda)\otimes
V(\mu)$ with respect to the $\mathfrak{[}g]^{\prime}$-action. In the next
section, we make use of the Virasoro algebra to more closely examine the
structure of such $W^{\nu}$.

##  4\. Virasoro algebra and Goddard–Kent–Olive Construction

In this section, we recall the basics of the Virasoro algebra and its
representation theory. In particular, we give an overview of the
Goddard–Kent–Olive (GKO) construction of the Virasoro algebra and its action
on tensor products of affine Lie algebra representations. We follow the
exposition of [KRR].

######  Definition 4.1.

The Virasoro algebra $Vir$ is a Lie algebra over $\mathbb{C}$ with basis
$\\{c,L_{k}|k\in\mathbb{Z}\\}$ with commutation relations

(2) $[L_{k},L_{j}]=(k-j)L_{k+j}+\frac{1}{12}(k^{3}-k)\delta_{k,-j}c,\
[Vir,c]=0.$

We set $Vir_{0}:=\mathbb{C}L_{0}\oplus\mathbb{C}c$, and the dual space
$Vir_{0}^{\ast}:=\mathbb{C}h\oplus\mathbb{C}z$, where $\\{h,z\\}$ is the basis
dual to that of $\\{L_{0},c\\}$. Similar to the case of Kac–Moody
representations, we have the notion of highest weight representations of
$Vir$:

######  Definition 4.2.

A representation $V$ of $Vir$ is defined to be a highest weight representation
with highest weight $\lambda\in Vir_{0}^{\ast}$ if there is a vector $v\in V$
such that

$X.v=\lambda(X)v\ \forall X\in Vir_{0},\ \ L_{k}.v=0\ \forall k\geq 1,\ \
V=U\left(\bigoplus_{k<0}\mathbb{C}L_{k}\right).v.$

The structure of highest weight $Vir$ representations is in many ways parallel
to that of Kac–Moody representations. Denote by $V_{\mu}$ the $\mu$-weight
space of $V$. Then, by the defining relations (2) for $Vir$, we have
$L_{-k}:V_{\mu}\to V_{\mu+kh}$. Further, if $V$ has highest weight $\lambda$,
then for any $v\in V$ (not necessarily highest weight vector) by the equation
(2) we again have $c.v=\lambda(c)v$. We refer to the value $\lambda(c)$ as the
central charge of the representation $V$.

######  Definition 4.3.

A $Vir$ representation $V$ is called unitarizable if there is a positive-
definite Hermitian form $\langle\cdot|\cdot\rangle$ on $V$ satisfying $\langle
L_{k}.v|w\rangle=\langle v|L_{-k}.w\rangle$ and $\langle cv|w\rangle=\langle
v|cw\rangle$ for all $k\in\mathbb{Z}$ and $v,w\in V$.

The existence of a positive-definite form on a highest weight representation
$V$ allows us to investigate the weight spaces of $V$. More specifically, the
following lemma from [BrKu] determines when the weight spaces $\lambda+kh$ of
an irreducible, highest weight representation with highest weight $\lambda$
are nontrivial.

######  Lemma 4.4.

Let $V$ be a unitarizable, highest weight (irreducible) representation of
$Vir$ with highest weight $\lambda$. Then, we have the following:

  1. (1)

If $\lambda(L_{0})\neq 0$, then $V_{\lambda+kh}\neq 0$ for any
$k\in\mathbb{Z}_{\geq 0}$.

  2. (2)

If $\lambda(L_{0})=0$ and $\lambda(c)\neq 0$, then $V_{\lambda+kh}\neq 0$ for
all $k\geq 2$, and $V_{\lambda+h}=0$.

  3. (3)

If $\lambda(L_{0})=\lambda(c)=0$, then $V$ is one-dimensional.

###### Proof.

Let $v$ be the highest weight vector of $V$. Then, we have (for all $k>0$):

$0\leq\langle L_{-k}v|L_{-k}v\rangle=\langle
L_{k}L_{-k}v|v\rangle=(2k\lambda(L_{0})+\frac{1}{12}(k^{3}-k)\lambda(c))\langle
v|v\rangle.$

Thus, $\lambda(L_{0})$ and $\lambda(c)$ both must be nonnegative numbers. So,
if $\lambda(L_{0})\neq 0$, we have $L_{-k}v\neq 0$ for any $k\geq 0$. If
$\lambda(L_{0})=0$, then we have $(k^{3}-k)\lambda(c)\geq 0$; if
$\lambda(c)\neq 0$, this is nonzero for $k>1$ and zero for $k=1$, so that
$L_{-1}v=0$ and $L_{-k}v\neq 0$ for $k>1$. Finally, if
$\lambda(L_{0})=\lambda(c)=0$, then we get that $L_{-k}v=0$ for all $k>0$, so
$V$ must be one-dimensional. ∎

The Goddard–Kent–Olive construction: The similarity between representations of
Kac–Moody algebras and the Virasoro algebra is not coincidental. A
foundational result linking the two theories is given by the Sugawara
construction, which embeds $Vir\hookrightarrow\hat{U}(\mathfrak{[}g])$ in a
certain completion $\hat{U}(\mathfrak{[}g])$ of the enveloping algebra
${U}(\mathfrak{[}g])$ of an affine Lie algebra $\mathfrak{[}g]$ such that an
integrable highest weight (irreducible) representation $V(\lambda)$ of
$\mathfrak{[}g]$ becomes a unitarizable representation of $Vir$. We make use
of a related construction, known as the Goddard–Kent–Olive (for short GKO)
construction, to produce an action of $Vir$ on a tensor product
$V(\lambda)\otimes V(\mu)$ of $\mathfrak{[}g]$-modules. At its core, the GKO
construction is ”relative” and relies on the diagonal inclusion
$\mathring{\mathfrak{[}g]}\hookrightarrow\mathring{\mathfrak{[}g]}\oplus\mathring{\mathfrak{[}g]}$,
but we will not give complete details of the construction here; see Lecture 10
of [KRR] for a more in-depth treatment. The following proposition is the
primary computational tool we need from this construction.

######  Proposition 4.5.

([KRR], Proposition 10.3) Let $\mathfrak{[}g]$ be an affine Lie algebra and
$\lambda,\mu\in\mathscr{P}^{+}$ be weights with levels $l$, $m$ repsectively.
Then,

(1) $V(\lambda)\otimes V(\mu)$ is a unitarizable $Vir$ representation with
non-negative central charge

$(\dim\mathring{\mathfrak{[}g]})\left(\frac{l}{l+h^{\vee}}+\frac{m}{m+h^{\vee}}-\frac{l+m}{l+m+h^{\vee}}\right),$

where $h^{\vee}$ is the dual Coxeter number of $\mathfrak{[}g]$ ([Kac],
$\S$6.1).

(2) $L_{0}$ acts on $V(\lambda)\otimes V(\mu)$ by

$\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{\Omega}{l+m+h^{\vee}}\right),$

where $\Omega$ is the Casimir operator of $\mathfrak{[}g]$ ([Ku3], $\S$1.5)
and $(\cdot|\cdot)$ is the normalized form on $\mathfrak{[}h]^{\ast}$ as in
[Ku3], Lemma 13.1.8.

(3) For all $k$, $[L_{k},\mathfrak{[}g]^{\prime}]=0$; i.e., the $L_{k}$ are
intertwining operators for the representation of $\mathfrak{[}g]^{\prime}$ on
$V(\lambda)\otimes V(\mu)$.

######  Remark 4.6.

As the Casimir $\Omega$ acts on the $\mathfrak{[}g]$-module $V(\nu)$ by
$(\nu|\nu+2\rho)$ (cf., [Ku3], 2.1.16), we can easily compute the action of
$L_{0}$ on any component $V(\nu)$ of $V(\lambda)\otimes V(\mu)$; it will act
via a scalar depending only on $\lambda,\mu,\nu$.

Now, let $V(\nu)\subset V(\lambda)\otimes V(\mu)$ be a $\delta$-maximal
component as in Definition 3.2 and consider the subspace $W^{\nu}$ as in
Definition 3.3. By Proposition 4.5 (3), we can conclude the following
immediate corollary.

######  Corollary 4.7.

$W^{\nu}$ is a unitarizable $Vir$ representation.

In general, $W^{\nu}$ is not an irreducible $Vir$ representation. However, by
Remark 4.6, $L_{0}$ acts on each $V(\nu-k\delta)$ as a scalar, so that each of
these summands corresponds to a single $Vir$ weight space.

######  Lemma 4.8.

For any $k\in\mathbb{Z}$, $L_{-k}:V(\nu)\to V(\nu-k\delta)$, corresponding to
shifting the $Vir$ weight by $kh$, where $\nu$ is not necessarily a
$\delta$-maximal weight.

###### Proof.

As all of $W^{\nu}$ has the same non-negative central charge, we need only
compute the change in the weight with respect to $h$. By Proposition 4.5 and
Remark 4.6, we have that $L_{0}$ acts on $V(\nu-k\delta)$ via

$\displaystyle\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu-k\delta|\nu-k\delta+2\rho)}{l+m+h^{\vee}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu|\nu+2\rho)-2k(\nu|\delta)-2k(\rho|\delta)}{l+m+h^{\vee}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu|\nu+2\rho)-2k(l+m)-2k(h^{\vee})}{l+m+h^{\vee}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu|\nu+2\rho)}{l+m+h^{\vee}}\right)+k.$

Now, as $L_{0}$ acts on $V(\nu)$ by
$\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu|\nu+2\rho)}{l+m+h^{\vee}}\right)$
and we have that $L_{-k}$ adds $kh$ to a weight space, we get the desired
result. ∎

Therefore, combining Lemmas 4.8 and 4.4 we get the following key proposition,
which allows us to determine if the multiplicities
$m_{k}=m_{\lambda,\mu}^{\nu-k\delta}$ as in Definition 3.3 are nonzero.

######  Proposition 4.9.

Let $\lambda,\mu\in\mathscr{P}^{+}$ be of positive levels $l,m$, respectively.
Let $V(\nu)\subset V(\lambda)\otimes V(\mu)$ be a $\delta$-maximal component,
and consider the $Vir$ subrepresentation $W^{\nu}$. Then, we have the
following:

(1) If
$\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\nu|\nu+2\rho)}{l+m+h^{\vee}}\neq
0$, we have $m_{k}\geq 1$ for all $k\geq 0$.

(2) Else, we have $m_{1}=0$ and $m_{k}\geq 1$ for $k=0$, $k\geq 2$.

###### Proof.

Choose a nonzero highest weight vector $v_{\nu}\in V(\nu)\subset W^{\nu}$. By
Lemma 4.8, we have $L_{k}(v_{\nu})=0$ for all $k>0$, as there are no
components of the form $V(\nu+k\delta)$ in $W^{\nu}$. Thus, $v_{\nu}$ is a
highest weight vector for the $Vir$ action, and generates a highest weight,
unitarizable (irreducible) $Vir$-submodule of $W^{\nu}$. Applying Lemma 4.4,
we can determine if $L_{-k}(v_{\nu})$ is nonzero by computing the
$L_{0}$-action on $v_{\nu}$ (done in the proof of Lemma 4.8). But,
$L_{-k}(v_{\nu})\in V(\nu-k\delta)$ by Lemma 4.8 and if nonzero it is the
highest weight vector of $V(\nu-k\delta)$ under the
$U(\mathfrak{[}g]^{\prime})$ action, since
$[L_{-k},\mathfrak{[}g]^{\prime}]=0$. Therefore, Lemma 4.4 gives the result.
(Observe that $c$ acts by a nonzero scalar on $V(\lambda)\otimes V(\mu)$ by
Proposition 4.5 (1).) ∎

We conclude this section with two examples; the first is an alternate approach
to [Kac], Exercise 12.16, and the second is an interpretation of root
components for affine Lie algebras with respect to imaginary roots.

######  Example 4.10.

Consider $\mathfrak{[}g]=\widehat{\mathfrak{[}sl]}_{2}$, and let
$\lambda=\mu=\Lambda_{0}$. Then, of course $V(2\Lambda_{0})\subset
V(\Lambda_{0})\otimes V(\Lambda_{0})$ is a $\delta$-maximal component. By
Proposition 4.5 and Remark 4.6, the $L_{0}$ action on $V(2\Lambda_{0})$ is
given by the scalar:

$\frac{1}{2}\left(\frac{(\Lambda_{0}|\Lambda_{0}+2\rho)}{3}+\frac{(\Lambda_{0}|\Lambda_{0}+2\rho)}{3}-\frac{(2\Lambda_{0}|2\Lambda_{0}+2\rho)}{4}\right).$

But, we have $(\Lambda_{0}|\Lambda_{0})=(\Lambda_{0}|\rho)=0$ (cf., [Kac],
$\S$6.2), so that $L_{0}$ acts by $0$ on $V(2\Lambda_{0})$. Thus, by
Proposition 4.9, we have $V(2\Lambda_{0}-\delta)\not\subset
V(\Lambda_{0})\otimes V(\Lambda_{0})$. But, $V(2\Lambda_{0}-k\delta)\subset
V(\Lambda_{0})\otimes V(\Lambda_{0})$ for any $k>1$; this agrees with Exercise
12.16 of [Kac].

######  Example 4.11.

For any affine Lie algebra $\mathfrak{[}g]$, let $\lambda=\mu=\rho$. Then, of
course, $V(2\rho)\subset V(\rho)\otimes V(\rho)$ is $\delta$-maximal. We then
get, since $\rho$ is of level $h^{\vee}$, that $L_{0}$ acts on $V(2\rho)$ via

$\frac{1}{2}\left(\frac{2(\rho|\rho+2\rho)}{2h^{\vee}}-\frac{(2\rho|2\rho+2\rho)}{3h^{\vee}}\right)=\frac{(\rho|\rho)}{6h^{\vee}}>0.$

Thus, by Proposition 4.9, we have $V(2\rho-k\delta)\subset V(\rho)\otimes
V(\rho)$ for any $k\geq 0$.

##  5\. Some general results on tensor product decomposition

The following proposition is crucial to our work on root components in the
case of affine Lie algebras. For the semisimple Lie algebras, it is due to
Kostant [Kos].

######  Proposition 5.1.

Let $\mathfrak{[}g]$ be a symmetrizable Kac–Moody algebra, and fix
$\lambda,\mu,\nu\in\mathscr{P}^{+}$. Then, the multiplicity of $V(\nu)$ in
$V(\lambda)\otimes V(\mu)$ is given by:

$m_{\lambda,\mu}^{\nu}=\dim\\{v\in
V(\mu)_{\nu-\lambda}:e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.v=0\ \text{for all
simple roots }\alpha_{i}\\},$

where $e_{i}$ is a root vector corresponding to the root $\alpha_{i}$.

###### Proof.

Define an increasing filtration $\\{F_{n}(\lambda)\\}_{n\geq 0}$ of
$V(\lambda)$ by

$F_{n}(\lambda):=\sum_{\beta\in Q^{+}:|\beta|\leq
n}\,V(\lambda)_{\lambda-\beta},$

where $Q^{+}:=\oplus_{i}\mathbb{Z}_{\geq 0}\alpha_{i}$ , for any $\beta=\sum
n_{i}\alpha_{i}$, we define $|\beta|:=\sum n_{i}$ and $V(\lambda)_{\gamma}$
denotes the $\gamma$-weight space. Then, clearly, each $F_{n}(\lambda)$ is
$\mathfrak{[}b]$-stable. We also set

$V(\lambda)^{\vee}:=\oplus_{\beta\in
Q^{+}}\,\left(V(\lambda)_{\lambda-\beta}\right)^{*}.$

Let $\mathbb{C}_{\nu}$ denote the one-dimensional $\mathfrak{[}b]$-module with
the Cartan subalgebra $\mathfrak{[}h]$ acting via the weight $\nu$ and of
course the commutator $\mathfrak{[}n]:=[\mathfrak{[}b],\mathfrak{[}b]]$ acts
trivially. Now,

$\displaystyle\operatorname{Hom}_{\mathfrak{[}g]}\left(V(\nu),V(\lambda)\otimes
V(\mu)\right)$
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{[}b]}\left(\mathbb{C}_{\nu},V(\lambda)\otimes
V(\mu)\right)$
$\displaystyle\simeq\varinjlim_{n}\,\operatorname{Hom}_{\mathfrak{[}b]}\left(\mathbb{C}_{\nu},F_{n}(\lambda)\otimes
V(\mu)\right)$
$\displaystyle\simeq\varinjlim_{n}\,\operatorname{Hom}_{\mathfrak{[}b]}\left(F_{n}(\lambda)^{*}\otimes\mathbb{C}_{\nu},V(\mu)\right),$
since $F_{n}(\lambda)$ is finite-dimensional (3)
$\displaystyle\simeq\operatorname{Hom}_{\mathfrak{[}b]}\left(V(\lambda)^{\vee}\otimes\mathbb{C}_{\nu},V(\mu)\right),$

where the last isomorphism is induced from the surjection
$V(\lambda)^{\vee}\to F_{n}(\lambda)^{*}$ obtained from dualizing the
embedding $F_{n}(\lambda)\hookrightarrow V(\lambda)$. It is indeed an
isomorphism since any $\mathfrak{[}b]$-module homomorphism
$f:V(\lambda)^{\vee}\otimes\mathbb{C}_{\nu}\to V(\mu)$ clearly descends to a
homomorphism $\bar{f}:F_{N}(\lambda)^{*}\otimes\mathbb{C}_{\nu}\to V(\mu)$ for
some large $N$ because the weights of
$V(\lambda)^{\vee}\otimes\mathbb{C}_{\nu}$ lie in $-\lambda+\nu+Q^{+}$ whereas
the weights of $V(\mu)$ lie in $\mu-Q^{+}$.

From the definition of the integrable highest weight module (cf. [Ku3],
Definition 2.1.5 and Corollary 2.2.6) (rather its analogue for the integrable
lowest module $V(\lambda)^{\vee}$), we get that $V(\lambda)^{\vee}$ is
obtained from the Verma module for the negative Borel
$M_{\mathfrak{[}b]^{-}}(-\lambda)$ (with lowest weight $-\lambda$) by taking
its quotient by the submodule generated by
$\\{e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.v_{-\lambda}\\}$, where $e_{i}$ runs
over all the (positive) simple root vectors and $v_{-\lambda}$ is the lowest
weight vector of $M_{\mathfrak{[}b]^{-}}(-\lambda)$. In particular, the map

$U(\mathfrak{[}n])\to V(\lambda)^{\vee},\,\,\,a\mapsto a\cdot v_{-\lambda},$

is surjective with kernel the left ideal generated by
$\\{e_{i}^{\lambda(\alpha_{i}^{\vee})+1}\\}_{i}$. From this and the above
isomorphism 5, we get the proposition. ∎

The above result also follows from the work of Kashiwara [Kas1] [Kas2] on the
existence of crystal base and global base for quantum group representations.

We can deduce from Proposition 5.1 the following well-known fact, which we
refer to as additivity in the tensor decomposition. Following [Ku1], Corollary
1.5, we give a purely algebraic proof, but remark that this can also be
deduced from the Kac-Moody analogue of the Borel–Weil theorem.

######  Corollary 5.2.

Let $\lambda$, $\lambda^{\prime}$, $\mu$, $\mu^{\prime}$,
$\nu\in\mathscr{P}^{+}$. Then, $m_{\lambda,\mu}^{\nu}\leq
m_{\lambda+\lambda^{\prime},\mu+\mu^{\prime}}^{\nu+\lambda^{\prime}+\mu^{\prime}}$.

###### Proof.

Define a map $\xi:V(\mu)\to V(\mu+\mu^{\prime})$ via $\xi(v)=\pi(v\otimes
v_{\mu^{\prime}})$, where $v_{\mu^{\prime}}$ is a nonzero highest weight
vector of $V(\mu^{\prime})$ and $\pi:V(\mu)\otimes V(\mu^{\prime})\to
V(\mu+\mu^{\prime})$ is the projection onto the Cartan component. For any
nonzero $v\in V(\mu)$, $v_{\mu}\otimes v_{\mu^{\prime}}\in
U(\mathfrak{[}g]).(v\otimes v_{\mu^{\prime}})$, so the map $\xi$ is clearly
injective. Now, by Proposition 5.1, choose a vector $v\in
V(\mu)_{\nu-\lambda}$ satisfying $e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.v=0$
for all the simple roots $\alpha_{i}$. Then, $\xi(v)\in
V(\mu+\mu^{\prime})_{\nu+\mu^{\prime}-\lambda}$ and satisfies
$e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.(\xi(v))=0$. Thus, by Proposition 5.1,
we get $m_{\lambda,\mu}^{\nu}\leq
m_{\lambda,\mu+\mu^{\prime}}^{\nu+\mu^{\prime}}$. Repeating the argument a
second time for $\lambda$ and $\lambda^{\prime}$ gives the result. ∎

##  6\. Existence of root components for imaginary roots

Now, let $\beta=k\delta$ be a positive imaginary root. By the regularity
condition (P2) for root components as in the Introduction, if
$(\lambda,\mu,\beta)$ is a Wahl triple, then necessarily $\lambda$ and $\mu$
are both regular dominant weights. Combining Example 4.11 and additivity in
the tensor decomposition (Corollary 5.2), we get the first example of root
components via the following corollary.

######  Corollary 6.1.

Let $\beta=k\delta$ be a positive imaginary root, and $(\lambda,\mu,\beta)$ a
Wahl triple. Then, $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$.

######  Remark 6.2.

In fact, the stronger statement that $V(\lambda+\mu-k\delta)\subset
V(\lambda)\otimes V(\mu)$ is true if at least one of $\lambda$ or $\mu$ is
regular dominant follows immediately from the same approach as in Example
4.11.

##  7\. Existence of root components for $\beta=\gamma+k\delta$, for
$\gamma\in\mathring{\Phi}^{+}$

Fix for the remainder of this section a simple untwisted affine Lie algebra
$\mathfrak{[}g]$ and a Wahl triple
$(\lambda,\mu,\beta)\in({\mathscr{P}^{+}})^{2}\times\Phi^{+}_{Re}$. Without
loss of generality, assume that $\lambda(d)=\mu(d)=0$. Let $l$, $m$ be the
levels of $\lambda$ and $\mu$, respectively. Then, by Proposition 4.5 (2), we
know precisely how $L_{0}\in Vir$ acts on $V(\lambda)\otimes V(\mu)$ via the
GKO construction. The goal of this section is to understand how this action
interacts with potential root components $V(\lambda+\mu-\beta)$. As a first
step, we consider the Cartan component $V(\lambda+\mu)\subset
V(\lambda)\otimes V(\mu)$ in the following lemma.

######  Lemma 7.1.

For $\lambda,\mu\in\mathscr{P}^{+}$ with levels $l$, $m$, respectively, we
have

$\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\lambda+\mu|\lambda+\mu+2\rho)}{l+m+h^{\vee}}\right)\geq
0.$

###### Proof.

Let $v:=v_{\lambda}\otimes v_{\mu}\in V(\lambda)\otimes V(\mu)$, where
$v_{\lambda}$, $v_{\mu}$ are nonzero highest weight vectors in $V(\lambda)$,
$V(\mu)$, respectively. Then, $v$ is the highest weight vector of the Cartan
component $V(\lambda+\mu)\subset V(\lambda)\otimes V(\mu)$. Now, consider the
Virasoro action $Vir.v$ coming from the GKO construction. As $V(\lambda+\mu)$
is a $\delta$-maximal component of $V(\lambda)\otimes V(\mu)$, by Lemma 4.8,
$L_{k}.v=0$ for all $k\geq 1$. So, consider the highest weight, unitarizable
(irreducible) Virasoro submodule of $V(\lambda)\otimes V(\mu)$ generated by
$v$. By Proposition 4.5, we know that the action of $L_{0}$ on $v$ is given
precisely by the scalar value above. Further, by the proof of Lemma 4.4, we
know that this value must be nonnegative.

∎

Now, we can obtain from this the following key proposition pertaining to Wahl
triples.

######  Proposition 7.2.

Let $(\lambda,\mu,\beta)\in(\mathscr{P}^{+})^{2}\times\Phi^{+}_{Re}$ be a Wahl
triple. Then,

$\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\lambda+\mu-\beta|\lambda+\mu-\beta+2\rho)}{l+m+h^{\vee}}\right)>0.$

###### Proof.

We use the bilinearity of the form $(\cdot|\cdot)$ to expand the third term as

$\frac{(\lambda+\mu-\beta|\lambda+\mu-\beta+2\rho)}{l+m+h^{\vee}}=\frac{(\lambda+\mu|\lambda+\mu+2\rho)}{l+m+h^{\vee}}-\frac{2(\lambda+\mu|\beta)+2(\rho|\beta)-(\beta|\beta)}{l+m+h^{\vee}}.$

In total, we therefore have

$\displaystyle\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\lambda+\mu-\beta|\lambda+\mu-\beta+2\rho)}{l+m+h^{\vee}}\right)$
$\displaystyle=\frac{1}{2}\left(\frac{(\lambda|\lambda+2\rho)}{l+h^{\vee}}+\frac{(\mu|\mu+2\rho)}{m+h^{\vee}}-\frac{(\lambda+\mu|\lambda+\mu+2\rho)}{l+m+h^{\vee}}\right)+$
$\displaystyle\,\,\,\,\,\,\,\frac{1}{2}\left(\frac{2(\lambda+\mu|\beta)+2(\rho|\beta)-(\beta|\beta)}{l+m+h^{\vee}}\right).$

By Lemma 7.1, we know that the first term is nonnegative. Further, we know
that $2(\rho|\beta)-(\beta|\beta)\geq 0$, as
$1\leq\rho(\beta^{\vee})=\frac{2(\rho|\beta)}{(\beta|\beta)}$. Finally, by the
property (P1) of the Wahl triple, we have $(\lambda+\mu|\beta)>0$ (since
$\lambda+\mu-\beta\in\mathscr{P}^{+}$, we get $(\lambda+\mu-\beta|\beta)\geq
0$ and hence $(\lambda+\mu|\beta)>0$), so that in total the expression as a
whole is strictly positive. ∎

As a consequence of Propositions 7.2 and 4.9, we get the following corollary.

######  Corollary 7.3.

Let $(\lambda,\mu,\beta)\in(\mathscr{P}^{+})^{2}\times\Phi^{+}_{Re}$ be a Wahl
triple and let $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ be a
$\delta$-maximal root component. In particular, $\lambda,\mu$ have strictly
positive levels. Then, for any $k\geq 0$, we have
$V(\lambda+\mu-\beta-k\delta)\subset V(\lambda)\otimes V(\mu).$

###### Proof.

By Propositions 7.2, 4.5 and Remark 4.6, $L_{0}$ acts on
$V(\lambda+\mu-\beta)$ via a scalar $>0$. Thus, by Proposition 4.9,

$V(\lambda+\mu-\beta-k\delta)\subset V(\lambda)\otimes V(\mu),\,\,\,\text{for
all $k\geq 0$}.$

This proves the corollary. ∎

Note that while the above corollary allows us to conclude the existence of
infinitely many root components from the existence of its associated
$\delta$-maximal component, the action of the Virasoro does not a priori say
anything about the appearance of the $\delta$-maximal root components in the
tensor product decomposition. Nevertheless, to conclude the proof of Theorem I
of Introduction, our work is reduced by Corollary 7.3 to showing the existence
of these $\delta$-maximal root components.

Recall that, for any $\beta\in\Phi^{+}_{Re}$, we can write
$\beta=\gamma+k\delta$ for some $\gamma\in\mathring{\Phi}$ and
$k\in\mathbb{Z}_{\geq 0}$. This can further be separated into two cases: If
$\gamma\in\mathring{\Phi}^{+}$, then we can take any $k\geq 0$. If
$\gamma\in\mathring{\Phi}^{-}$, then necessarily $k\geq 1$. Therefore, when
considering the $\delta$-maximal root components $V(\lambda+\mu-\beta)$, it
suffices to consider the two possibilities: $V(\lambda+\mu-\gamma)$ and
$V(\lambda+\mu-(-\gamma+\delta))$, where $\gamma\in\mathring{\Phi}^{+}$. The
first of these cases follows immediately from Kumar’s result in the finite
case [Ku1].

######  Proposition 7.4.

Let $(\lambda,\mu,\gamma)$ be a Wahl triple with
$\gamma\in\mathring{\Phi}^{+}$. Then,
$m_{\lambda,\mu}^{\lambda+\mu-\gamma}\neq 0$.

###### Proof.

Let $\mathring{V}(\lambda)\subset V(\lambda)$ denote the
$\mathring{\mathfrak{[}g]}$-submodule generated by the highest weight vector,
and similarly for $\mathring{V}(\mu)$. Then, by [Ku1], Theorem 1.1, we have

$\mathring{V}(\lambda+\mu-\gamma)\subset\mathring{V}(\lambda)\otimes\mathring{V}(\mu).$

Denote by $v_{\gamma}\in V(\lambda)\otimes V(\mu)$ the highest weight vector
for $\mathring{\mathfrak{[}g]}$ that generates
$\mathring{V}(\lambda+\mu-\gamma)$. We claim that this is in fact a highest
weight vector for $\mathfrak{[}g]$. Indeed, all that remains to check is that
$e_{0}.v_{\gamma}=0$. By the assumption that $\lambda(d)=\mu(d)=0$, we have
that no vector in $V(\lambda)\otimes V(\mu)$ can have weight $\nu$ with
$\nu(d)\geq 1$. However, since $\gamma\in\mathring{\Phi}^{+}$, if
$e_{0}.v_{\gamma}\neq 0$, we have
$(\lambda+\mu-\gamma+\alpha_{0})(d)=\alpha_{0}(d)=1$, a contradiction. Thus,
$e_{0}.v_{\gamma}=0$ and generates a $\mathfrak{[}g]$-submodule
$V(\lambda+\mu-\gamma)\subset V(\lambda)\otimes V(\mu)$.

∎

From this, via Corollary 7.3, we have the following corollary.

######  Corollary 7.5.

Let $(\lambda,\mu,\beta)$ be a Wahl triple with $\beta=\gamma+k\delta$ for
some $\gamma\in\mathring{\Phi}^{+}$ and $k\geq 0$. Then,
$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$.

##  8\. Existence of root components for $\beta=\gamma+k\delta$, for
$\gamma\in\mathring{\Phi}^{-}$

We now show the existence of the $\delta$-maximal root components of the form
$V(\lambda+\mu-(-\gamma+\delta))$ for $\gamma\in\mathring{\Phi}^{+}$. The
results and arguments of this section parallel those in [Ku1]; we reproduce
some of the original arguments therein here for completeness.

Recall from Proposition 5.1 that

$m_{\lambda,\mu}^{\lambda+\mu-\beta}=\text{dim}\\{v\in
V(\mu)_{\mu-\beta}:e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.v=0\ \text{for all the
simple roots }\alpha_{i}\\}.$

Our goal is to explicitly construct such a vector $v\in V(\mu)_{\mu-\beta}$
for $(\lambda,\mu,\beta)\in(\mathscr{P}^{+})^{2}\times\Phi^{+}_{Re}$ a Wahl
triple. To do so, we begin with the following preparatory lemma.

######  Lemma 8.1.

Let $\beta\in\Phi^{+}_{Re}$, with $\beta=\gamma+k\delta$ for some
$\gamma\in\mathring{\Phi}$ and $k\in\mathbb{Z}_{\geq 0}$, and suppose
$\beta-2\alpha_{i}$ is a root for some $0\leq i\leq\ell$. Then, neither of
$\beta-2\alpha_{j}$ or $\beta-2\alpha_{j}-\alpha_{i}$ is a root or zero for
$j\neq i$.

###### Proof.

If $\beta\in\mathring{\Phi}^{+}$ (that is, if $k=0$), then the lemma holds by
explicit knowledge of the finite root systems; see [Bou] for example. Else,
first suppose $\beta-2\alpha_{0}$ is a root. Then, we have
$\beta-2(\delta-\theta)=(\gamma+2\theta)+(k-2)\delta$ is a root, where
$\theta\in\mathring{\Phi}^{+}$ is the highest root of the underlying
semisimple Lie algebra. Thus, $\beta-2\alpha_{0}$ is a root if and only if
$\gamma+2\theta$ is a root in $\mathring{\Phi}$, so that $\gamma=-\theta$. In
this case, neither of $\beta-2\alpha_{j}$ or $\beta-2\alpha_{j}-\alpha_{0}$ is
a root for any $j\neq 0$, as the first would mean that
$-\theta-2\alpha_{j}\in\mathring{\Phi}$, and the second would mean that
$-2\alpha_{j}\in\mathring{\Phi}$, a clear contradiction. Also, clearly,
$\beta-2\alpha_{j}$ or $\beta-2\alpha_{j}-\alpha_{i}$ can not be zero.

Now, suppose $\beta-2\alpha_{i}$ is a root for $i\neq 0$. Then, we have
$(\gamma-2\alpha_{i})+k\delta$ is a root, so that
$\gamma-2\alpha_{i}\in\mathring{\Phi}$. By the finite case, we get that
$\beta-2\alpha_{j}$ and $\beta-2\alpha_{j}-\alpha_{i}$ are not roots or zero
for $j\neq 0,i$, as this would have to hold for $\gamma$. (For $\gamma$
positive, we have seen this above and for $\gamma$ negative, we can similarly
see using the explicit knowledge of the root system as in [Bou].) As above,
$\beta-2\alpha_{0}\in\Phi$ would imply that $\gamma=-\theta$, contradicting
$\beta-2\alpha_{i}$ is a root. Finally, $\beta-2\alpha_{0}-\alpha_{i}\in\Phi$
would imply that $\gamma-\alpha_{i}+2\theta\leq\theta$, so that
$\gamma=-\theta$ or $\gamma=-\theta+\alpha_{i}$; in each of these cases,
$\gamma-2\alpha_{i}$ is not a root, contradicting $\beta-2\alpha_{i}$ being a
root. Further, it is easy to see that $\beta-2\alpha_{j}-\alpha_{i}$ can not
be zero for any $j\neq i$. ∎

We next introduce the following set of indices, which will play a role in the
construction of sufficient Wahl triples associated to a positive root $\beta$.

######  Definition 8.2.

For $\beta\in\Phi^{+}$, define $F_{\beta}:=\\{0\leq
i\leq\ell:\beta-\alpha_{i}\not\in\Phi\sqcup\\{0\\}\\}$.

######  Proposition 8.3.

Let $(\lambda,\mu,\beta)$ be a Wahl triple with $\beta$ a real root, and
suppose that $\beta-2\alpha_{i}\not\in\Phi^{+}$ for any $0\leq i\leq\ell$, or
else $\mu(\beta^{\vee})=1$. Then, $V(\lambda+\mu-\beta)\subset
V(\lambda)\otimes V(\mu)$.

Thus, for simply-laced $\mathfrak{[}g]$ (i.e., $\mathring{\mathfrak{[}g]}$ is
simply-laced), for any Wahl triple,

(4) $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu).$

###### Proof.

Let $v_{\mu}$ denote a nonzero highest weight vector of $V(\mu)$, and choose a
nonzero root vector $X_{-\beta}\in\mathfrak{[}g]_{-\beta}$. Set
$v:=X_{-\beta}.v_{\mu}$. For any $0\neq X_{\beta}\in\mathfrak{[}g]_{\beta}$,
we have (using [Ku3], Theorem 1.5.4),

$X_{\beta}.v=X_{\beta}(X_{-\beta}.v_{\mu})=[X_{\beta},X_{-\beta}]v_{\mu}=(X_{\beta}|X_{-\beta})\nu^{-1}(\beta)\cdot
v_{\mu}=(X_{\beta}|X_{-\beta})\mu(\nu^{-1}(\beta))v_{\mu},$

where $\nu:\mathfrak{[}h]\to\mathfrak{[}h]^{*}$ is the map induced by
$(\cdot|\cdot)$. This is nonzero as $(X_{\beta}|X_{-\beta})\neq 0$ and by the
regularity condition $(P2)$, $\mu(\nu^{-1}(\beta))>0$. Then, $v\in
V(\mu)_{\mu-\beta}$ is a nonzero vector.

By the dominance of $\lambda$ we have $\lambda(\alpha_{i}^{\vee})+1\geq 1$.
Then, if $i\in F_{\beta}$, we get

$e_{i}.v=e_{i}(X_{-\beta}.v_{\mu})=[e_{i},X_{-\beta}].v_{\mu}=0,$

as by definition $\alpha_{i}-\beta\not\in\Phi\cup\\{0\\}$. Else, if $i\not\in
F_{\beta}$, we have by the regularity condition $(P2)$ that
$\lambda(\alpha_{i}^{\vee})+1\geq 2$, and

$e_{i}^{2}.v=((\operatorname{ad}_{e_{i}})^{2}(X_{-\beta})).v_{\mu}=0$

by the assumption that $\beta-2\alpha_{i}\not\in\Phi^{+}$. Thus, for all $i$,
$e_{i}^{\lambda(\alpha_{i}^{\vee})+1}.v=0$. This proves the proposition in the
case $\beta-2\alpha_{i}\not\in\Phi^{+}$ for any $i$ by using Proposition 5.1.

For the proof in the case $\mu(\beta^{\vee})=1$, observe first that for any
weight $\nu$ of $V(\mu)$, $(\mu|\mu)\geq(\nu|\nu)$ (cf. [Kac], Proposition
11.4 (a)). Denoting $n:=\lambda(\alpha_{i}^{\vee})+1$, we have:

$\displaystyle(\mu-\beta+n\alpha_{i}|\mu-\beta+n\alpha_{i})$
$\displaystyle=(\mu|\mu)+n^{2}(\alpha_{i}|\alpha_{i})+2n(\mu-\beta|\alpha_{i}),\,\,\text{since
$s_{\beta}\mu=\mu-\beta$}$
$\displaystyle=(\mu|\mu)+n^{2}(\alpha_{i}|\alpha_{i})+n(\alpha_{i}|\alpha_{i})(\mu-\beta|\alpha_{i}^{\vee})$
$\displaystyle=(\mu|\mu)+n(\alpha_{i}|\alpha_{i})\left((\lambda+\mu-\beta|\alpha_{i}^{\vee})+1\right)$
$\displaystyle>(\mu|\mu),\,\,\,\text{since
$\lambda+\mu-\beta\in\mathscr{P}^{+}$ by (P1)}.$

Thus, $V(\mu)_{\mu-\beta+n\alpha_{i}}=0.$

To prove the equation (4), by Corollaries 6.1, 7.3 and 7.5, it suffices to
assume that $\beta=\delta-\gamma$, for a positive root
$\gamma\in\mathring{\Phi}^{+}$. Any such root $\beta$ does satisfy the
condition that $\beta-2\alpha_{i}\notin\Phi^{+}$ for any simple root
$\alpha_{i}$. Hence the first part of the proposition proves 4. ∎

######  Remark 8.4.

This proof holds more generally for any symmetrizable Kac-Moody algebra
$\mathfrak{[}g]$ and triple $(\lambda,\mu,\beta)$ satisfying $(P1)$ and $(P2)$
under the condition on $\beta$ as in Proposition 8.3.

Next, we consider triples $(\lambda,\mu,\beta)$ where $\beta$ is a real root
with $\beta-2\alpha_{i}\in\Phi^{+}$ for some $0\leq i\leq\ell$. By Lemma 8.1,
this $i$ is unique. For technical reasons, we assume that $\mathfrak{[}g]\neq
G_{2}^{(1)}$; this will be handled separately.

######  Proposition 8.5.

Let $\mathfrak{[}g]\neq G_{2}^{(1)}$, and let $(\lambda,\mu,\beta)$ be a Wahl
triple with $\beta$ a real root. Suppose $\alpha_{i}$ is the unique simple
root such that $\beta-2\alpha_{i}\in\Phi^{+}$. In particular,
$\mu(\beta^{\vee})\geq 1$ by the condition (P2). Then,
$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ assuming at least one
of the following:

  1. (1)

$\lambda\in\mathcal{P}^{+}$ is regular dominant, or

  2. (2)

$F_{\beta}=F_{\beta-\alpha_{i}}$

###### Proof.

Again, let $v_{\mu}\in V(\mu)$ denote a nonzero highest weight vector,
$X_{-\beta}\in\mathfrak{[}g]_{-\beta}$ a nonzero root vector, and define
$X_{-\beta+\alpha_{i}}:=[e_{i},X_{-\beta}]$,
$X_{-\beta+2\alpha_{i}}:=[e_{i},X_{-\beta+\alpha_{i}}]$. Since
$\mathfrak{[}g]\neq G_{2}^{(1)}$, we have that $|\beta(\alpha_{i}^{\vee})|\leq
2$, so that $\beta-3\alpha_{i},\beta+\alpha_{i}\not\in\Phi$. Set

$v:=(X_{-\beta+\alpha_{i}}f_{i}-2\mu(\alpha_{i}^{\vee})X_{-\beta})\cdot
v_{\mu}\in V(\mu)_{\mu-\beta}.$

Then, by a similar computation as in the proof of Proposition 8.3 (making use
of the condition (P2) since $i\notin F_{\beta}$), we have $X_{\beta}.v\neq 0$
for any $0\neq X_{\beta}\in\mathfrak{[}g]_{\beta}$, so that $v\neq 0$. (Since
$\beta+\alpha_{i}\notin\Phi,\beta(\alpha_{i}^{\vee})\geq 0$.) Now, for any
$j\neq i$, we have

$\displaystyle e_{j}^{2}.v$
$\displaystyle=e_{j}^{2}(X_{-\beta+\alpha_{i}}f_{i}.v_{\mu})-2\mu(\alpha_{i}^{\vee})e_{j}^{2}(X_{-\beta}.v_{\mu})$
$\displaystyle=((\operatorname{ad}_{e_{j}})^{2}(X_{-\beta+\alpha_{i}})f_{i}).v_{\mu}-2\mu(\alpha_{i}^{\vee})((\operatorname{ad}_{e_{j}})^{2}(X_{-\beta})).v_{\mu}$
$\displaystyle=0$

as $(\operatorname{ad}_{e_{j}})^{2}(X_{-\beta+\alpha_{i}})=0$ and
$(\operatorname{ad}_{e_{j}})^{2}(X_{-\beta})=0$ by Lemma 8.1. Further, we have

$\displaystyle e_{i}.v$
$\displaystyle=X_{-\beta+2\alpha_{i}}f_{i}v_{\mu}+\mu(\alpha_{i}^{\vee})X_{-\beta+\alpha_{i}}v_{\mu}-2\mu(\alpha_{i}^{\vee})X_{-\beta+\alpha_{i}}v_{\mu}$
$\displaystyle=X_{-\beta+2\alpha_{i}}f_{i}v_{\mu}-\mu(\alpha_{i}^{\vee})X_{-\beta+\alpha_{i}}v_{\mu}$
$\displaystyle\implies e_{i}^{2}.v$
$\displaystyle=\mu(\alpha_{i}^{\vee})X_{-\beta+2\alpha_{i}}v_{\mu}-\mu(\alpha_{i}^{\vee})X_{-\beta+2\alpha_{i}}v_{\mu}$
$\displaystyle=0$

Therefore $e_{j}^{2}.v=0$ for all $j$.

If $\lambda\in\mathcal{P}^{+}$ is regular dominant, then
$\lambda(\alpha_{j}^{\vee})+1\geq 2$ for all $j$. So, by Proposition 5.1 we
get $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$.

Else, for any $j$ such that $\lambda(\alpha_{j}^{\vee})=0$, by condition (P2)
we have $j\in F_{\beta}$; note that $i\not\in F_{\beta}$. Thus, if
$F_{\beta}=F_{\beta-\alpha_{i}}$, we have

$e_{j}.v=[e_{j},X_{-\beta+\alpha_{i}}]f_{i}\cdot
v_{\mu}-2\mu(\alpha_{i}^{\vee})[e_{j},X_{-\beta}]\cdot v_{\mu}=0,$

so, again by Proposition 5.1, we have $V(\lambda+\mu-\beta)\subset
V(\lambda)\otimes V(\mu)$.

∎

To end this section, recall the set of indices $F_{\beta}$ for
$\beta\in\Phi^{+}$ from Definition 8.2. The following dominant weights will
play a crucial role for the remaining constructions.

######  Definition 8.6.

Given $\beta\in\Phi^{+}_{Re}$, define $\rho_{\beta}:=\sum_{i\not\in
F_{\beta}}\Lambda_{i}$.

By construction, the weight $\rho_{\beta}$ satisfies condition (P2) for the
root $\beta$ and is the minimal dominant weight with $\rho_{\beta}(d)=0$ that
can do so. Therefore, we can conclude the following lemma by the additivity
property of the tensor decomposition Corollary 5.2.

######  Lemma 8.7.

If $2\rho_{\beta}-\beta\in\mathscr{P}^{+}$ is such that
$m_{\rho_{\beta},\rho_{\beta}}^{2\rho_{\beta}-\beta}\not=0$, then
$m_{\lambda,\mu}^{\lambda+\mu-\beta}\neq 0$ for any Wahl triple
$(\lambda,\mu,\beta)$.

Further, it is easy to see that $2\rho_{\beta}-\beta\in\mathcal{P}^{+}$ for
any $\mathfrak{[}g]$ except possibly for $G_{2}$-type
$\mathring{\mathfrak{[}g]}$.

##  9\. Exceptional Maximal Components

In light of Corollaries 6.1, 7.3 and 7.5, all that remains to show to complete
the proof of Theorem I is the existence of root components
$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ for
$\beta=-\gamma+\delta$, where $\gamma\in\mathring{\Phi}^{+}$. We can reduce
further, by making use of Proposition 8.3 and Lemma 8.1, to the case that
$\beta-2\alpha_{i}\in\Phi^{+}$ for some unique $i$ (in particular,
$\mathring{\mathfrak{[}g]}$ is not simply-laced). We can exclude the case
$i=0$, via the following easy lemma.

######  Lemma 9.1.

Let $(\lambda,\mu,\beta)$ be a Wahl triple such that $\beta=-\gamma+k\delta$
for some $\gamma\in\mathring{\Phi}^{+}$ and $k\geq 1$ and such that
$\beta-2\alpha_{0}\in\Phi$. Then, $V(\lambda+\mu-\beta)\subset
V(\lambda)\otimes V(\mu)$.

###### Proof.

Direct computation shows that

$\beta-2\alpha_{0}=-\gamma+k\delta-2\alpha_{0}=2\theta-\gamma+(k-2)\delta.$

This is a root if and only if $2\theta-\gamma\in\mathring{\Phi}$, so
necessarily $\gamma=\theta$. But, then we have $\beta=\alpha_{0}+(k-1)\delta$.
Thus, we can reduce to the $\delta$-maximal component associated to
$\beta=\alpha_{0}$; the existence of $V(\lambda+\mu-\alpha_{0})$ in
$V(\lambda)\otimes V(\mu)$ is clear by taking any linear combination:
$c(f_{0}\cdot v_{\lambda})\otimes v_{\mu}+dv_{\lambda}\otimes f_{0}\cdot
v_{\mu}$, for $c,d\in\mathbb{C}$. ∎

The remaining options for such $\beta=-\gamma+\delta$ (where
$\gamma\in\mathring{\Phi}^{+}$), with $i\neq 0$, correspond to roots
$\gamma\in\mathring{\Phi}^{+}$ such that
$\gamma+2\alpha_{i}\in\mathring{\Phi}^{+}$. We collect these possibilities,
organized by the type $\mathfrak{[}g]$, $\gamma\in\mathring{\Phi}^{+}$, and
the associated unique simple root denoted $\alpha_{i}(\gamma)$, in Table 1
below.

$\mathfrak{[}g]$ | $\gamma$ | $\alpha_{j}(\gamma)$
---|---|---
$B_{\ell}^{(1)}$ | $\sum_{i\leq k<\ell}\alpha_{k},\ 1\leq i<\ell$ | $\alpha_{\ell}$
$C_{\ell}^{(1)}$ | $\left(2\sum_{i\leq k<\ell}\alpha_{k}\right)+\alpha_{\ell},\ 2\leq i\leq\ell$ | $\alpha_{i-1}$
$F_{4}^{(1)}$ | $\alpha_{2}$ | $\alpha_{3}$
$F_{4}^{(1)}$ | $\alpha_{1}+\alpha_{2}$ | $\alpha_{3}$
$F_{4}^{(1)}$ | $\alpha_{2}+2\alpha_{3}$ | $\alpha_{4}$
$F_{4}^{(1)}$ | $\alpha_{1}+\alpha_{2}+2\alpha_{3}$ | $\alpha_{4}$
$F_{4}^{(1)}$ | $\alpha_{1}+2\alpha_{2}+2\alpha_{3}$ | $\alpha_{4}$
$F_{4}^{(1)}$ | $\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$ | $\alpha_{3}$
$G_{2}^{(1)}$ | $\alpha_{2}$ | $\alpha_{1}$
$G_{2}^{(1)}$ | $\alpha_{1}+\alpha_{2}$ | $\alpha_{1}$
Table 1. Exceptional Components

We refer to these roots $\beta$ and their associated root components as
”exceptional,” since root behavior of these do not appear in the semisimple
setting and as the methods to demonstrate their existence in the tensor
decomposition do not uniformly fit into the general approach of [Ku1].
Instead, we will primarily make use of the PRV components to construct these
final $\delta$-maximal components. Throughout the remainder of this section,
we deal with Wahl triples $(\lambda,\mu,\beta=\delta-\gamma)$ for each
specified $\gamma$ in the above table. We first recall the following results:

######  Theorem 9.2.

([Ku2], Theorem 3.7; [M2], Corollaire 3)(PRV components) Let $\mathfrak{[}g]$
be any symmetrizable Kac-Moody Lie algebra. For any
$\lambda,\mu\in\mathscr{P}^{+}$ and the weyl group elements $v,w\in W$ such
that $\eta:=v\lambda+w\mu\in\mathscr{P}^{+}$, we have $V(\eta)\subset
V(\lambda)\otimes V(\mu)$.

###  9.1. $B_{\ell}^{(1)}$

For $\gamma=\sum_{i\leq k<\ell}\alpha_{k},\ 1\leq i<\ell$,
$\beta=-\gamma+\delta$, we have $\alpha_{\ell}$ as the unique simple root such
that $\beta-2\alpha_{\ell}\in\Phi^{+}$. Setting
$[\ell]:=\\{0,1,2,\dots,\ell\\}$, we have:

$F_{\beta}=\begin{cases}[\ell]\backslash\\{i-1,\ell\\}&i\neq 2\\\
[\ell]\backslash\\{0,1,\ell\\}&i=2\end{cases}$

Comparing this to $F_{\beta-\alpha_{\ell}}$, we see that
$F_{\beta}=F_{\beta-\alpha_{\ell}}$ in all the cases; thus we can apply
Proposition 8.5.

###  9.2. $C_{\ell}^{(1)}$

For any $\gamma=\gamma_{i}=(2\sum_{i\leq
k<\ell}\alpha_{k})+\alpha_{\ell}\,(2\leq i\leq\ell)$ in the above table,
$F_{\beta_{i}}=[\ell]\setminus\\{i-1\\}$, where
$\beta_{i}:=\delta-\gamma_{i}$. In this case
$\rho_{\beta_{i}}(\beta_{i}^{\vee})=1$. Thus, Proposition 8.3 and Lemma 8.7
take care of this case.

###  9.3. $F_{4}^{(1)}$

  * •

$\gamma=\alpha_{2}$: Here,
$\beta=-\alpha_{2}+\delta=\alpha_{0}+2\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$,
$F_{\beta}=\\{0,2,4\\}$, and we can take
$\lambda=\mu=\rho_{\beta}=\Lambda_{1}+\Lambda_{3}$. We use the following
variant of the construction from [Ku1]: define $v\in
V(\rho_{\beta})_{\rho_{\beta}-\beta}$ by

$v:=\left(X_{-\beta+\alpha_{3}}f_{3}-2X_{-\beta}-X_{-\beta+\alpha_{3}+\alpha_{4}}[f_{3},f_{4}]\right)v_{\Lambda_{1}+\Lambda_{3}}.$

Then, one can check directly as previously that $X_{\beta}.v\neq 0$, so that
$v\neq 0$, and that $e_{0}.v=e_{2}.v=e_{4}.v=e_{1}^{2}.v=e_{3}^{2}.v=0$, as
required to apply Proposition 5.1.

  * •

$\gamma=\alpha_{1}+\alpha_{2}$: We have
$\beta=-\gamma+\delta=\alpha_{0}+\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$,
$F_{\beta}=\\{1,2,4\\}$, and we can take
$\lambda=\mu=\rho_{\beta}=\Lambda_{0}+\Lambda_{3}$. Set
$w=s_{3}s_{2}s_{4}s_{3}s_{2}s_{4}s_{3}$, $v=s_{1}s_{0}$. Then,

$v\rho_{\beta}=\rho_{\beta}-\alpha_{0}-\alpha_{1},\
w\rho_{\beta}=\rho_{\beta}-2\alpha_{2}-4\alpha_{3}-2\alpha_{4}.$

Thus, we have $V(w\rho_{\beta}+v\rho_{\beta})=V(2\rho_{\beta}-\beta)\subset
V(\rho_{\beta})\otimes V(\rho_{\beta})$ as a PRV component.

  * •

$\gamma=\alpha_{2}+2\alpha_{3}$: In this case, we have
$\beta=-\gamma+\delta=\alpha_{0}+2\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
and $\alpha_{4}$ is the unique simple root such that
$\beta-2\alpha_{4}\in\Phi^{+}$. Then,
$F_{\beta}=\\{0,2,3\\}=F_{\beta-\alpha_{4}}$ and we can apply Proposition 8.5.

  * •

$\gamma=\alpha_{1}+\alpha_{2}+2\alpha_{3}$: We have
$\beta=-\gamma+\delta=\alpha_{0}+\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
and $\alpha_{4}$ is the unique simple root such that
$\beta-2\alpha_{4}\in\Phi^{+}$. As in the previous case, we compute
$F_{\beta}=\\{1,3\\}=F_{\beta-\alpha_{4}}$, so that we can apply Proposition
8.5.

  * •

$\gamma=\alpha_{1}+2\alpha_{2}+2\alpha_{3}$: For this case, we get
$\beta=-\gamma+\delta=\alpha_{0}+\alpha_{2}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$.
Set $w=s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}$, $v=s_{0}$, and use
$\lambda=\mu=\rho_{\beta}=\Lambda_{0}+\Lambda_{4}$. Then,

$v\rho_{\beta}=\Lambda_{0}+\Lambda_{4}-\alpha_{0},\,\,w\rho_{\beta}=\rho_{\beta}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4},$

so that we get $V(w\rho_{\beta}+v\rho_{\beta})=V(2\rho_{\beta}-\beta)\subset
V(\rho_{\beta})\otimes V(\rho_{\beta})$ as a PRV component.

  * •

$\gamma=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$: In this final case,
we have $\beta=-\gamma+\delta=\alpha_{0}+\alpha_{1}+\alpha_{2}+2\alpha_{3}$,
and $\alpha_{3}$ is the unique simple root such that
$\beta-2\alpha_{3}\in\Phi^{+}$. Then, we see that
$F_{\beta}=\\{1,2,4\\}=F_{\beta-\alpha_{3}}$, so we can apply Proposition 8.5.

###  9.4. $G_{2}^{(1)}$

  * •

$\gamma=\alpha_{2}$: We have
$\beta=-\alpha_{2}+\delta=\alpha_{0}+3\alpha_{1}+\alpha_{2}$. Then, the
minimal dominant $\lambda,\mu\in\mathcal{P}^{+}$ satisfying the necessary
conditions (P1) and (P2) are $\lambda=\Lambda_{0}+\Lambda_{1}$,
$\mu=\Lambda_{0}+2\Lambda_{1}$. Setting $w=s_{1}s_{2}s_{1}$, $v=s_{0}$, we
have

$v\mu=\Lambda_{0}+2\Lambda_{1}-\alpha_{0},\,\,w\lambda=s_{1}s_{2}s_{1}(\Lambda_{0}+\Lambda_{1})=\lambda-3\alpha_{1}-\alpha_{2},$

so that we have $V(w\lambda+v\mu)=V(\lambda+\mu-\beta)\subset
V(\lambda)\otimes V(\mu)$ as a PRV component.

  * •

$\gamma=\alpha_{1}+\alpha_{2}$: We have
$\beta=-\alpha_{1}-\alpha_{2}+\delta=\alpha_{0}+2\alpha_{1}+\alpha_{2}$,
$F_{\beta}=\\{2\\}$. Then, we have $\rho_{\beta}=\Lambda_{0}+\Lambda_{1}$, and
$2\rho_{\beta}-\beta\in\mathcal{P}^{+}$. Setting $w=s_{2}s_{1}$ and
$v=s_{0}s_{1}$, we have
$w\rho_{\beta}=\Lambda_{0}+\Lambda_{1}-\alpha_{1}-\alpha_{2}$ and
$v\rho_{\beta}=\Lambda_{0}+\Lambda_{1}-\alpha_{0}-\alpha_{1}$, so that we have
$V(w\rho_{\beta}+v\rho_{\beta})=V(2\rho_{\beta}-\beta)\subset
V(\rho_{\beta})\otimes V(\rho_{\beta})$ as a PRV component.

##  10\. Main Theorem on Root Components

Combining Corollaries 6.1, 7.3 and 7.5, Lemma 8.7 and Proposition 8.3 and the
results from Section 9, we get the following main result of our paper (cited
as Theorem I in Introduction):

Recall that by a Wahl triple we mean a triple
$(\lambda,\mu,\beta)\in({\mathscr{P}^{+}})^{2}\times\Phi^{+}$ such that

  * (P1)

$\lambda+\mu-\beta\in\mathscr{P}^{+}$, and

  * (P2)

If $\lambda(\alpha_{i}^{\vee})=0$ or $\mu(\alpha_{i}^{\vee})=0$, then
$\beta-\alpha_{i}\not\in\Phi\sqcup\\{0\\}$.

######  Theorem 10.1.

For any affine Lie algebra $\mathfrak{[}g]$ and Wahl triple
$(\lambda,\mu,\beta)\in({\mathscr{P}^{+}})^{2}\times\Phi^{+}$,

$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu).$

##  11\. Study of the cohomology
$H^{*}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with its standard
Borel subalgebra $\mathfrak{b}$ and standard Cartan subalgebra $\mathfrak{h}$
([Ku3], $\S$1.1.2). Let $\Phi\subset\mathfrak{h}^{*}$ be the set of roots and
$\Phi^{+}$ (resp. $\Phi^{-}$) be the subset of positive (resp. negative) roots
and let $\Delta=\\{\alpha_{1},\dots,\alpha_{\ell}\\}\subset\Phi^{+}$ be the
subset of simple roots with $\alpha_{i}^{\vee}$ the corresponding simple
coroots. Let $\mathscr{P}^{+}$ be the set of dominant integral weights. The
integrable, highest weight (irreducible) $\mathfrak{[}g]$-modules are
parameterized by $\mathscr{P}^{+}$. For $\lambda\in\mathscr{P}^{+}$, let
$V(\lambda)$ be the corresponding integrable, highest weight
$\mathfrak{[}g]$-module. For any suset $S\subset\\{1,\dots,\ell\\}$, let
$\mathfrak{p}=\mathfrak{p}_{S}:=\mathfrak{b}\oplus(\oplus_{\alpha\in\Phi^{+}_{S}}\mathfrak{g}_{-\alpha})$
be the corresponding standard parabolic subalgebra, where
$\mathfrak{g}_{\alpha}$ is the root space corresponding to the root $\alpha$
and $\Phi^{+}_{S}:=\Phi^{+}\cap(\oplus_{k\in S}\,\mathbb{Z}_{\geq
0}\alpha_{k})$. Then,

$\mathfrak{u}:=\oplus_{\alpha\in\Phi^{+}(S)}\,\mathfrak{g}_{\alpha}$

is the nil-radical of $\mathfrak{p}$, where
$\Phi^{+}(S):=\Phi^{+}\setminus\Phi^{+}_{S}.$ Let
$\mathcal{G},\mathcal{B},\mathcal{P}=\mathcal{P}_{S},\mathcal{U},\mathcal{H}$
be the corresponding ‘maximal’ groups associated to the Lie algebras
$\mathfrak{g},\mathfrak{b},\mathfrak{p},\mathfrak{u},\mathfrak{h}$
respectively (cf. [Ku3], Section 6.1). Let $W$ be the Weyl group of
$\mathfrak{g}$ and let $W_{\mathcal{P}}^{\prime}$ be the set of smallest
length coset representatives in $W/W_{\mathcal{P}}$, where $W_{\mathcal{P}}$
is the subgroup of $W$ generated by the simple reflections $\\{s_{k}\\}_{k\in
S}$.

For any pro-algebraic $\mathcal{P}$-module $M$, by $\mathscr{L}(M)$ we mean
the corresponding homogeneous vector bundle (locally free sheaf) on
$\mathcal{X}_{\mathcal{P}}:=\mathcal{G}/\mathcal{P}$ associated to the
principal $\mathcal{P}$-bundle: $\mathcal{G}\to\mathcal{G}/\mathcal{P}$ by the
representation $M$ of $\mathcal{P}$ (cf. [Ku3], Corollary 8.2.5). For any
integral weight $\lambda\in\mathfrak{h}^{*}$, such that
$\lambda(\alpha_{k}^{\vee})=0$ for all $k\in S$, the one dimensional
$\mathcal{H}$-module $\mathbb{C}_{\lambda}$ (given by the character $\lambda$)
admits a unique $\mathcal{P}$-module structure (extending the
$\mathcal{H}$-module structure); in particular, we have the line bundle
$\mathscr{L}(\mathbb{C}_{\lambda})$ on $\mathcal{X}_{\mathcal{P}}$. We
abbreviate the line bundle $\mathscr{L}(\mathbb{C}_{-\lambda})$ by
$\mathscr{L}(\lambda)$ and its restriction to the Schubert variety
$X_{w}^{\mathcal{P}}:=\overline{\mathcal{B}w\mathcal{P}/\mathcal{P}}$ by
$\mathscr{L}_{w}(\lambda)$ (for any $w\in W_{\mathcal{P}}^{\prime}$). Given
two line bundles $\mathscr{L}(\lambda)$ and $\mathscr{L}(\mu)$, we can form
their external tensor product to get the line bundle
$\mathscr{L}(\lambda\boxtimes\mu)$ on
$\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$.

A dominant integral weight $\mu$ is called $S$-regular if
$\mu(\alpha_{k}^{\vee})=0$ if and only if $k\in S$. The set of such weights is
denoted by ${\mathscr{P}^{+}_{S}}^{o}$. With this definition, we have the
following:

######  Proposition 11.1.

For any $\mu\in{\mathscr{P}^{+}_{S}}^{o}$ and $w\in W_{\mathcal{P}}^{\prime}$
such that $X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable unde the left
multiplication (i.e., $w^{-1}\alpha_{k}\in\Phi_{S}^{+}\sqcup\Phi^{-}$ for all
the simple roots $\alpha_{k}$ with $k\in S$), (a)
$H^{p}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}\otimes\mathscr{L}_{w}(\mu))=0$, for
all $p>0$,

(b)
$H^{p}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))=0$,
for all $p>0$ , and (c)
$H^{0}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))$
(resp.
$H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu))$
is canonically isomorphic (as $\mathcal{P}$-modules) with
$\left(V_{w}(\mu)/(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot v_{\mu})\right)^{*}$
(resp. $(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot v_{\mu})^{*})$, where
$\mathscr{I}_{e}$ denotes the ideal sheaf of the base point
$e=\mathcal{P}/\mathcal{P}\subset X_{w}^{\mathcal{P}},\mathscr{O}_{w}$ denotes
the structure sheaf of $X_{w}^{\mathcal{P}}$, $v_{\mu}\neq 0\in[V(\mu)]_{\mu}$
, $V_{w}(\mu)\subset V(\mu)$ is the Demazure module (which is the
$\mathfrak{b}$-submodule of $V(\mu)$ generated by the extremal weight vector
$v_{w\mu}$) and $\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu}:=\mathbb{C}v_{\mu}\oplus T_{e}(X_{w}^{\mathcal{P}})\cdot v_{\mu}$
(here $T_{e}(X_{w}^{\mathcal{P}})$ is the Zariski tangent space of
$X_{w}^{\mathcal{P}}$ at $e$).

(Observe that, since $\mu(\alpha_{k}^{\vee})=0$ for $k\in S$,
$\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot v_{\mu}\subset V(\mu)$ is stable under
$\mathcal{P}$, and also the point $e$ being $\mathcal{P}$-fixed,
$H^{0}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{k}\otimes\mathscr{L}_{w}(\mu))$
and
$H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e}^{k})\otimes\mathscr{L}_{w}(\mu))$
have canonical $\mathcal{P}$-module structures for any $k\geq 1$.)

###### Proof.

(a) Consider the sheaf exact sequence:

(5)
$0\to\mathscr{I}_{e}\otimes\mathscr{L}_{w}(\mu)\to\mathscr{L}_{w}(\mu)\to(\mathscr{O}_{w}/\mathscr{I}_{e})\otimes\mathscr{L}_{w}(\mu)\to
0.$

Clearly,
$H^{p}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e})\otimes\mathscr{L}_{w}(\mu))=H^{p}(e,\mathscr{L}(\mu)_{|e})=0$,
for all $p>0$. Also, $H^{p}(X_{w}^{\mathcal{P}},\mathscr{L}_{w}(\mu))=0$, for
$p>0$, and the restriction map:
$H^{0}(X_{w}^{\mathcal{P}},\mathscr{L}_{w}(\mu))\to
H^{0}(e,\mathscr{L}_{w}(\mu)_{|e})$ is surjective (cf. [Ku3], Theorem 8.2.2).
So, the long exact cohomology sequence associated to the sheaf sequence (5)
gives (a).

(b) Similarly, consider the sheaf exact sequence:

(6)
$0\to\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu)\to\mathscr{L}_{w}(\mu)\to(\mathscr{O}_{w}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu)\to
0.$

We write a part of the corresponding cohomology exact sequence:

$\displaystyle 0\to
H^{0}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))\to$
$\displaystyle
H^{0}(X_{w}^{\mathcal{P}},\mathscr{L}_{w}(\mu))\overset{\pi}{\to}H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu))$
(7) $\displaystyle\to
H^{1}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))\to
0.$

The sheaf exact sequence:

(8)
$0\to(\mathscr{I}_{e}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu)\to(\mathscr{O}_{w}/\mathscr{I}^{2}_{e})\otimes\mathscr{L}_{w}(\mu)\to(\mathscr{O}_{w}/\mathscr{I}_{e})\otimes\mathscr{L}_{w}(\mu)\to
0$

gives:

(9) $0\to T_{e}(X_{w}^{\mathcal{P}})^{*}\otimes\mathbb{C}_{\mu}^{*}\to
H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}^{2}_{e})\otimes\mathscr{L}_{w}(\mu))\to\mathbb{C}_{\mu}^{*}\to
0.$

Now, we assert that the map $\pi$ is surjective: To prove this, observe first
that the map

$\theta:\mathfrak{u}^{-}\to V(\mu),\,\,\,X\mapsto X\cdot v_{\mu},$

is injective, where $\mathfrak{u}^{-}$ is the opposite nil-radical of
$\mathfrak{p}$: For any root $\beta\in\Phi^{+}(S)$, pick root vectors
$X_{\beta}\in\mathfrak{g}_{\beta}$ and $Y_{\beta}\in\mathfrak{g}_{-\beta}$.
Then,

$X_{\beta}Y_{\beta}\cdot
v_{\mu}=(X_{\beta}|Y_{\beta})(\mu|\beta)v_{\mu},\,\,\,\text{by
\cite[cite]{[\@@bibref{}{Ku3}{}{}]}, Theorem 1.5.4}.$

From the non-degeneracy of $(\cdot|\cdot)$ over $\mathfrak{g}$ and the weight
consideration, we get the injectivity of $\theta$. (Here we have used the
assumption that $\mu\in{\mathscr{P}^{+}_{S}}^{o}$.) Since $\pi$ is a
$\mathfrak{p}$ and hence $\mathfrak{h}$-module map, from the exact sequence
(9) and the following identification with the dual Demazure module:

$H^{0}(X_{w}^{\mathcal{P}}\mathscr{L}_{w}(\mu))\simeq
V_{w}(\mu)^{*}\,\,\,\text{(cf. \cite[cite]{[\@@bibref{}{Ku3}{}{}]}, Corollary
8.1.26)},$

to prove the surjectivity of $\pi$, it suffices to show that
$\theta_{w}:T_{e}(X_{w}^{\mathcal{P}})\to V_{w}(\mu),\,X\mapsto X\cdot
v_{\mu},$ is injective, where we identify
$T_{e}(X_{w}^{\mathcal{P}})\hookrightarrow
T_{e}(X^{\mathcal{P}})\simeq\mathfrak{u}^{-}$. But, the injectivity of
$\theta_{w}$ clearly follows from the injectivity of $\theta$.

So, the exact sequence (11) (and the surjectivity of $\pi$) establishes the
vanishing of
$H^{1}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))$.
Vanishing of
$H^{p}(X_{w}^{\mathcal{P}},\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu))$,
for $p>1$, follows by considering the cohomology exact sequence associated to
the sheaf exact sequence:

(10)
$0\to\mathscr{I}_{e}^{2}\otimes\mathscr{L}_{w}(\mu)\to\mathscr{I}_{e}\otimes\mathscr{L}_{w}(\mu)\to(\mathscr{I}_{e}/\mathscr{I}_{e}^{2})\otimes\mathscr{L}_{w}(\mu)\to
0,$

and the (a)-part of the proposition. So, (b) follows.

(c) It is easy to see, from the description of the map $\pi$, that

(11) $\operatorname{Ker}\pi=\\{f\in
V_{w}(\mu)^{*}:f_{|(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})}=0\\}\simeq\left(V_{w}(\mu)/(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})\right)^{*}.$

So, using the exact sequences (11) and (9), (c) follows. ∎

From the above proposition together with [Ku3], Theorem 8.2.2, we immediately
get:

######  Corollary 11.2.

Under the assumptions of Proposition 11.1,
$H^{p}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}^{k}_{e})\otimes\mathscr{L}_{w}(\mu))=0$,
for all $p>0$ and $k=1,2$.

##  12\. Surjectivity of the Gaussian map

We continue to follow the notation and assumptions from Section 11. For any
$w\in W_{\mathcal{P}}^{\prime}$ such that the Schubert variety
$X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable, set

$\hat{\mathcal{X}}_{w}^{\mathcal{P}}:=\mathcal{G}\times^{\mathcal{P}}X_{w}^{\mathcal{P}}.$

Consider the isomorphism:

$\delta:\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}\simeq\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\,\,\,[g,x]\mapsto(g\mathcal{P},gx),\,\,\text{for
$g\in\mathcal{G}$ and $x\in\mathcal{X}_{\mathcal{P}}$}.$

We have an embedding

$\hat{\mathcal{X}}_{w}^{\mathcal{P}}\hookrightarrow\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}.$

In particular, we can restrict the line bundle
$\mathscr{L}(\lambda\boxtimes\mu)$ via the above isomorphism $\delta$ to
$\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ to get the line bundle denoted
$\mathscr{L}_{w}(\lambda\boxtimes\mu)$.

######  Lemma 12.1.

For $\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o},k=1,2,w\in
W_{\mathcal{P}}^{\prime}$ as above, and any $p\geq 0$, there are canonical
isomorphisms:

$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}^{k}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\simeq
H^{p}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}(H^{0}(X_{w}^{\mathcal{P}},\mathscr{I}^{k}_{e}\otimes\mathscr{L}_{w}(\mu)))),$

and

$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}^{k}_{e})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\simeq
H^{p}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}(H^{0}(X_{w}^{\mathcal{P}},(\mathscr{O}_{w}/\mathscr{I}^{k}_{e})\otimes\mathscr{L}_{w}(\mu)))),$

where $\hat{\mathscr{I}}_{e}$ denotes the ideal sheaf of
$\widehat{\mathcal{X}}_{e}^{\mathcal{P}}$ in
$\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ and $\hat{\mathscr{O}}_{w}$ is the
structure sheaf of $\hat{\mathcal{X}}_{w}^{\mathcal{P}}$.

Further,

(12)
$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,\,\,\,\text{for
all $p>0$}.$

(We will see later that
$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,$
for all $p>0$.)

###### Proof.

Consider the fibration

$\sigma:\hat{\mathcal{X}}_{w}^{\mathcal{P}}\to\mathcal{X}_{\mathcal{P}},\,\,\,[g,x]\mapsto
g\mathcal{P}$

and the associated Leray spectral sequence. Then, it is easy to see that
$R^{p}\sigma_{*}(\hat{\mathscr{I}}_{e}^{k}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))$
is canonically isomorphic with
$\mathscr{L}(\lambda)\otimes\mathscr{L}(H^{p}(X_{w}^{\mathcal{P}},\mathscr{I}^{k}_{e}\otimes\mathscr{L}_{w}(\mu)))$
for any $p\geq 0$. (A similar statement is true for
$\hat{\mathscr{I}}_{e}^{k}$ replaced by
$\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e}^{k}$.) In particular, by
Proposition 11.1, Corollary 11.2 and the (degenerate) Leray spectral sequence,
first part of the lemma follows.

To prove the vanishing (12), use the cohomology exact sequence associated to
the sheaf exact sequence:

$0\to\hat{\mathscr{I}}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to\mathscr{L}_{w}(\lambda\boxtimes\mu)\to(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to
0,$

and [Ku2], Theorem 2.7 (also see [M2], Lemmas 21, 22). ∎

The following lemma is used in proving our crucial Proposition 12.4.

######  Lemma 12.2.

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (cf. Section 3). Take
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ and $\theta\in\mathscr{P}^{+}$.
Then, if
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)$ is nonzero, then $\theta=\lambda+\mu-\beta$, for some
$\beta\in\Phi^{+}(S)\cup\\{0\\}$.

If $\beta$ is zero, then the above space is one dimensional.

If such a $\beta$ is a real root, then the above space is nonzero if and only
if $S\subset F_{\beta}$ (cf Definition 8.2), in which case it is one
dimensional.

If $\beta$ is an imaginary root, then this space is of dimension
$(\dim\mathfrak{g}_{\beta})-|S|$.

###### Proof.

It is easy to see that, as a $\mathfrak{p}$-module,

$\mathfrak{g}\cdot
v_{\mu}\simeq(\mathfrak{g}/\operatorname{Ann}v_{\mu})\otimes\mathbb{C}_{\mu},$

where $\operatorname{Ann}v_{\mu}$ is the annihilator of $v_{\mu}$ in
$\mathfrak{g}$ and the right side is given the tensor product
$\mathfrak{p}$-module structure. Hence, using the nondegenerate invariant form
$(.|.)$ on $\mathfrak{g}$, we get:

(13) $(\mathfrak{g}\cdot
v_{\mu})^{\vee}\simeq(\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp})\otimes\mathbb{C}_{\mu}^{*}\,,$

where $\operatorname{Ker}\mu:=\\{h\in\mathfrak{h}:\mu(h))=0\\}$,
$(\operatorname{Ker}\mu)^{\perp}:=\\{h\in\mathfrak{h}:(h|\operatorname{Ker}\mu)=0\\}$
and $\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp}$ is a
$\mathfrak{p}$-module under the adjoint representation. By the presentation of
$V(\theta)^{\vee}$ as a $U(\mathfrak{b})$-module as in the proof of
Proposition 5.1, we get:

$\displaystyle\operatorname{Hom}_{\mathfrak{b}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)\simeq\operatorname{Hom}_{\mathfrak{b}}\left(V(\theta)^{\vee},\mathbb{C}_{(\lambda+\mu)}^{*}\otimes(\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp})\right),\,\,\text{by}\,\eqref{eqn12.2}$
$\displaystyle\simeq\\{v\in[\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp}]_{\lambda+\mu-\theta}:(\operatorname{ad}e_{i})^{\theta(\alpha_{i}^{\vee})+1}v=0\\},\,\,\text{for
all the simple roots\, $\alpha_{i}$},$

where $e_{i}$ is a root vector corresponding to the simple root $\alpha_{i}$.
It is easy to see, from the above description, that

(14) $\displaystyle\operatorname{Hom}$
${}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)\simeq\\{v\in[\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp}]_{\lambda+\mu-\theta}:(\operatorname{ad}e_{i})^{\theta(\alpha_{i}^{\vee})+1}v=0,$
$\displaystyle\,\,\text{for all the simple roots $\alpha_{i}$ and
$(\operatorname{ad}f_{k})v=0$ for all $k\in S$}\\},$

where $f_{k}$ is a root vector corresponding to the negative simple root
$-\alpha_{k}$. Clearly, the following conditions (1) and (2) are both
satisfied if the right side of (14) is nonzero:

(1) $\lambda+\mu-\theta=\beta$, for some $\beta\in\Phi^{+}(S)\cup\\{0\\}$, and
(2) If $\beta\neq 0$ is real, $F_{\beta}\supset S$.

Conversely, for any dominant $\theta$ which satisfies the conditions (1) and
(2) as above, the right side of (14) (and hence
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)$ is one dimensional if either $\beta=0$ or $\beta$ is a
real root:

If $\beta=0$,
$(\operatorname{ad}e_{i})^{\theta(\alpha_{i}^{\vee})+1}((\operatorname{Ker}\mu)^{\perp})=0$
for all $i\notin S$, since $\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$.
Moreover,
$\operatorname{ad}e_{k}((\operatorname{Ker}\mu)^{\perp})=\operatorname{ad}f_{k}((\operatorname{Ker}\mu)^{\perp})=0$
for all $k\in S$ (since $\mu(\alpha_{k}^{\vee})=0$).

So, assume now that $\beta\in\Phi^{+}(S)$ is a real root and pick $0\neq
X_{\beta}\in\mathfrak{g}_{\beta}$. By the above condition (2),
$(\operatorname{ad}f_{k})X_{\beta}=0$, for $k\in S$. So, by the $sl(2)$-module
theory, $\beta(\alpha_{k}^{\vee})\leq 0$ and
$(\operatorname{ad}e_{k})^{1-\beta(\alpha_{k}^{\vee})}X_{\beta}=0$. In
particular,
$(\operatorname{ad}e_{k})^{1+\theta(\alpha_{k}^{\vee})}X_{\beta}=0.$ For
$i\notin S$, we again claim that
$(\operatorname{ad}e_{i})^{\theta(\alpha_{i}^{\vee})+1}X_{\beta}=0$, since
$\beta+(\theta(\alpha_{i}^{\vee})+1)\alpha_{i}\notin\Phi^{+}$: We have

$(\beta+(\theta(\alpha_{i}^{\vee})+1)\alpha_{i})(\alpha_{i}^{\vee})=(\lambda+\mu+\theta)(\alpha_{i}^{\vee})+2\geq
4\,,$

since $\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ and
$\theta\in\mathscr{P}^{+}$. But, this contradicts [Bou], Page 278, Fact 6.
(This is where we use the assumption on $\mathfrak{g}$ being an affine Kac-
Moody Lie algebra.) This completes the proof of the lemma in the case $\beta$
is either zero or a real root.

Assume now that $\beta=\lambda+\mu-\theta$ is an imaginary root $p\delta,p>0$.
For any $h\in\mathfrak{h}$ take $v(h)=h\otimes
t^{p}\in\mathfrak{g}_{p\delta}$. Then, for any $k\in S$,

$(\operatorname{ad}e_{k})v(h)=(\operatorname{ad}f_{k})v(h)=0\,\,\text{if and
only if}\,\,\alpha_{k}(h)=0,$

where $\alpha_{0}$ is interpreted as the negative of the highest root.
Moreover, for $i\notin S$,

(15)
$(\operatorname{ad}e_{i})^{(\lambda+\mu-p\delta)(\alpha_{i}^{\vee})+1}v(h)=0,\,\,\,\text{since
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ and
$p\delta(\alpha_{i}^{\vee})=0$}.$

Thus, by the equation (14), for $\theta=\lambda+\mu-p\delta$,

(16)
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)\simeq\\{h\otimes t^{p}:\alpha_{k}(h)=0\ \forall k\in
S\\}.$

In particular, it has dimension $\dim\mathfrak{g}_{p\delta}-|S|$. ∎

######  Lemma 12.3.

Let the notation and assumptions be as in Lemma 12.2 and let $M$ be a
$\mathfrak{p}$-module quotient $\pi:R\twoheadrightarrow M$, where
$R:=\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot v_{\mu})^{\vee}$. Then,
for any $\theta\in\mathscr{P}^{+}$, the canonical map

(17)
$\hat{\pi}_{M}:\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},R\right)\to\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$

is surjective.

###### Proof.

By the proof of Lemma 12.2, both the sides of (17) are zero unless
$\theta=\lambda+\mu-\beta$, for some $\beta\in\Phi^{+}(S)\cup\\{0\\}$. We
identify $\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},R\right)$
with the right side of (14) and a similar identification for
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$. To prove
the lemma, we consider the three cases separately:

Case I, $\beta=0$: In this case
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},R\right)\simeq(\operatorname{Ker}\mu)^{\perp}$
(cf. Proof of Lemma 12.2) and thus the lemma follows easily in this case.

Case II, $\beta$ is a real root: In this case, following the proof of Lemma
12.2,

$\displaystyle\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},R\right)$
$\displaystyle\simeq\mathfrak{g}_{\beta},\,\,\,\text{if $F_{\beta}\supset S$}$
$\displaystyle=0,\,\,\,\text{otherwise}.$

Identifying $R$ with $\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp}$ via
the equation (13), it suffices to prove that if
$0\neq\pi(\mathfrak{g}_{\beta})\subset\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$,
then $F_{\beta}\supset S$. Since
$0\neq\pi(\mathfrak{g}_{\beta})\subset\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$,
we get that, for all $k\in S$, $[f_{k},\pi(\mathfrak{g}_{\beta})]=0$, which
implies $\beta(\alpha_{k}^{\vee})\leq 0$. Assune, if possible, that
$F_{\beta}\not\supset S$, i.e., for some $k\in S,\beta-\alpha_{k}\in\Phi^{+}$
(it can not be zero since $\beta\in\Phi^{+}(S)$). Then,
$[e_{k},\mathfrak{g}_{\beta-\alpha_{k}}]$ is nonzero and hence,
$\mathfrak{g}_{\beta}$ being one dimensional,
$[e_{k},\mathfrak{g}_{\beta-\alpha_{k}}]=\mathfrak{g}_{\beta}$ giving rise to
$[e_{k},\pi(\mathfrak{g}_{\beta-\alpha_{k}})]\neq 0$. From this, by the
$sl_{2}$-module theory, we easily see that
$[f_{k},\pi(\mathfrak{g}_{\beta})]\neq 0$. A contradiction (to the equation
(14)) since
$\pi(\mathfrak{g}_{\beta})\subset\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$.
Thus, $F_{\beta}\supset S$. From this the lemma follows in this case.

Case III, $\beta$ is an imaginary root $p\delta$ for $p>0$: Take $h\otimes
t^{p}$ (for $h\in\mathfrak{h}$) such that $\pi(h\otimes
t^{p})\subset\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},M\right)$.
Define the subset $S^{\prime}\subset S$ by

$S^{\prime}:=\\{k\in S:\pi(\alpha_{k}^{\vee}\otimes t^{p})=0\\},$

where we set $\alpha_{0}^{\vee}=-\theta^{\vee}$ for the highest root $\theta$
of $\mathring{\mathfrak{g}}$. Consider the decomposition

(18) $\mathring{\mathfrak{h}}=\left(\oplus_{k\in
S^{\prime}}\,\mathbb{C}\alpha_{k}^{\vee}\right)\oplus\mathring{\mathfrak{h}}(S^{\prime}),$

where
$\mathring{\mathfrak{h}}(S^{\prime}):=\\{x\in\mathring{\mathfrak{h}}:\alpha_{k}(x)=0\
\forall k\in S^{\prime}\\}.$ Let $h^{\prime}$ be the projection of $h$ in the
$\mathring{\mathfrak{h}}(S^{\prime})$-factor (under the above decomposition).
Then, from the definition of $S^{\prime}$, we see that $\pi(h\otimes
t^{p})=\pi(h^{\prime}\otimes t^{p}).$ Moreover, it is easy to see that
$h^{\prime}\otimes
t^{p}\in\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},R\right)$,
i.e.,

$(\operatorname{ad}e_{k})(h^{\prime}\otimes
t^{p})=(\operatorname{ad}f_{k})(h^{\prime}\otimes t^{p})=0,\,\,\,\text{for all
$k\in S$}$

(cf. the equation (15)). ∎

We prove the following crucial result towards the proof of Theorem 12.7. We
follow the notation as in Lemma 12.1.

######  Proposition 12.4.

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (cf. Section 3). Then,
for any $\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ (where $S$ is an arbitrary
subset of the simple roots of $\mathfrak{g}$) and any $w\in
W_{\mathcal{P}}^{\prime}$ such that the Schubert variety $X_{w}^{\mathcal{P}}$
is $\mathcal{P}$-stable, consider the following two conditions:

(a)
$H^{1}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0.$

(b) For all the real roots $\beta\in\Phi^{+}(S)$, satisfying $F_{\beta}\supset
S$ and $\lambda+\mu-\beta\in\mathscr{P}^{+}$ (in particular,
$(\lambda,\mu,\beta)$ is a Wahl triple), there exists a
$f_{\beta}\in\operatorname{Hom}_{\mathfrak{b}}\left(\mathbb{C}_{\lambda+\mu-\beta}\otimes
V(\lambda)^{\vee},V(\mu)\right)$ such that

(19) $X_{\beta}(f_{\beta}(\mathbb{C}_{\lambda+\mu-\beta}\otimes
v_{\lambda}^{*}))\neq 0,\,\,\text{for $X_{\beta}\neq
0\in\mathfrak{g}_{\beta}$},\,\,\,\text{ where $v_{\lambda}^{*}\neq
0\in[V(\lambda)^{\vee}]_{-\lambda}$}.$

Then, the condition (b) implies the condition (a).

###### Proof.

Consider the sheaf exact sequence over $\hat{\mathcal{X}}_{w}^{\mathcal{P}}$:

(20)
$0\to\hat{\mathscr{I}}^{2}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to\mathscr{L}_{w}(\lambda\boxtimes\mu)\to(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to
0.$

From the cohomology exact sequence associated to the above sheaf exact
sequence and the vanishing of
$H^{1}(\hat{\mathcal{X}}^{\mathcal{P}}_{w},\mathscr{L}_{w}(\lambda\boxtimes\mu))$
(cf. [Ku2], Theorem 2.7), we get that (a) is equivalent to the surjectivity of
the canonical map

$\tau_{w}:H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\mathscr{L}_{w}(\lambda\boxtimes\mu))\to
H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)).$

By Lemma 12.1 and Proposition 11.1 we get:

$H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}^{2}_{e})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\simeq
H^{0}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}((\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*})).$

Further, by [Ku3], Theorem 8.2.2,

$H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\mathscr{L}_{w}(\lambda\boxtimes\mu))\simeq
H^{0}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}(V_{w}(\mu)^{*})).$

Transporting the map $\tau_{w}$ under these identifications, we get the map

$\tilde{\tau}_{w}:H^{0}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}(V_{w}(\mu)^{*}))\to
H^{0}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}((\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*})),$

which is induced from the canonical restriction:
$\mathscr{L}(V_{w}(\mu)^{*})\to\mathscr{L}((\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*})$.

By [M1], Proposition 12, for any finite dimensional pro-algebraic
$\mathcal{P}$-module $M$, we have a canonical $\mathcal{G}$-module
isomorphism:

$H^{0}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(M))\simeq\oplus_{\theta\in\mathscr{P}^{+}}\,V(\theta)^{*}\otimes[V(\theta)\otimes
M]^{\mathfrak{p}},$

where $\mathcal{G}$ acts trivially on $[V(\theta)\otimes M]^{\mathfrak{p}}$.
So, the surjectivity of the map $\tilde{\tau}_{w}$ (and hence of $\tau_{w}$)
is equivalent to the surjectivity of the canonical restriction maps

$\gamma_{\theta}^{w}:\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V_{w}(\mu)^{*}\right)\to\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}\right)$

for all $\theta\in\mathscr{P}^{+}$.

For $\theta=\lambda+\mu$, the map $\gamma^{w}_{\theta}$ is clearly nonzero and
hence surjective by Lemmas 12.2 and 12.3.

Making use of Lemmas 12.2 and 12.3 again, we can assume (for the surjectivity
of $\gamma_{\theta}^{w}$) that $\theta$ is of the form
$\theta=\lambda+\mu-\beta$ for $\beta\in\Phi^{+}(S)$ and if $\beta$ is a real
root, then $F_{\beta}\supset S$.

We first analyze the surjectivity of

$\gamma_{\theta}:\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V(\mu)^{\vee}\right)\to\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right).$

Choose a positive definite Hermitian form $\\{\cdot|\cdot\\}$ on $V(\lambda)$
(and $V(\mu)$) satisfying

$\\{Xv|w\\}=\\{v|\sigma(X)w\\},\,\,\text{for $v,w\in V(\lambda)$ and
$X\in\mathfrak{g}$},$

where $\sigma$ is a conjugate-linear involution of $\mathfrak{g}$ which takes
$\mathfrak{g}_{\alpha}$ to $\mathfrak{g}_{-\alpha}$ for any root $\alpha$ (cf.
[Ku3], Theorem 2.3.13). Now, set the tensor product form (again denoted by)
$\\{\cdot|\cdot\\}$ on $V(\lambda)\otimes V(\mu)$.

Consider the following diagram (for $\theta=\lambda+\mu-\beta$):

$\textstyle{\operatorname{Hom}_{\mathfrak{g}}(V(\theta),V(\lambda)\otimes
V(\mu))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\simeq}$$\scriptstyle{\gamma_{2}}$$\textstyle{\operatorname{Hom}_{\mathfrak{g}}(V(\lambda)\otimes
V(\mu),V(\theta))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\gamma_{3}}$$\textstyle{\operatorname{Hom}_{\mathfrak{p}}(\mathbb{C}_{\lambda}\otimes
V(\mu),V(\theta))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\gamma}_{\theta}}$$\textstyle{\operatorname{Hom}_{\mathfrak{b}}(\mathbb{C}_{\theta}\otimes
V(\lambda)^{\vee},V(\mu))}$$\textstyle{\operatorname{Hom}_{\mathfrak{p}}(\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot
v_{\mu}),V(\theta))\,,}$

where $\gamma_{1}$ and $\gamma_{3}$ are the canonical restriction maps, which
are isomorphisms (as can be easily seen). We now describe the map
$\gamma_{2}$: For any nonzero
$f\in\operatorname{Hom}_{\mathfrak{g}}(V(\theta),V(\lambda)\otimes V(\mu))$,
write $V(\lambda)\otimes
V(\mu)=\operatorname{Image}f\oplus(\operatorname{Image}f)^{\perp}$ and set
$\gamma_{2}(f)$ as the projection on the first factor (identifying it with
$V(\theta)$ under $f$). It is easy to see that $\gamma_{2}$ is a bijection
(though non-linear). Further, $\hat{\gamma}_{\theta}$ is the canonical
restriction map.

We identify

$\operatorname{Hom}_{\mathfrak{p}}\left(\mathbb{C}_{\lambda}\otimes
V(\mu),V(\theta)\right)\simeq\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V(\mu)^{\vee}\right)$

and

$\operatorname{Hom}_{\mathfrak{p}}\left(\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot
v_{\mu}),V(\theta)\right)\simeq\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right).$

We first take $\theta=\lambda+\mu-\beta$ for a real root $\beta\in\Phi^{+}(S)$
such that $S\subset F_{\beta}$. Following through the various isomorphisms in
the above diagram, it can be seen that (denoting
$\mathfrak{n}^{+}=[\mathfrak{b},\mathfrak{b}]$ and $\mathfrak{n}^{-}$ is the
opposite subalgebra to $\mathfrak{n}$)

$\displaystyle\hat{\gamma}_{\theta}$ is nonzero (and hence surjective since
the range has dimension $1$) $\displaystyle\Leftrightarrow\,\text{there exists
a $\mathfrak{b}$-morphism $f=f_{\beta}:\mathbb{C}_{\theta}\otimes
V(\lambda)^{\vee}\to V(\mu)$}$ such that
$\\{(\gamma_{1}^{-1}f)V(\theta)|\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot
v_{\mu})\\}\neq 0$
$\displaystyle\Leftrightarrow\\{U(\mathfrak{n}^{-})((\gamma_{1}^{-1}f)\mathbb{C}_{\theta})|\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot
v_{\mu})\\}\neq 0$
$\displaystyle\Leftrightarrow\\{(\gamma_{1}^{-1}f)\mathbb{C}_{\theta}|\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot
v_{\mu})\\}\neq 0\,\,\text{by the invariance of}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\\{\cdot|\cdot\\};\,\text{since
$\mathbb{C}_{\lambda}\otimes(\mathfrak{g}\cdot v_{\mu})$ is
$U(\mathfrak{n})$-stable}$
$\displaystyle\Leftrightarrow\\{f(\mathbb{C}_{\theta}\otimes
v_{\lambda}^{*})|\mathfrak{g}\cdot v_{\mu}\\}\neq 0$
$\displaystyle\Leftrightarrow\\{f(\mathbb{C}_{\theta}\otimes
v_{\lambda}^{*})|X_{-\beta}\cdot v_{\mu}\\}\neq 0,\,\text{for $0\neq
X_{-\beta}\in\mathfrak{g}_{-\beta}$}$
$\displaystyle\Leftrightarrow\\{X_{\beta}(f(\mathbb{C}_{\theta}\otimes
v_{\lambda}^{*}))|v_{\mu}\\}\neq 0,\,\text{by the invariance of
$\\{\cdot|\cdot\\}$}$ $\displaystyle\Leftrightarrow
X_{\beta}(f(\mathbb{C}_{\theta}\otimes v_{\lambda}^{*}))\neq 0.$

For $\beta=0$, $\hat{\gamma}_{\theta}$ is clearly surjective.

So, let us now take $\beta$ to be an imaginary root $p\delta$, for $p>0$:

Recall from the isomorphism (16) that, under the identification (13),

$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)\simeq\\{h\otimes
t^{p}:h\in\mathfrak{h}\,\text{and}\,\alpha_{k}(h)=0\ \forall k\in S\\}.$

We next show that, under the restriction map

$V(\mu)^{\vee}\to(\mathfrak{g}\cdot
v_{\mu})^{\vee}\simeq(\mathfrak{u}\oplus(\operatorname{Ker}\mu)^{\perp})\otimes\mathbb{C}_{\mu}^{*},$

for any $h\in\mathfrak{h}$, the element $x_{h}:=(-\frac{h\otimes
t^{p}}{p\mu(K)})\cdot v_{\mu}^{*}$

(21) $x_{h}\mapsto(h\otimes t^{p})\otimes v_{\mu}^{*}\,.$

To see this, take any $h^{\prime}\in\mathfrak{h}$. Then,

$\displaystyle x_{h}(h^{\prime}\otimes t^{-p})\cdot v_{\mu})$
$\displaystyle=v_{\mu}^{*}\left(\left[\frac{h\otimes
t^{p}}{p\mu(K)},h^{\prime}\otimes t^{-p}\right]\cdot v_{\mu}\right)$
$\displaystyle=(h|h^{\prime}).$

Similarly,

$\displaystyle\left((h\otimes t^{p})\otimes
v_{\mu}^{*}\right)((h^{\prime}\otimes t^{-p})\cdot v_{\mu})$
$\displaystyle=(h\otimes t^{p}|h^{\prime}\otimes t^{-p})$
$\displaystyle=(h|h^{\prime}).$

This proves (21). Now, for any $h\in\mathfrak{h}$ such that $\alpha_{k}(h)=0$
for all $k\in S$, the element $x_{h}$ satisfies

(22) $e_{i}^{\theta(\alpha_{i}^{\vee})+1}\cdot x_{h}=0\,\forall
i,\,\,\,\text{and}\,\,f_{k}\cdot x_{h}=0\,\forall k\in S.$

To prove this, observe that (in $V(\mu)^{\vee}$)

$\displaystyle e_{i}^{\theta(\alpha_{i}^{\vee})+1}\cdot(h\otimes t^{p})\cdot
v_{\mu}^{*}$
$\displaystyle=\sum_{j=0}^{\theta(\alpha_{i}^{\vee})+1}\,\begin{pmatrix}\theta(\alpha_{i}^{\vee})+1\\\
j\end{pmatrix}\left((\operatorname{ad}e_{i})^{j}(h\otimes
t^{p})\right)\cdot(e_{i}^{\theta(\alpha_{i}^{\vee})+1-j}\cdot v_{\mu}^{*}),$
cf. the proof of [Ku3], Lemma 1.3.3 $\displaystyle=0,\,\,\,\text{since
$(\operatorname{ad}e_{i})^{2}(h\otimes t^{p})=0$ and
$e_{i}^{\mu(\alpha_{i}^{\vee})+1}\cdot v_{\mu}^{*}=0$},$

by the presentation of $V(\mu)^{\vee}$ as a $\mathfrak{b}$-module given in the
proof of Proposition 5.1. Further, for $k\in S$,

$\displaystyle f_{k}\cdot(h\otimes t^{p})\cdot v_{\mu}^{*}$
$\displaystyle=[f_{k},h\otimes t^{p}]\cdot v_{\mu}^{*}$
$\displaystyle=0,\,\,\,\text{since $\alpha_{k}(h)=0$ by assumption.}$

Thus, $v_{\theta}^{*}\mapsto x_{h}$ extends uniquely to a
$\mathfrak{p}$-module morphism
$V(\theta)^{\vee}\to\mathbb{C}_{\lambda}^{*}\otimes V(\mu)^{\vee}$. This
proves the surjectivity of the restriction map

$\gamma_{\theta}:\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V(\mu)^{\vee}\right)\to\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)$

for $\theta=\lambda+\mu-p\delta$.

Consider the commutative diagram for any $\theta\in\mathscr{P}^{+}$ of the
form $\theta=\lambda+\mu-\beta$, for $\beta\in\Phi^{+}(S)\cup\\{0\\}$ and
$w\in W_{\mathcal{P}}^{\prime}$ such that $X_{w}^{\mathcal{P}}$ is
$\mathcal{P}$-stable:

$\textstyle{\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V(\mu)^{\vee}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\theta}}$$\textstyle{\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}_{M}}$$\textstyle{\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes
V_{w}(\mu)^{*}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\theta}^{w}}$$\textstyle{\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}\right),}$

where the vertical maps are the canonical restriction maps and
$M:=\mathbb{C}_{\lambda}^{*}\otimes(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}$. By Lemma 12.3, $\hat{\pi}_{M}$ is surjective. Further, by the
above proof, $\gamma_{\theta}$ is surjective for all
$\theta=\lambda+\mu-\beta$ such that $\beta=0$ or an imaginary root. Moreover,
as proved above, for a real root $\beta\in\Phi^{+}(S)$ with $F_{\beta}\supset
S$, $\gamma_{\theta}$ is surjective if and only if there exists a
$\mathfrak{b}$-morphism $f_{\beta}:\mathbb{C}_{\theta}\otimes
V(\lambda)^{\vee}\to V(\mu)$ such that
$X_{\beta}(f_{\beta}(\mathbb{C}_{\theta}\otimes v_{\lambda}^{*}))\neq 0$.

By Lemma 12.2, for any real root $\beta\in\Phi^{+}(S)$ satisfying
$F_{\beta}\not\supset S$, we have
$\operatorname{Hom}_{\mathfrak{p}}\left(V(\theta)^{\vee},\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot
v_{\mu})^{\vee}\right)=0$. In particular, $\gamma_{\theta}^{w}$ is surjective
for $\beta=0$ or an imaginary root or a real root $\beta\in\Phi^{+}(S)$ such
that $F_{\beta}\not\supset S$ and it is surjective for a real root
$\beta\in\Phi^{+}(S)$ with $F_{\beta}\supset S$ if the condition (b) of the
proposition is satisfied for $\beta$. Thus, $\tilde{\tau}_{w}$ (equivalently
$\tau_{w}$) is surjective if the condition (b) of the proposition is
satisfied. This proves the proposition. ∎

######  Lemma 12.5.

Let $(\lambda,\mu,\beta)$ be a Wahl triple for a real root $\beta$ and let
$V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ be a $\delta$-maximal
root component. Observe that $\beta\in\mathring{\Phi}^{+}$ or
$\beta=\delta-\gamma$ for $\gamma\in\mathring{\Phi}^{+}$. Then,

  1. (1)

If $\beta\in\mathring{\Phi}^{+}$, the validity of condition (b) of Proposition
12.4 for $V(\lambda+\mu-\beta)$ (i.e., the validity of (19)) implies its
validity for $V(\lambda+\mu-\beta-k\delta)$ for any $k\geq 0$.

  2. (2)

If $\beta=\delta-\gamma$ for $\gamma\in\mathring{\Phi}^{+}$, then we have
(with notation as introduced below in the proof , see the identity (12))

$\displaystyle(l+m+h^{\vee})X_{-\gamma}(k+1)\cdot$ $\displaystyle
v_{\lambda}^{*}(L_{-k}w)=l(k+1)X_{-\gamma}(1)\cdot w_{\mu-\beta}$
$\displaystyle-$
$\displaystyle(\lambda_{|\mathring{\mathfrak{h}}},\gamma_{|\mathring{\mathfrak{h}}})X_{-\gamma}(1)\cdot
w_{\mu-\beta}+\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta_{i}+\gamma\in\mathring{\Phi}^{+}\end{subarray}}[X_{-\gamma}(1),X_{-\beta_{i}}]X_{\beta_{i}}\cdot
w_{\mu-\beta}.$

###### Proof.

Let $w\in V(\lambda+\mu-\beta)\subset V(\lambda)\otimes V(\mu)$ be the highest
weight vector. Then, $L_{-k}(w)$ is a highest weight vector in
$V(\lambda)\otimes V(\mu)$ with weight $\lambda+\mu-\beta-k\delta$ (cf. §4
together with Proposition 7.2). Let $l$, $m$ be the central charges of
$V(\lambda)$ and $V(\mu)$, respectively, and $h^{\vee}$ the dual Coxeter
number of $\mathfrak{[}g]$. Write

$w=v_{\lambda}\otimes w_{\mu-\beta}+\sum_{q\in
Q^{+}\backslash\\{0\\}}v^{\clubsuit}_{\lambda-q}\otimes
w^{\clubsuit}_{\mu-\beta+q},$

where $v^{\clubsuit}_{\lambda-q}$ is a basis of weight vectors of $V(\lambda)$
of weight $\lambda-q$, $w^{\clubsuit}_{\mu-\beta+q}\in V(\mu)_{\mu-\beta+q}$
(possibly zero), and $Q^{+}:=\bigoplus_{j=0}^{\ell}\mathbb{Z}_{\geq
0}\alpha_{j}$. Observe that $v^{\clubsuit}_{\lambda-q}$ occurs as many times
(possibly zero) as the dimension of the weight space $V(\lambda)_{\lambda-q}$.

Denote by $v^{\ast}_{\lambda}:V(\lambda)\otimes V(\mu)\to V(\mu)$ the
contraction via the highest weight vector $v_{\lambda}$. Then, by the proof of
Proposition 5.1, we have that $v^{\ast}_{\lambda}(w)$, for $w$ the highest
weight vector as above, satisfies the conditions
$e_{i}^{\lambda(\alpha_{i}^{\vee})+1}\cdot(v^{\ast}_{\lambda}(w))=0$ for all
simple roots $\alpha_{i}$.

By [KRR], Proposition 10.3, for any $k>0$, the GKO operator (cf. Proposition
4.5):

$(l+m+h^{\vee})L_{-k}(w)=m(L_{-k}^{\mathfrak{[}g]}\otimes
Id)(w)+lv_{\lambda}\otimes(L_{-k}^{\mathfrak{[}g]}w_{\mu-\beta})+l\sum_{q\neq
0}v^{\clubsuit}_{\lambda-q}\otimes(L_{-k}^{\mathfrak{[}g]}w^{\clubsuit}_{\mu-\beta+q})\\\
-\sum_{j\in\mathbb{Z},i}(u_{i}(-j)\cdot v_{\lambda})\otimes(u^{i}(j-k)\cdot
w_{\mu-\beta})-\sum_{j\in\mathbb{Z},\,i}\sum_{0\neq q\leq\beta}(u_{i}(-j)\cdot
v^{\clubsuit}_{\lambda-q})\otimes(u^{i}(j-k)\cdot
w^{\clubsuit}_{\mu-\beta+q}),$

where $\\{u_{i}\\}$ is a basis of $\mathring{\mathfrak{[}g]}$, $\\{u^{i}\\}$
is its dual basis, and $L_{-k}^{\mathfrak{[}g]}$ is the Virasoro operator of
$\mathfrak{[}g]$ (cf [KRR], Corollary 10.1). Now, take a root basis
$\\{X_{\beta_{i}}\\}$ of
$\mathring{\mathfrak{[}n]}^{+}:=\oplus_{\alpha\in\mathring{\Phi}^{+}}\,\mathring{\mathfrak{g}}_{\alpha}$
and the dual basis $\\{X_{-\beta_{i}}\\}$ of
$\mathring{\mathfrak{[}n]}^{-}:=\oplus_{\alpha\in\mathring{\Phi}^{+}}\,\mathring{\mathfrak{g}}_{-\alpha}$,
and a basis $\\{h_{j}\\}$ of $\mathring{\mathfrak{[}h]}$ and its dual basis
$\\{h^{j}\\}$, with respect to the normalized form. That is,
$(X_{\beta_{i}},X_{-\beta_{j}})=\delta_{ij}$, $(h_{i},h^{j})=\delta_{ij}$.
Then, we have

$\displaystyle(l+m+h^{\vee})v_{\lambda}^{\ast}(L_{-k}(w))=l\
L_{-k}^{\mathfrak{[}g]}(w_{\mu-\beta})$
$\displaystyle-\sum_{i}v_{\lambda}^{\ast}(u_{i}\cdot
v_{\lambda})(u^{i}(-k)\cdot w_{\mu-\beta})$ (23) $\displaystyle-\sum_{i,0\neq
q\leq\beta}v_{\lambda}^{\ast}(u_{i}\cdot
v^{\clubsuit}_{\lambda-q})(u^{i}(-k)\cdot w^{\clubsuit}_{\mu-\beta+q}).$

Now, we consider the two cases separately. First, assume
$\beta\in\mathring{\Phi}^{+}$. Then, for $X_{\beta}(k):=X_{\beta}\otimes
t^{k}$, we have by the equation (12):

$\displaystyle(l+m+h^{\vee})X_{\beta}(k)\cdot
v_{\lambda}^{\ast}(L_{-k}(w))=lkX_{\beta}\cdot
w_{\mu-\beta}+\sum_{j}\lambda(h_{j})\beta(h^{j})X_{\beta}\cdot w_{\mu-\beta}$
$\displaystyle-\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta-\beta_{i}\in\mathring{\Phi}^{+}\end{subarray}}v_{\lambda}^{\ast}(X_{\beta_{i}}\cdot
v^{\clubsuit}_{\lambda-\beta_{i}})[X_{\beta},X_{-\beta_{i}}]\cdot
w^{\clubsuit}_{\mu-\beta+\beta_{i}}-v_{\lambda}^{\ast}(X_{\beta}\cdot
v^{\clubsuit}_{\lambda-\beta})((\mu,\beta)+km)w^{\clubsuit}_{\mu},$ by [KRR],
Corollary 10.1, Identity 10.13(b) $\displaystyle=lkX_{\beta}\cdot
w_{\mu-\beta}+(\lambda_{|\mathring{\mathfrak{[}h]}},\beta_{|\mathring{\mathfrak{[}h]}})X_{\beta}\cdot
w_{\mu-\beta}-\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta-\beta_{i}\in\mathring{\Phi}^{+}\end{subarray}}v^{\ast}_{\lambda}(X_{\beta_{i}}\cdot
v^{\clubsuit}_{\lambda-{\beta_{i}}})[X_{\beta},X_{-\beta_{i}}]\cdot
w^{\clubsuit}_{\mu-\beta+\beta_{i}}$ (24)
$\displaystyle-v^{\ast}_{\lambda}(X_{\beta}\cdot
v^{\clubsuit}_{\lambda-\beta})((\mu,\beta)+km)w^{\clubsuit}_{\mu}.$

For any root $\beta_{i}\in\mathring{\Phi}^{+}$, choose a basis of
$V(\lambda)_{\lambda-\beta_{i}}$ consisting of $X_{-\beta_{i}}.v_{\lambda}$
(which we will denote simply by $v^{\circ}_{\lambda-\beta_{i}}$) and any basis
$v^{\clubsuit^{\prime}}_{\lambda-\beta_{i}}$ of
$V(\lambda)_{\lambda-\beta_{i}}$ annihilated by $X_{\beta_{i}}$. With such a
basis, the above equation (12) becomes:

(25) $(l+m+h^{\vee})X_{\beta}(k)\cdot
v^{\ast}_{\lambda}(L_{-k}(w))=lkX_{\beta}\cdot
w_{\mu-\beta}+(\lambda_{|\mathring{\mathfrak{[}h]}},\beta_{|\mathring{\mathfrak{[}h]}})X_{\beta}\cdot
w_{\mu-\beta}\\\ -\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta-\beta_{i}\in\mathring{\Phi}^{+}\end{subarray}}v^{\ast}_{\lambda}(X_{\beta_{i}}X_{-\beta_{i}}\cdot
v_{\lambda})[X_{\beta},X_{-\beta_{i}}]\cdot
w^{\circ}_{\mu-\beta+\beta_{i}}-v^{\ast}_{\lambda}(X_{\beta}X_{-\beta}\cdot
v_{\lambda})((\mu,\beta)+km)w^{\circ}_{\mu}.$

Since $X_{\beta_{i}}$ annihilates $w$, we get

(26) $-X_{\beta_{i}}\cdot
w_{\mu-\beta}=(\lambda,\beta_{i})w^{\circ}_{\mu-\beta+\beta_{i}}.$

Combining the equations (25) and (26), we get taking $X_{\beta}\cdot
w_{\mu-\beta}=w_{\mu}$ (which is possible by the assumption that condition (b)
of Proposition 12.4 is valid for the $\delta$-maximal component
$V(\lambda+\mu-\beta)$):

(27) $(l+m+h^{\vee})X_{\beta}(k)\cdot
v^{\ast}_{\lambda}(L_{-k}(w))=lkw_{\mu}+(\lambda_{|\mathring{\mathfrak{[}h]}},\beta_{|\mathring{\mathfrak{[}h]}})w_{\mu}\\\
+\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta-\beta_{i}\in\mathring{\Phi}^{+}\end{subarray}}[X_{\beta},X_{-\beta_{i}}]X_{\beta_{i}}\cdot
w_{\mu-\beta}+((\mu,\beta)+km)w_{\mu}.$

We next claim that for any $\beta_{i}\in\mathring{\Phi}^{+}$ such that
$\beta-\beta_{i}\in\mathring{\Phi}^{+}$,

(28)
$[X_{\beta},X_{-\beta_{i}}]X_{\beta_{i}}+[X_{\beta},X_{-(\beta-\beta_{i})}]X_{\beta-\beta_{i}}=d_{\beta_{i}}X_{\beta},\,\,\text{for
some $d_{\beta_{i}}\in\mathbb{Z}_{>0}$.}$

To prove this, write

(29) $[X_{\beta},X_{-\beta_{i}}]=cX_{\beta-\beta_{i}},\,\,\text{for some
$c\neq 0$. }$

Then,

$\displaystyle([X_{\beta},X_{-(\beta-\beta_{i})}],X_{-\beta_{i}})$
$\displaystyle=-c(X_{-(\beta-\beta_{i})},X_{\beta-\beta_{i}}),\ \text{by
\eqref{eqn12.5.4}}$ $\displaystyle=-c.$

Thus,

(30) $[X_{\beta},X_{-(\beta-\beta_{i})}]=-cX_{\beta_{i}}.$

Combining (29) and (30), we get

$\displaystyle\hskip
5.0pt[X_{\beta},X_{-\beta_{i}}]X_{\beta_{i}}+[X_{\beta},X_{-(\beta-\beta_{i})}]X_{\beta-\beta_{i}}$
$\displaystyle=c[X_{\beta-\beta_{i}},X_{\beta_{i}}]$
$\displaystyle=[[X_{\beta},X_{-\beta_{i}}],X_{\beta_{i}}],\ \text{by
\eqref{eqn12.5.4}}$ $\displaystyle=d_{\beta_{i}}X_{\beta},\,\,\text{for some
$d_{\beta_{i}}\in\mathbb{Z}_{>0}$ by $sl_{2}$-module theory.}$

This proves (28). Substituting (28) into (27), we get

(31) $(l+m+h^{\vee})X_{\beta}(k)\cdot
v^{\ast}_{\lambda}(L_{-k}(w))=(lk+(\lambda_{|\mathring{\mathfrak{[}h]}},\beta_{|\mathring{\mathfrak{[}h]}})+(\mu,\beta)+km+\frac{1}{2}\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta-\beta_{i}\in\mathring{\Phi}^{+}\end{subarray}}d_{\beta_{i}})w_{\mu}\neq
0,$

since $\beta\in\mathring{\Phi}^{+}$. This proves the first part of the lemma.

Now, consider the case where $\beta=\delta-\gamma$ for some
$\gamma\in\mathring{\Phi}^{+}$. Recall that $\theta$ is the highest root of
$\mathring{\mathfrak{[}g]}$. By equation (12),

$\displaystyle(l+m+h^{\vee})X_{-\gamma}(k+1)\cdot
v^{\ast}_{\lambda}(L_{-k}(w))$ $\displaystyle=l(k+1)X_{-\gamma}(1)\cdot
w_{\mu-\beta}-\sum_{j}\lambda(h_{j})\gamma(h^{j})X_{-\gamma}(1)\cdot
w_{\mu-\beta}$
$\displaystyle-\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta_{i}+\gamma\in\mathring{\Phi}^{+}\end{subarray}}v^{\ast}_{\lambda}(X_{\beta_{i}}\cdot
v^{\clubsuit}_{\lambda-\beta_{i}})[X_{-\gamma},X_{-\beta_{i}}](1)\cdot
w^{\clubsuit}_{\mu-\beta+\beta_{i}}$ $\displaystyle=$ $\displaystyle
l(k+1)X_{-\gamma}(1)\cdot
w_{\mu-\beta}-(\lambda_{|\mathring{\mathfrak{[}h]}},\gamma_{|\mathring{\mathfrak{[}h]}})X_{-\gamma}(1)\cdot
w_{\mu-\beta}$
$\displaystyle-\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta_{i}+\gamma\in\mathring{\Phi}^{+}\end{subarray}}v^{\ast}_{\lambda}(X_{\beta_{i}}X_{-\beta_{i}}\cdot
v_{\lambda})[X_{-\gamma},X_{-\beta_{i}}](1)\cdot
w^{\circ}_{\mu-\beta+\beta_{i}}$ $\displaystyle=$ $\displaystyle
l(k+1)X_{-\gamma}(1)\cdot
w_{\mu-\beta}-(\lambda_{|\mathring{\mathfrak{[}h]}},\gamma_{|\mathring{\mathfrak{[}h]}})X_{-\gamma}(1)\cdot
w_{\mu-\beta}$
$\displaystyle+\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta_{i}+\gamma\in\mathring{\Phi}^{+}\end{subarray}}[X_{-\gamma},X_{-\beta_{i}}](1)X_{\beta_{i}}\cdot
w_{\mu-\beta},\,\,\text{using equation \eqref{eqn12.5.3}.}$

Finally, we get

$\displaystyle(l+m+h^{\vee})$ $\displaystyle X_{-\gamma}(k+1)\cdot
v_{\lambda}^{*}(L_{-k}w)=l(k+1)X_{-\gamma}(1)\cdot w_{\mu-\beta}$ (32)
$\displaystyle-(\lambda_{|\mathring{\mathfrak{h}}},\gamma_{|\mathring{\mathfrak{h}}})X_{-\gamma}(1)\cdot
w_{\mu-\beta}+\sum_{\begin{subarray}{c}\beta_{i}\in\mathring{\Phi}^{+}:\\\
\beta_{i}+\gamma\in\mathring{\Phi}^{+}\end{subarray}}[X_{-\gamma}(1),X_{-\beta_{i}}]X_{\beta_{i}}\cdot
w_{\mu-\beta}.$

∎

######  Proposition 12.6.

Let $(\lambda,\mu,\beta)$ be a Wahl triple for a real root $\beta$. Then, the
condition (b) of Proposition 12.4 (specifically, the identity (19)) is
satisfied for some embedding $V(\lambda+\mu-\beta)\subset V(\lambda)\otimes
V(\mu)$ in the following cases:

(a) $\mathring{\mathfrak{g}}$ is arbitrary but $\beta$ is of the form
$k\delta+\gamma$, for any $\gamma\in\mathring{\Phi}^{+}$ and $k\geq 0$.

(b) $\mathring{\mathfrak{g}}$ is arbitrary but $\beta$ is of the form
$(k+1)\delta-\theta$, for any $k\geq 0$, where $\theta$ is the highest root of
$\mathring{\mathfrak{g}}$.

(c) $\mathring{\mathfrak{g}}$ is simply-laced.

(d) $\mathring{\mathfrak{g}}$ is of type $B_{\ell}\,(\ell\geq 2)$.

(e) $\mathring{\mathfrak{g}}$ is of type $C_{\ell}\,(\ell\geq 2)$.

(f) $\mathring{\mathfrak{g}}$ is not of type $G_{2}$ and $\lambda$ is regular
dominant.

###### Proof.

(a) By [Ku1], $\S$2.7, the proposition is true for $\beta=\gamma$. Thus, (a)
follows follows from Lemma 12.5.

(b) In this case $\beta=\alpha_{0}+k\delta$ for $k\geq 0$. For
$\beta=\alpha_{0}$, the proposition follows from the proof of Lemma 9.1. Now,
to prove the proposition for $\beta=\alpha_{0}+k\delta$, by the identity (12)
of the proof of Lemma 12.5, we get that the right side of the identity (12)
equals $(lk+(\lambda,\alpha_{0}^{\vee}))w_{\mu}\neq 0$. This proves (b).

(c) Using (a) and (b), it suffices to prove the proposition for
$\beta=(k+1)\delta-\gamma$, for $k\geq 0$ and
$\gamma\in\mathring{\Phi}^{+}\setminus\\{\theta\\}.$ In this case the
proposition follows from the proof of Proposition 8.3.

(d) As in (c), it suffices to prove the proposition for
$\beta=(k+1)\delta-\gamma$, for $k\geq 0$ and
$\gamma\in\mathring{\Phi}^{+}\setminus\\{\theta\\}.$ Observe first that if
$\beta-2\alpha_{i}\notin\Phi^{+}$ for any simple root $\alpha_{i}\,(0\leq
i\leq\ell)$, then by the proof of Proposition 8.3, the proposition follows.
So, assume that $\beta-2\alpha_{i_{o}}\in\Phi^{+}$ for some (and hence unique
by Lemma 8.1) simple root $\alpha_{i_{o}}$. If $i_{o}\neq 0$, then by $\S$9.1
for $B_{\ell}^{(1)},F_{\beta}=F_{\beta-\alpha_{i_{o}}}$. (Observe that
$F_{\beta}=F_{\beta+k\delta}$ for any $k\in\mathbb{Z}$.) Hence, the
proposition in this case follows from the proof of Proposition 8.5. So, assume
now that $i_{o}=0$. In this case, by the proof of Lemma 9.1,
$\beta=(k+1)\delta-\theta$, which is already covered by (b). This completes
the proof of the proposition in the case (d).

(e) As in (d), it suffices to prove the proposition for
$\beta=(k+1)\delta-\gamma$, for $k\geq 0$ and
$\gamma\in\mathring{\Phi}^{+}\setminus\\{\theta\\}$ such that
$\beta-2\alpha_{j}\in\Phi^{+}$ for some (and hence unique) simple root
$\alpha_{j}\,(1\leq j\leq\ell)$. Thus, $\beta=(k+1)\delta-\gamma_{i},\,k\geq
0$, where $\gamma=\gamma_{i}\,(2\leq i\leq\ell)$ is as in Example of
$C_{\ell}^{(1)}$ in Section 9. Further, using Lemma 8.7, we can assume that
$\lambda=\mu=\rho_{\beta}=\Lambda_{i-1}$ (cf. Example of $C_{\ell}^{(1)}$ in
Section 9). (Observe that if the proposition is valid for the Wahl triple
$(\lambda,\mu,\beta)$, then the proposition is valid for
$(\lambda+\lambda^{\prime},\mu+\mu^{\prime},\beta)$ for any dominant weights
$\lambda^{\prime},\mu^{\prime}\in\mathscr{P}^{+}$ by the proof of Corollary
5.2.)

For $\beta=\delta-\gamma_{i}$, the proposition follows from the proof of
Proposition 8.3 since $\Lambda_{i-1}(\beta^{\vee})=1.$ We now prove the
proposition for $\beta=(k+1)\delta-\gamma_{i}$ for $k\geq 1$ (and
$\lambda=\mu=\Lambda_{i-1}$). We use the identity (12) of the proof of Lemma
12.5 to show that in this case $X_{-\gamma_{i}}(k+1)\cdot
v_{\lambda}^{*}(L_{-k}w)\neq 0:$

By the proof of Proposition 8.3, we can take
$w_{\Lambda_{i-1}-(\delta-\gamma_{i})}=X_{\gamma_{i}}(-1)\cdot
w_{\Lambda_{i-1}}$. We freely use the notation from [Bou], Planche III. Then,
for any $\beta_{j}\in\mathring{\Phi}^{+}$ such that
$\beta_{j}+\gamma_{i}\in\mathring{\Phi}^{+}$, we get that $\beta_{j}$ is
precisely of the form $\epsilon_{j}-\epsilon_{i}$ for $1\leq j<i$ (and
$\gamma_{i}=2\epsilon_{i}$). Thus,

$\displaystyle[X_{-\gamma_{i}}(1),X_{-\beta_{j}}]X_{\beta_{j}}X_{\gamma_{i}}(-1)\cdot
w_{\Lambda_{i-1}}$
$\displaystyle=\left[[X_{-\gamma_{i}}(1),X_{-\beta_{j}}],[X_{\beta_{j}},X_{\gamma_{i}}(-1)]\right]\cdot
w_{\Lambda_{i-1}}$
$\displaystyle=\left([X_{-\gamma_{i}}(1),X_{-\beta_{j}}],[X_{\beta_{j}},X_{\gamma_{i}}(-1)]\right)(\Lambda_{i-1},\delta-\gamma_{i}-\beta_{j})w_{\Lambda_{i-1}},$
by [Ku3], Theorem 1.5.4
$\displaystyle=\left(X_{-\gamma_{i}}(1),[X_{-\beta_{j}},[X_{\beta_{j}},X_{\gamma_{i}}(-1)]]\right)(\Lambda_{i-1},\delta-\gamma_{i}-\beta_{j})w_{\Lambda_{i-1}}$
$\displaystyle=\left(X_{-\gamma_{i}}(1),2(\gamma_{i}(\beta_{j}^{\vee})+3)X_{\gamma_{i}}(-1)\right)(\Lambda_{i-1},\delta-\gamma_{i}-\beta_{j})w_{\Lambda_{i-1}},$
since
$(\operatorname{ad}X_{-\beta_{j}})X_{\gamma_{i}}=(\operatorname{ad}X_{\beta_{j}})^{3}X_{\gamma_{i}}=0$
and $(\operatorname{ad}X_{\beta_{j}})^{2}X_{\gamma_{i}}\neq 0$ (33)
$\displaystyle=w_{\Lambda_{i-1}},\,\,\,\text{since
$(\Lambda_{i-1},\delta-\gamma_{i}-\beta_{j})=1/2$ and
$\gamma_{i}(\beta_{j}^{\vee})=2$}.$

Hence,

$\sum_{1\leq
j<i}\,[X_{-\gamma_{i}}(1),X_{-\beta_{j}}]X_{\beta_{j}}X_{\gamma_{i}}(-1)\cdot
w_{\Lambda_{i-1}}=(i-1)w_{\Lambda_{i-1}}.$

Further,

$X_{-\gamma_{i}}(1)X_{\gamma_{i}}(-1)\cdot
w_{\Lambda_{i-1}}=[X_{-\gamma_{i}}(1),X_{\gamma_{i}}(-1)]\cdot
w_{\Lambda_{i-1}}=w_{\Lambda_{i-1}},\,\,\text{since
$l=m:=\Lambda_{i-1}(K)=1$}.$

So, the right side of the identity (12) of the proof of Lemma 12.5 becomes

$\left((k+1)+i-1\right)w_{\Lambda_{i-1}}=(k+i)w_{\Lambda_{i-1}}\neq 0.$

This proves the proposition in this case.

(f) Proof of the proposition in this case follows from the proofs of
Propositions 8.3 and 8.5.

∎

Following is the main theorem of this section:

######  Theorem 12.7.

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra and let $w\in
W_{\mathcal{P}}^{\prime}$ be such that the Schubert variety
$X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable. Then, for any
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$ (where $S$ is an arbitrary subset of
the simple roots) such that the condition (b) of Proposition 12.4 is satisfied
for all the Wahl triples $(\lambda,\mu,\beta)$ for any real root
$\beta\in\Phi^{+}$ (cf. Proposition 12.6),

(34)
$H^{p}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,\,\,\,\text{for
all $p>0$}.$

In particular, the canonical Gaussian map

(35)
$H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\to
H^{0}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},(\hat{\mathscr{I}}_{e}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))$

is surjective.

In particular, (34) and (35) are valid for any simply-laced
$\mathring{\mathfrak{g}}$ and $\mathring{\mathfrak{g}}$ of types
$B_{\ell},C_{\ell}$. Moreover, they also are valid for
$\mathring{\mathfrak{g}}$ of type $F_{4}$ in the case $\mathcal{P}$ is the
Borel subgroup $\mathcal{B}$.

###### Proof.

By Proposition 12.4, we get the vanishing
$H^{1}(\hat{\mathcal{X}}_{w}^{\mathcal{P}},\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))=0.$

To prove the higher cohomology vanishing, consider the cohomology exact
sequence, associated to the sheaf exact sequence over
$\hat{\mathcal{X}}_{W}^{\mathcal{P}}$:

(36)
$0\to\hat{\mathscr{I}}_{e}^{2}\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to\mathscr{L}_{w}(\lambda\boxtimes\mu)\to(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu)\to
0.$

First of all by [Ku2], Theorem 2.7 (also see [M2], Corollaire 3),

(37)
$H^{p}(\hat{\mathcal{X}}^{\mathcal{P}}_{w},\mathscr{L}_{w}(\lambda\boxtimes\mu))=0,\,\,\,\text{for
all $p>0$}$

Further, by Lemma 12.1 and Proposition 11.1,

(38)
$H^{p}(\hat{\mathcal{X}}^{\mathcal{P}}_{w},(\hat{\mathscr{O}}_{w}/\hat{\mathscr{I}}_{e}^{2})\otimes\mathscr{L}_{w}(\lambda\boxtimes\mu))\simeq
H^{p}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}((\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*})).$

Now, we prove the vanishing:

(39)
$H^{p}({\mathcal{X}}_{\mathcal{P}},\mathscr{L}(\lambda)\otimes\mathscr{L}((\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}))=0,\,\,\,\text{for all $p>0$}:$

Take a $\mathcal{P}$-module filtration of the finite dimensional
$\mathcal{P}$-module
$M_{w}:=\mathbb{C}_{\lambda}^{*}\otimes(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}$:

$0=F_{0}\subset F_{1}\subset\cdots\subset F_{d}=M_{w},$

such that $V_{j}:=F_{j}/F_{j-1}$ is an irreducible $\mathcal{P}$-module for
each $1\leq j\leq d$. By Engel’s theorem, $V_{j}$ is a trivial
$\mathcal{U}$-module, thus it can be considered as an irreducible module for
the (finite dimensional) Levi algebra
$\mathfrak{l}:=\mathfrak{p}/\mathfrak{u}$. Let $\delta_{j}$ be the lowest
weight of $V_{j}$ as an $\mathfrak{l}$-module. Thus,

(40) $\delta_{j}(\alpha_{k}^{\vee})\leq 0,\,\,\,\text{for all $k\in S$}.$

To prove the vanishing (39), it suffices to show that

(41) $H^{p}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(V_{j}))=0,\,\,\,\text{for
all $p>0$}.$

From the classical Borel-Weil-Bott theorem for the finite dimensional
reductive Lie algebra $\mathfrak{l}$ and the Leray spectral sequence for the
fibration $\mathcal{G}/\mathcal{B}\to\mathcal{G}/\mathcal{P}$, we get

(42) $H^{p}(\mathcal{X}_{\mathcal{P}},\mathscr{L}(V_{j}))\simeq
H^{p}(\mathcal{X}_{\mathcal{B}},\mathscr{L}(-\delta_{j})),\,\,\,\text{for all
$p\geq 0$}.$

Observe further that $\delta_{j}$, in particular, is a weight of
$\mathbb{C}_{\lambda}^{*}\otimes(\hat{T}_{e}(X_{w}^{\mathcal{P}})\cdot
v_{\mu})^{*}$ which is a quotient of
$\mathbb{C}_{\lambda}^{*}\otimes(\mathfrak{g}\cdot v_{\mu})^{\vee}$. Thus,
$\delta_{j}$ is of the form $\delta_{j}=-(\lambda+\mu-\beta)$, for some
$\beta\in\Phi^{+}(S)\cup\\{0\\}$. Hence, for $i\notin S$, since
$\lambda,\mu\in{\mathscr{P}^{+}_{S}}^{o}$, by [Bou], Page 278, item 6,

(43) $-\delta_{j}(\alpha_{i}^{\vee})\geq-1.$

Thus, combining the equations (40), (43) and (42), the vanishing (41) follows
from the affine analogue of the Borel-Weil-Bott theorem (cf. [Ku3], Corollary
8.3.12). This proves the vanishing (39).

Thus, the cohomology exact sequence associated to the sheaf exact sequence
(36) and the above two vanishing results (37) and (39) (using the
identification (38)) prove (34) and (35) of the theorem.

The last ‘In particular’ statement of the theorem follows from Proposition
12.6. ∎

For any $w\in W_{\mathcal{P}}^{\prime}$ such that $X_{w}^{\mathcal{P}}$ is
$\mathcal{P}$-stable, we think of $\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ as a
sub ind-variety of $\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$
via the isomorphism
$\delta:\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}\to\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$
(as in the beginning of Section 12). From now on, we think of the sheaf
$\hat{\mathscr{I}}_{e}^{k}=\hat{\mathscr{I}}_{e}(w)^{k}$ over
$\hat{\mathcal{X}}_{w}^{\mathcal{P}}$ (for any $k\geq 1$) as a sheaf over
$\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$ by considering
$\delta_{*}(i_{w})_{*}(\hat{\mathscr{I}}_{e}(w)^{k})$, where
$i_{w}:\mathcal{G}\times^{\mathcal{P}}X_{w}^{\mathcal{P}}\to\mathcal{G}\times^{\mathcal{P}}\mathcal{X}_{\mathcal{P}}$
is the canonical embedding. Let $\mathscr{I}_{D}$ be the ideal sheaf of the
diagonal $D\subset\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}}$.
Then, from the sheaf exact sequence:

$0\to\hat{\mathscr{I}}_{e}(w)\to\hat{\mathscr{O}}_{w}\to\hat{\mathscr{O}}_{e}\to
0,$

we get that

$\mathscr{I}_{D}\simeq\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w).$

Define

$\tilde{\mathscr{I}}_{D}^{2}:=\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w)^{2}.$

Observe that $\mathscr{I}_{D}^{2}\subset\tilde{\mathscr{I}}_{D}^{2}$ and the
inclusion is strict in general.

As a corollary of the above theorem, we get the following. This was
conjectured in the finite case by Wahl [Wah] and proved in that case by Kumar
[Ku1].

######  Corollary 12.8.

Under the notation and assumptions of Theorem 12.7, the canonical Gaussian map

$H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\mathscr{I}_{D}\otimes\mathscr{L}(\lambda\boxtimes\mu))\to
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\mathscr{I}_{D}/\tilde{\mathscr{I}}_{D}^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))$

is surjective.

In particular, it is surjective for any simply-laced $\mathring{\mathfrak{g}}$
as well as $\mathring{\mathfrak{g}}$ of types $B_{\ell},C_{\ell}$. In
addition, it is surjective for $\mathring{\mathfrak{g}}$ of type $F_{4}$ in
the case $\mathcal{P}$ is the Borel subgroup $\mathcal{B}$.

###### Proof.

By Theorem 12.7, for any $w\in W_{\mathcal{P}}^{\prime}$ such that
$X_{w}^{\mathcal{P}}$ is $\mathcal{P}$-stable, the canonical map

$\eta_{w}:H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\hat{\mathscr{I}}_{e}(w)\otimes\mathscr{L}(\lambda\boxtimes\mu))\to
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))$

is surjective. Moreover, $\eta_{w}$ is a $\mathcal{G}$-module morphism between
$\mathcal{G}$-modules which are finite direct sums of $V(\theta)^{*}$ for some
$\theta\in\mathscr{P}^{+}$ (cf. proof of Proposition 12.4). Let $K_{w}$ be the
kernel of $\eta_{w}$. Then, $K_{w}$ being a finite direct sum of
$V(\theta)^{*}$, it is easy to see that $\\{K_{w}\\}_{w}$ satisfies the
Mittag-Leffler condition (cf. [H2], Chap. II, $\S$9). Thus, by [H2], Chap. II,
Proposition 9.1, the canonical map

(44)
$\varprojlim_{w}\,H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\hat{\mathscr{I}}_{e}(w)\otimes\mathscr{L}(\lambda\boxtimes\mu))\to\varprojlim_{w}\
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))$

is surjective. Further, by [H2], Chap. II, Proposition 9.2,

(45)
$\varprojlim_{w}\,H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\hat{\mathscr{I}}_{e}(w)\otimes\mathscr{L}(\lambda\boxtimes\mu))\simeq
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\mathscr{I}_{D}\otimes\mathscr{L}(\lambda\boxtimes\mu))$

and

(46) $\varprojlim_{w}\
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))\simeq
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu)).$

From the left exactness of $\varprojlim$ in the category of sheaves,

(47)
$j:\mathscr{I}_{D}/\tilde{\mathscr{I}}_{D}^{2}\hookrightarrow\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2}.$

Hence, by combining the equations (44)-(47), we get that the composite

$\displaystyle
H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\mathscr{I}_{D}\otimes$
$\displaystyle\mathscr{L}(\lambda\boxtimes\mu))\overset{\eta}{\to}H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\mathscr{I}_{D}/\tilde{\mathscr{I}}_{D}^{2}\otimes\mathscr{L}(\lambda\boxtimes\mu))$
$\displaystyle\overset{j_{*}}{\hookrightarrow}H^{0}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},(\varprojlim_{w}\,\hat{\mathscr{I}}_{e}(w)/\hat{\mathscr{I}}_{e}(w)^{2})\otimes\mathscr{L}(\lambda\boxtimes\mu))$

is surjective and hence so is $\eta$ (and $j_{*}$ is an isomorphism). This
proves the corollary. ∎

######  Remark 12.9.

(a) It is very likely that, under the assumptions of Corollary 12.8,
$H^{p}(\mathcal{X}_{\mathcal{P}}\times\mathcal{X}_{\mathcal{P}},\tilde{\mathscr{I}}_{D}^{2}\otimes\mathscr{L}(\lambda\boxtimes\mu))=0$