# Simplified presentations and embeddings of Demazure modules

Deniz Kus  University of Bochum, Faculty of Mathematics, Universitätsstr. 150,
44801 Bochum, Germany<EMAIL_ADDRESS>and  R. Venkatesh  Department of
Mathematics, Indian Institute of Science, Bangalore 560012<EMAIL_ADDRESS>
###### Abstract.

For an untwisted affine Lie algebra we prove an embedding of any higher level
Demazure module into a tensor product of lower level Demazure modules (e.g.
level one in type A) which becomes in the limit (for anti-dominant weights)
the well-known embedding of finite-dimensional irreducible modules of the
underlying simple Lie algebra into the tensor product of fundamental modules.
To achieve this goal, we first simplify the presentation of these modules
extending the results of [13] in the $\mathfrak{g}$-stable case. As an
application, we propose a crystal theoretic way to find classical
decompositions with respect to a maximal semi-simple Lie subalgebra by
identifying the Demazure crystal as a connected component in the corresponding
tensor product of crystals.

D.K. was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation)– grant 446246717. R.V. was partially funded by the grants
DST/INSPIRE/04/2016/000848, MTR/2017/000347, and an Infosys Young Investigator
Award.

##  1\. Introduction

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra with
corresponding untwisted affine Lie algebra $\widehat{\mathfrak{g}}$. Affine
Demazure modules are a family of finite-dimensional representations that are
characterized by an element in the affine Weyl group and a dominant integral
weight. They have been the subject of intense study with lots of applications
and conjectures (see for example [13, 18, 23, 29, 32, 36, 39] and the
references therein). Although there are character formulas in terms of
Demazure operators [29], it seems very complicated to give closed dimension
formulas let alone closed character formulas which are comparable to the Kac-
Weyl character formula.

The first connection to Macdonald polynomials was observed in [39] and later
extended to the simply-laced case in [23], where the level one Demazure
characters are identified with certain specializations of Macdonald
polynomials. For non simply-laced types a similar approach connects the
specialized Macdonald polynomials with (generalized) Weyl modules [9, 15]. In
the higher level case character or dimension formulas for Demazure modules are
generically not known and this problem has been partially solved for
$\mathfrak{sl}_{n+1}$-stable Demazure modules of level two in [1, 4]. We
emphasize at this point that the most critical ones are the prime Demazure
modules which do not admit a non-trivial splitting into a tensor product.

The article [1] uses the dual functional realization of the loop algebra and
the fact that level two Demazure modules for $\mathfrak{sl}_{n+1}$ appear in a
completely different context. They are the graded limits of an important
family of modules for the quantum affine algebra and the classical
decomposition is obtained via lattice points in convex polytopes. The article
[4] uses a different approach which was first introduced and developed in
[25], namely the theory of Demazure flags. This approach provides a further
deep and unexpected connection to combinatorics and number theory; see for
example [5] for the connection to the combinatorics of Dyck path or [2] and
[3] for the connection to Mock-Theta functions and hypergeometric series.

Another approach to the problem of finding dimension formulas, character
formulas or classical decompositions is given by crystal theory. It is known
that the Demazure module $V_{w}(\Lambda)$ admits a crystal base [27] which is
the full subgraph of the crystal base of the irreducible integrable highest
weight module whose vertices consist precisely of those elements that are
reachable by raising operators from the unique element of weight $w\Lambda$.
There is a connection to the tensor product of Kirillov-Reshetikhin crystals
which has been worked out by various authors in the last years (see for
example [33, 37, 40] and the references therein). The main result of [33],
which seems to be the most general version so far, states that the
$\mathfrak{g}$-stable level $k$ Demazure module for non-exceptional types
appear inside a tensor product of Kirillov-Reshetikhin crystals by removing
all edges that are not level $k$ Demazure edges (see [33, Section 2] for a
precise definition) and picking the connected component which contains the
unique element of extremal weight $w\Lambda$. Finding very explicit
combinatorial models for these crystals is still a very important and open
problem.

Most of the connections described above benefit enormously from the
presentation of Demazure modules proved by Mathieu in [34]. However the
relations are sometimes very hard to check. After simplifying the defining
relations of $\mathfrak{g}$-stable Demazure modules in [13] many new
isomorphisms and maps were proved afterwards. By way of example, in the same
article, the authors proved that the Q-system introduced by Kirillov and
Reshetikhin [28] extends to a canonical short exact sequence of fusion
products. In the study of global Demazure modules and arc schemes [14] the
authors used the simplified presentation to show that rectangular
$\mathfrak{g}$-stable Demazure modules can be obtained as localizations of
global Demazure modules at zero. The same simplification was used in [31] to
show that the fusion product of finite-dimensional irreducible
$\mathfrak{g}$-modules of the same highest weight is isomorphic to the
truncated Weyl module or in [6] to identify level two $\mathfrak{g}$-stable
Demazure modules with graded limits of important families of modules for the
quantum affine algebra.

The crystal theoretic approach to Demazure modules and the realization inside
a tensor product obtained in [33] is one motivation of the article to embed
any higher level Demazure crystal (no restrictions on the type nor on the
extremal weight) into a tensor product of Demazure crystals which are in some
way easier to understand. We attack this problem algebraically which we will
explain now in more details.

For an element $w$ in the affine Weyl group and a dominant integral weight
$\Lambda$ let $V_{w}(\Lambda)=\mathbf{D}_{\mu}^{k}$ the affine Demazure module
where $w\Lambda=\mu+k\Lambda_{0}$ (modulo the null root). For such an element
$\mu\in P$ we define the set

$\mathcal{P}(\mu,k)=\\{(\mu_{1},\dots,\mu_{k})\in P^{\times
k}:\mu_{1}+\dots+\mu_{k}=\mu\\}$

and introduce the notion of $r$-admissible elements in Section 4. We show that
for an $r$-admissible element $(\mu_{1},\dots,\mu_{k})$ (sometimes the unique
element described in [16] is the correct choice) we have an embedding of
graded modules

$\mathbf{D}_{\mu}^{rk}\hookrightarrow\mathbf{D}_{\mu_{1}}^{r}\otimes\cdots\otimes\mathbf{D}_{\mu_{k}}^{r}.$

It is clear that this embedding becomes in the limit $k\rightarrow\infty$ for
anti-dominant weights the well-known embedding of finite-dimensional
irreducible modules of the underlying simple Lie algebra into the tensor
product of fundamental modules. To achieve this embedding we first need to
simplify the presentation of Demazure modules for any weight $\mu$ (see
Section 4.1) extending the $\mathfrak{g}$-stable case from [13]. We believe
that this fact is of independent interest and finds several more applications
in the future.

Finally we propose, similar to the approach taken in [33], a way to come up
with classical decompositions. For any weight $\mu\in P$ we define a subset
$R^{-}(\mu)$ (see Section 2) and if this set is non-empty we can find a
maximal semi-simple subalgebra $\mathfrak{g}_{0}\subseteq\mathfrak{g}$
corresponding to the nodes in $R^{-}(\mu)$; the $\mathfrak{g}$-stable case
corresponds to $R^{+}=R^{-}(\mu)$. The $\mathfrak{g}_{0}$ decompostion of the
Demazure module “seems” (we use this wording because it is not made precise in
this article) to be encoded in the connected component containing the unique
element of weight $w\Lambda$ inside the tensor product (according to the
splitting above), after removing the arrows not corresponding to the nodes in
$R^{-}(\mu)$. The problem of describing the corresponding crystal inside the
tensor product in a very explicit way will be discussed elsewhere.

Organization of the paper: In Section 2 we establish the basic notation and
elementary results needed in the rest of the paper. In Section 3, we define
three classes of modules $M_{\mu,\mathbf{p}},M^{\prime}_{\mu,\mathbf{p}}$ and
$M^{\prime\prime}_{\mu,\mathbf{p}}$, $\mu$ an arbitrary integral weight and
$\mathbf{p}$ a sequence of functions indexed by the positive roots, and prove
surjective maps among them. This allows us to simplify the presentation of all
Demazure modules in Section 4 which was obtained earlier in [13] in the
$\mathfrak{g}$-stable case. Furthermore, we prove an embedding of Demazure
modules and propose a way to calculate the classical decompositions with
respect to a maximal semi-simple Lie subalgebra.

Acknowledgement: The first author thanks Daisuke Sagaki for many helpful
discussions.

##  2\. Preliminaries

### 2.1.

We denote by $\mathbb{Z}$, $\mathbb{Z}_{+}$, and $\mathbb{N}$ the set of
integers, the set of non-negative integers and the set of positive integers
respectively. The base field will be the complex numbers $\mathbb{C}$
throughout. For a given Lie algebra $\mathfrak{a}$, we denote by
$\mathbf{U}(\mathfrak{a})$ the universal enveloping algebra of $\mathfrak{a}$
and by $\mathfrak{a}[t]=\mathfrak{a}\otimes\mathbb{C}[t]$ its current algebra
with Lie bracket

$[x\otimes t^{r},y\otimes t^{s}]=[x,y]\otimes t^{r+s},\ \ x,y\in\mathfrak{a},\
\ r,s\in\mathbb{Z}_{+}.$

### 2.2.

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra with
Cartan subalgebra $\mathfrak{h}$ and root system $R$ spanned by the simple
roots $\\{\alpha_{1},\dots,\alpha_{n}\\}$ with index set $I=\\{1,\dots,n\\}$.
Let $\mathfrak{b}$ be a Borel subalgebra and $R^{\pm}$ be the corresponding
set of positive and negative roots respectively and denote by $\theta\in
R^{+}$ the highest root of $\mathfrak{g}$. Let $(\cdot,\cdot)$ be the non-
degenerate bilinear form on $\mathfrak{h}^{*}$ with $(\theta,\theta)=2$
induced by the restriction of the (suitably normalized) Killing form of
$\mathfrak{g}$ to $\mathfrak{h}$ and set $d_{\alpha}=2/(\alpha,\alpha)$ for
$\alpha\in R$. We denote by $P$ the set of integral weights with basis
$\\{\varpi_{1},\dots,\varpi_{n}\\}$ and let $P^{+}$ be the subset of dominant
integral weights. We further fix a Chevalley basis
$\\{x_{\alpha}^{\pm},h_{\alpha}:\alpha\in R^{+}\\}$ of $\mathfrak{g}$ and set

$R^{+}(\mu)=\\{\alpha\in R^{+}:\mu(h_{\alpha})\geq 0\\},\ \
R^{-}(\mu)=\\{\alpha\in R^{+}:\mu(h_{\alpha})\leq 0\\},\ \ \ \mu\in P.$

We denote by $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody
algebra with underlying simple Lie algebra $\mathfrak{g}$. The canonical
central element is denoted by $c$ and the derivation by $d$ and let
$\widehat{\mathfrak{b}}$ be the standard Borel subalgebra. The induced set of
positive and negative roots respectively is denoted by $\widehat{R}^{\pm}$ and
let $\widehat{P}$ and $\widehat{P}^{+}$ respectively the set of integral and
dominant integral weights respectively. Let $W$ be the Weyl group of $R$ and
recall that the affine Weyl group $\widehat{W}$ is given by the semi-direct
product

$\widehat{W}=W\ltimes t_{M},\ \ t_{M}=\\{t_{\mu}:\mu\in M\\}$

where $M$ is the lattice generated by the elements $w(\theta)$ for all $w\in
W$ (see [26, Proposition 6.5]).

### 2.3.

Given $\Lambda\in\widehat{P}^{+}$, let $V(\Lambda)$ be the irreducible,
integrable $\widehat{\mathfrak{g}}$-module with highest weight $\Lambda$ and
highest weight vector $v_{\Lambda}$. We have a weight space decomposition

$V(\Lambda)=\bigoplus_{\gamma}V(\Lambda)_{\Lambda-\gamma},\ \ \
\mathrm{dim}V(\Lambda)_{w\Lambda}=1,\ \ \forall w\in\widehat{W}.$

An important class of representations for the Borel subalgebra is given by the
so-called Demazure modules. To each pair
$(\Lambda,w)\in\widehat{P}^{+}\times\widehat{W}$ we define
$V_{w}(\Lambda)=\mathbf{U}(\widehat{\mathfrak{b}})V(\Lambda)_{w\Lambda}$ and
note that

$w\Lambda=w^{\prime}\Lambda\Rightarrow
V_{w}(\Lambda)=V_{w^{\prime}}(\Lambda).$

We denote these modules alternatively by $\mathbf{D}^{k}_{\mu}[i]$ whenever

$w\Lambda=\mu+k\Lambda_{0}+i\delta$ (2.1)

where $\Lambda_{0}\in\widehat{\mathfrak{h}}^{*}$ is the $0$-th fundamental
weight determined by

$\Lambda_{0}(c)=1,\ \Lambda_{0}(h_{\alpha_{i}})=0=\Lambda_{0}(d),\ \
i=1.\dots,n.$

Moreover, for each triple $(\mu,k,i)$ there exists a Weyl group element $w$
and a dominant affine weight $\Lambda$ such that (2.1) holds. This follows
immediately from the fact that $P+\mathbb{N}\Lambda_{0}+\mathbb{Z}\delta$ lies
in the Tits cone [26, Proposition 5.8].

######  Remark.

The modules $\mathbf{D}^{k}_{\mu}[i]$ are $\mathfrak{g}$-stable if and only if
$\mu$ is an anti-dominant weight, and in this case they are often denoted by
$D(k,w_{0}\mu)[i]$ in the literature where $w_{0}$ is the longest element in
the Weyl group $W$ (see for example [6, 13, 36]). However $D(k,w_{0}\mu)[i]$
or $D_{k,w_{0}\mu}[i]$ is also used sometimes to denote the module
$\mathbf{D}_{k}^{k\mu}[i]$ of lowest weight $k\mu$ (see for example [14])
which leads to lot of confusion. We decided to use the notation introduced
above to avoid any confusion with the literature and to emphasize that we
allow arbitrary weights.

A presentation for Demazure modules has been proved in [34], which we record
now in the affine setting. In the finite–dimensional setting this result was
obtained earlier in [24].

######  Theorem 1.

The module $\mathbf{D}^{k}_{\mu}[i]$ is as a
$\mathbf{U}\big{(}\widehat{\mathfrak{b}}\big{)}$–module isomorphic to the
cyclic module generated by a non-zero vector $v$ with the following defining
relations: For $h\in\mathfrak{h}$ and $\alpha\in R^{+}$, we have

  * •

$(h\otimes t^{s})v=\delta_{s,0}\mu(h)v$, $dv=iv$, and $cv=kv$,

  * •

$(x_{\alpha}^{\pm}\otimes t^{s^{\pm}})^{p^{\pm}_{\alpha}+1}v=0$,
$p^{\pm}_{\alpha}=\max\\{0,\mp\mu(h_{\alpha})-d_{\alpha}s^{\pm}k\\}$,
$s^{+}\geq 0$, $s^{-}>0.$

The above presentation of these modules has been vastly simplified in [13]
(see [32] for the twisted case) when $\mu$ is anti-dominant. This
simplification turned out to be quite useful in proving the existence of maps
from Demazure modules to other classes of modules. By way of example see [13,
Section 5] where the authors proved the fact that the Q-system introduced by
Kirillov and Reshetikhin [28] extends to a canonical short exact sequence of
fusion products or in the study of global Demazure modules and arc schemes
[14] where the authors showed that rectangular Demazure modules can be
obtained as localizations of global Demazure modules at zero. For more
applications of the simplified presentation we refer the reader to [19, 31,
38, 42].

##  3\. The main modules and existence of maps

One motivation of the paper is to simplify the defining relations of a class
of representations for the Iwahori subalgebras
$\mathfrak{I}[t]:=\mathfrak{b}\oplus(\mathfrak{g}\otimes t\mathbb{C}[t])$ in
untwisted affine Kac-Moody algebras. This class will include in particular all
Demazure modules and we focus on the simplification of them in the next
section. As an application we will prove an embedding of any higher level
Demazure module into a tensor product of lower level Demazure modules (see
Section 4). This allows us to obtain lower bounds for the dimensions of
certain irreducible modules for the quantum affine algebra in simply-laced
type.

### 3.1.

We first introduce this class of representations. Let
$\mathbf{p}=(p_{\alpha}^{\pm})$ be a sequence of functions indexed by the
positive roots

$p^{+}_{\alpha}:\mathbb{Z}_{+}\rightarrow\mathbb{Z}_{+},\ \
p^{-}_{\alpha}:\mathbb{N}\rightarrow\mathbb{Z}_{+},\ \ \alpha\in R^{+}$

such that there exists a minimal $s^{\pm}_{\alpha}$ satisfying
$p^{\pm}_{\alpha}(a)=0$ for all $a\geq s^{\pm}_{\alpha}$. Given $\mu\in P$ we
sometimes extend without further comment the function $p_{\alpha}^{-}$ to the
non-negative integers by setting
$p_{\alpha}^{-}(0)=\max\\{0,\mu(h_{\alpha})\\}$.

######  Definition.

For an integral weight $\mu\in P$ we denote by $M_{\mu,\mathbf{p}}$ the cyclic
module for the Iwahori subalgebra $\mathfrak{I}[t]$ generated by a non-zero
vector $v$ with the following defining relations. For $\alpha\in R^{\mp}(\mu)$
and $h\in\mathfrak{h}$ we have:

  1. (1)

$\big{(}h\otimes t^{s}\big{)}v=\delta_{s,0}\mu(h)v$, $s\geq 0$,
$\big{(}x_{\alpha}^{\mp}\otimes t^{s^{\mp}}\big{)}v=0,\ \ s^{+}\geq 0,\ \
s^{-}>0$

  2. (2)

$\big{(}x_{\alpha}^{\pm}\otimes
t^{\epsilon^{\pm}}\big{)}^{p^{\pm}_{\alpha}(\epsilon^{\pm})+1}v$, where
$\epsilon^{+}=0$ and $\epsilon^{-}=1$

  3. (3)

For all $1\leq i\leq s_{\alpha}^{\pm}$ and tuples
$(a_{i},...,a_{s_{\alpha}^{\pm}})$ satisfying
$\displaystyle\sum_{k=0}^{s_{\alpha}^{\pm}-i}(k+1)a_{i+k}\geq
p^{\pm}_{\alpha}(i)+1$ we have

$\big{(}x_{\alpha}^{\pm}\otimes
t^{s_{\alpha}^{\pm}}\big{)}^{a_{s_{\alpha}^{\pm}}}\big{(}x_{\alpha}^{\pm}\otimes
t^{s_{\alpha}^{\pm}-1}\big{)}^{a_{s_{\alpha}^{\pm}-1}}\cdots\big{(}x_{\alpha}^{\pm}\otimes
t^{i}\big{)}^{a_{i}}v=0.$

These modules are needed later to simplify the relations of the class of
modules $M^{\prime\prime}_{\mu,\boldsymbol{p}}$ which we define now.

### 3.2.

Given $\mu\in P$ and $\mathbf{p}$ as above, consider the tuple

$_{\pm}\boldsymbol{\xi}^{\alpha}=(_{\pm}\xi_{1}^{\alpha},_{\pm}\xi_{2}^{\alpha},\dots),\
\ _{\pm}\xi_{i}^{\alpha}=p_{\alpha}^{\pm}(i-1)-p_{\alpha}^{\pm}(i),\ \alpha\in
R^{\mp}(\mu)$ (3.1)

######  Definition.

Define $M^{\prime}_{\mu,\mathbf{p}}$ to be the cyclic $\mathfrak{I}[t]$-module
with defining relations given by Definition Definition(1)-(2) and imposing the
relations in Definition Definition(3) (for fixed $i$) only if

${}_{\pm}\xi_{i+1}^{\alpha}<\ _{\pm}\xi_{i}^{\alpha}\geq\text{
$a_{i}+\cdots+a_{s_{\alpha}^{\pm}}$}.$

Moreover, denote by $M^{\prime\prime}_{\mu,\mathbf{p}}$ to be the cyclic
$\mathfrak{I}[t]$-module with defining relations given by Definition
Definition(1)-(2) and

$(x_{\alpha}^{\pm}\otimes t^{i})^{p_{\alpha}^{\pm}(i)+1},\ \ \alpha\in
R^{\mp}(\mu),\ \ 1\leq i\leq s_{\alpha}^{\pm}$

For example, we always include the relation $(x_{\alpha}^{\pm}\otimes
t^{s_{\alpha}^{\pm}})v=0$ in $M^{\prime}_{\mu,\mathbf{p}}$ and
$M^{\prime\prime}_{\mu,\mathbf{p}}$ since

$0=\ _{\pm}\xi_{s^{\pm}_{\alpha}+1}^{\alpha}<\
_{\pm}\xi_{s_{\alpha}^{\pm}}^{\alpha}\geq 1$

by the minimality of $s_{\alpha}^{\pm}$. The importance of
$M^{\prime\prime}_{\mu,\boldsymbol{p}}$ will become clear by the following
example. They appear as many well-studied representations hence giving a
strong connection to combinatorics, graded limits of representations for
quantum affine algebras [6, 12] and the theory of Macdonald polynomials [9,
15, 23].

######  Example.

We discuss three examples.

  1. (1)

Let $\mu\in P$ be an anti-dominant weight. If we set $p_{\alpha}^{-}(s)=0$ for
all $s\in\mathbb{N}$ and $p_{\alpha}^{+}(s)=\max\\{0,-\mu(h_{\alpha})-s\\}$
for all $s\in\mathbb{Z}_{+}$ we recover the local Weyl module, namely we have
an isomorphism $W_{\mathrm{loc}}(w_{0}\mu)\cong
M_{\mu,\mathbf{p}}^{\prime\prime}$ as $\mathfrak{g}[t]$-modules; we can extend
the $\mathfrak{I}[t]$-structure on $M_{\mu,\mathbf{p}}^{\prime\prime}$ by
requiring that $(\mathfrak{n}^{-}\otimes 1)$ acts as zero on $v$.

  2. (2)

Let $\mu\in P$ and $k\in\mathbb{N}$. We set

$p_{\alpha}^{\pm}(s)=\mathrm{max}\\{0,\mp\mu(h_{\alpha})-d_{\alpha}sk\\},\ \
s\in\mathbb{N},\ \ p_{\alpha}^{+}(0)=\mathrm{max}\\{0,-\mu(h_{\alpha})\\}$

Then, by Theorem 1 we have an isomorphism
$M^{\prime\prime}_{\lambda,\mathbf{p}}\cong\mathbf{D}_{\mu}^{k}$, where we
drop the dependence on $i$ since
$\mathbf{D}_{\mu}^{k}[i]\cong\mathbf{D}_{\mu}^{k}[i^{\prime}]$ as
$\mathfrak{I}[t]$-modules.

  3. (3)

Choosing the data

$p_{\alpha}^{+}(0)=0,\ p_{\alpha}^{-}(1)=\mu(h_{\alpha}),\ \ \alpha\in
R^{+}(\mu)$

and

$p_{\alpha}^{+}(0)=-\mu(h_{\alpha}),\ p_{\alpha}^{-}(1)=0,\ \ \alpha\in
R^{-}(\mu)$

we get an isomorphism to the generalized Weyl modules studied in [15]; in fact
we have to choose the remaining integers as well, but we can simply choose the
remaining $p_{\alpha}^{\pm}(s)$ so that they become redundant.

The following simple observation is needed later.

######  Lemma.

Let $\mathbf{p}$ be the sequence of functions from Example Example(2). Then,
we have

$2p_{\alpha}^{\pm}(i)\leq p_{\alpha}^{\pm}(i+1)+p_{\alpha}^{\pm}(i-1)$ (3.2)

and equality holds if one of the following two conditions hold

  * •

$1\leq i\leq s_{\alpha}^{\pm}-2$

  * •

$i=s_{\alpha}^{\pm}-1$ and $\pm\mu(h_{\alpha})=d_{\alpha}s_{\alpha}^{\pm}k$.

###### Proof.

If one of the two conditions hold, then the maximum in $p_{\alpha}^{\pm}(s)$
is always given by $\mp\mu(h_{\alpha})-d_{\alpha}sk$ for all $s\in\\{i,i\pm
1\\}$ which implies equality in (3.2). The general case follows from several
case considerations and we illustrate the case
$\mp\mu(h_{\alpha})-d_{\alpha}(i+1)k\leq 0$ and
$\mp\mu(h_{\alpha})-d_{\alpha}ik\geq 0$ only. In this case the inequality is
equivalent to

$\mp
2\mu(h_{\alpha})-2d_{\alpha}ik\leq\mp\mu(h_{\alpha})-d_{\alpha}(i-1)k\Leftrightarrow\mp\mu(h_{\alpha})\leq
d_{\alpha}(i+1)k.$

∎

### 3.3.

Now we prove several maps among the aforementioned modules and show in the
subsequent section how they can be used to simplify their presentation. First
we introduce some well-known sets which have been used already in [13] and
[32] to simplify the relations of stable Demazure modules. Roughly speaking,
our proof follows the proofs of [13] and [32] in a modified way. For
$r\in\mathbb{N}$ and $s\in\mathbb{Z}_{+}$, let

$\mathbf{S}(r,s)=\left\\{(b_{p})_{p\geq 0}:b_{p}\in\mathbb{Z}_{+},\ \
\sum_{p\geq 0}b_{p}=r,\ \ \sum_{p\geq 0}pb_{p}=s\right\\}.$ (3.3)

For $k\in\mathbb{Z}_{+}$, we denote by $\mathbf{S}(r,s)_{k}$ (resp.
${}_{k}\mathbf{S}(r,s)$) the subset of $\mathbf{S}(r,s)$ consisting of
elements satisfying

$b_{p}=0,\ \ p\geq k,\ \ ({\rm{resp.}}\ \ b_{p}=0\ \ p<k).$

Given $x\in\mathfrak{g}$ define the following elements

$\displaystyle\mathbf{x}(r,s)_{k}$
$\displaystyle=\sum_{\mathbf{b}\in\mathbf{S}(r,s)_{k}}(x\otimes
1)^{(b_{0})}\cdots(x\otimes t^{k-1})^{(b_{k-1})},$ ${}_{k}\mathbf{x}(r,s)$
$\displaystyle=\sum_{\mathbf{b}\in_{k}\mathbf{S}(r,s)}(x\otimes
t^{k})^{(b_{k})}\cdots(x\otimes t^{s})^{(b_{s})}$
$\displaystyle~{}^{t}\\!\mathbf{x}(r,s)$
$\displaystyle=\sum_{\mathbf{b}\in\mathbf{S}(r,s)}(x\otimes
t)^{(b_{0})}(x\otimes t^{2})^{(b_{1})}\cdots(x\otimes t^{s+1})^{(b_{s})},$

where for any integer $p$ and any $y\in\mathfrak{g}[t]$, we set
$y^{(p)}=y^{p}/p!$. Moreover, we set for simplicity
${}_{0}\mathbf{x}(r,s)=\mathbf{x}(r,s)$ and define similarly as above the
elements $~{}^{t}\\!\mathbf{x}(r,s)_{k}$ and ${}^{t}_{k}\mathbf{x}(r,s)$. We
obtain,

$\mathbf{x}(1,s)=x\otimes t^{s},\ \ ~{}^{t}\\!\mathbf{x}(1,s)=x\otimes
t^{s+1},\ \ _{k}\mathbf{x}(r,kr)=(x\otimes t^{k})^{(r)}.$ (3.4)

The following proposition is an easy modification of the analogues result
proved in [13].

######  Proposition.

Let $V$ be any representation of $\mathfrak{I}[t]$ and let $v\in V$,
$x\in\mathfrak{g}$, and $K,k\in\mathbb{Z}_{+}$. Then,

$\displaystyle{}^{t}\mathbf{x}(r,s)v=0\ {\rm{for\ all}}\
s\in\mathbb{Z}_{+},r\in\mathbb{N}\ \ {\rm{with}}\ s+r\geq 1+kr+K\ \iff$
$\displaystyle{}^{t}_{k}\mathbf{x}(r,s)v=0\ {\rm{for\ all}}\
s\in\mathbb{Z}_{+},r\in\mathbb{N}\ {\rm{with}}\ s+r\geq 1+kr+K.$

Moreover, if $V$ is stable under the action with $(x\otimes 1)$, the same
equivalence holds if we erase the $t$ in the superscript. ∎

### 3.4.

For a Lie algebra $\mathfrak{a}$ we denote by $\mathfrak{a}[t]_{+}$ the
subspace spanned by the positive degree elements, i.e.,
$\mathfrak{a}[t]_{+}=\bigoplus_{r>0}\mathbb{C}(\mathfrak{a}\otimes t^{r})$.
Furthermore, let

$\mathbf{U}_{+}=\mathbf{U}(\mathfrak{g}[t])\cdot\mathfrak{n}^{+}[t]\oplus\mathbf{U}\big{(}\mathfrak{n}^{-}[t]\oplus\mathfrak{h}[t]\big{)}\cdot\mathfrak{h}[t]_{+}$

and define $\mathbf{U}_{++}$ similarly as above by replacing
$\mathfrak{n}^{+}[t]$ in $\mathbf{U}_{+}$ by $\mathfrak{n}^{+}[t]_{+}$.
Consider the automorphism $\Phi$ on $\mathfrak{g}[t]$ given by

$x^{+}_{\alpha}\otimes t^{s}\mapsto x^{-}_{\alpha}\otimes t^{s},\ \
x^{-}_{\alpha}\otimes t^{s}\mapsto x^{+}_{\alpha}\otimes t^{s},\ \ s\geq 0$

and note that

$\Phi(\mathbf{U}_{++})=\mathbf{U}(\mathfrak{g}[t])\cdot\mathfrak{n}^{-}[t]_{+}\oplus\mathbf{U}(\mathfrak{n}^{+}[t]\oplus\mathfrak{h}[t])\cdot\mathfrak{h}[t]_{+}.$

The following result is a slight modification of the Garland identities [21,
Lemma 7.5/7.8]. For a current algebra reformulation of the above lemma see
also [12, Lemma 1.3].

######  Lemma.

Given $s\in\mathbb{N}$, $r\in\mathbb{Z}_{+}$ and $\alpha\in R^{+}$ we have the
following identities

  1. (1)

$(x_{\alpha}^{-}\otimes t)^{(s)}(x_{\alpha}^{+}\otimes
1)^{(s+r)}\equiv(-1)^{s}\ \mathbf{x}^{+}_{\alpha}(r,s)\ \ \mathrm{mod}\
\big{(}\Phi(\mathbf{U}_{++})\cap\mathbf{U}(\mathfrak{I}[t])\big{)}$

  2. (2)

$(x_{\alpha}^{+}\otimes 1)^{(s)}(x_{\alpha}^{-}\otimes
t)^{(s+r)}\equiv(-1)^{s}\ {}^{t}\mathbf{x}^{-}_{\alpha}(r,s)\ \ \mathrm{mod}\
\big{(}\mathbf{U}_{+}\cap\mathbf{U}(\mathfrak{I}[t])\big{)}$

∎

### 3.5.

The key result which will lead to the aforementioned simplified presentation
is the following.

######  Proposition.

Let $\mu\in P$ and suppose that the tuple ${}_{\pm}\xi^{\alpha}$ from (3.1) is
a partition. Then, the following relations are true in the module
$M^{\prime}_{\mu,\mathbf{p}}$. For all $\alpha\in R^{+}(\mu)$ (resp.
$\alpha\in R^{-}(\mu)$) and $r\in\mathbb{N},s,k\in\mathbb{Z}_{+}$ such that

$s+r\geq 1+rk+p_{\alpha}^{-}(k+1)\ \ \ (\text{resp. }s+r\geq
1+rk+p_{\alpha}^{+}(k))$

we have ${}^{t}\mathbf{x}^{-}_{\alpha}(r,s)v=0,\ (\text{resp.
}\mathbf{x}^{+}_{\alpha}(r,s)v=0).$

###### Proof.

We give the proof of the proposition only for $\alpha\in R^{+}(\mu)$. The
other case is treated similarly and we omit the details. For simplicity, we
set in the rest of the proof ${}_{-}\xi^{\alpha}_{j}=\xi^{\alpha}_{j}$,
$s_{\alpha}=s^{-}_{\alpha}$ and $p_{\alpha}=p^{-}_{\alpha}$. If
$r\geq\xi_{1}^{\alpha}=\mu(h_{\alpha})-p_{\alpha}(1)$, then we get

$s+r\geq 1+k\xi_{1}^{\alpha}+p_{\alpha}(k+1)\geq
1+\mu(h_{\alpha})-\xi_{k+1}^{\alpha}\geq
1+p_{\alpha}(1)+\xi_{1}^{\alpha}-\xi_{k+1}^{\alpha}\geq p_{\alpha}(1)+1$

and the claim follows from Lemma Lemma since $(x_{\alpha}^{-}\otimes
t)^{a}v=0$ if $a\geq p_{\alpha}(1)+1$. So assume that $r<\xi^{\alpha}_{1}$. If
additionally $r\leq\xi^{\alpha}_{s_{\alpha}}=p_{\alpha}(s_{\alpha}-1)$ we can
deduce $s+r\geq 1+(s_{\alpha}-1)r$ which we will explain now. To see the above
inequality let $k<s_{\alpha}-1$ (otherwise the statement is clear) and we get

$s+r\geq 1+kr+p_{\alpha}(k+1)\geq
1+kr+(s_{\alpha}-k-1)\xi_{s_{\alpha}}^{\alpha}\geq 1+kr+(s_{\alpha}-k-1)r$

and the claim follows. Moreover, we obtain that any
$\mathbf{b}\in\mathbf{S}(r,s)$ has the property $b_{m}>0$ for some $m\geq
s_{\alpha}-1$, since otherwise we would have

$s=\sum_{p\leq s_{\alpha}-2}pb_{p}\leq r(s_{\alpha}-2).$

In particular, the claim follows from the relation $(x_{\alpha}^{-}\otimes
t^{s_{\alpha}})v=0$ in $M^{\prime}_{\mu,\mathbf{p}}$. So we can assume in the
rest of the proof that $\xi^{\alpha}_{s_{\alpha}}<r<\xi^{\alpha}_{1}$ and let
$i\in\\{1,\dots,s_{\alpha}-1\\}$ be such that
$\xi^{\alpha}_{i+1}<r\leq\xi^{\alpha}_{i}$. Furthermore, let
$\mathbf{b}\in\mathbf{S}(r,s)_{s_{\alpha}-1}$ (otherwise we are done). Since
$s=(s_{\alpha}-2)b_{s_{\alpha}-2}+\cdots+b_{1}$ we obtain

$(s_{\alpha}-2)b_{s_{\alpha}-2}+\cdots+(i-1)b_{i-1}+(r-\sum_{j\geq
i-1}b_{j})(i-2)\geq s\geq 1+r(k-1)+p_{\alpha}(k+1),$

and hence

$(s_{\alpha}-i)b_{s_{\alpha}-2}+\cdots+b_{i-1}\geq
1+r(k-i+1)+p_{\alpha}(k+1).$

Since $r>\xi^{\alpha}_{i+1}$, we see that the right hand side of the
inequality is bigger or equal to $p_{\alpha}(i)$ provided $k\geq i-1$:

$\displaystyle 1+r(k-i+1)+$ $\displaystyle p_{\alpha}(k+1)\geq
1+(k-i+1)(p_{\alpha}(i)-p_{\alpha}(i+1))+p_{\alpha}(k+1)$ $\displaystyle\geq
1+(p_{\alpha}(i)-p_{\alpha}(i+1))+\sum^{k-i}_{j=1}(p_{\alpha}(i+j)-p_{\alpha}(i+j+1))+p_{\alpha}(k+1)$
$\displaystyle\geq 1+p_{\alpha}(i).$

If $k<i-1$, then

$(s_{\alpha}-i)b_{s_{\alpha}-2}+\cdots+b_{i-1}\geq 1+\sum_{i\geq j\geq
k+2}(\xi^{\alpha}_{j}-r)+\xi^{\alpha}_{i+1}+\cdots+\xi^{\alpha}_{s_{\alpha}}\geq
1+p_{\alpha}(i).$

So each summand of ${}^{t}\mathbf{x}^{-}_{\alpha}(r,s)$ acts as zero by the
defining relations of $M_{\mu,\mathbf{p}}^{\prime}$ and the calculations
above. This finishes the proof of the proposition. ∎

######  Corollary.

Let $k\in\mathbb{N}$, $\alpha\in R^{\mp}(\mu)$ and
$a_{k},\dots,a_{s_{\alpha}^{\pm}}$ non-negative integers such that

$a_{k}+2a_{k+1}+\cdots+(s_{\alpha}^{\pm}-k+1)a_{s_{\alpha}^{\pm}}\geq
1+p_{\alpha}^{\pm}(k).$

We set

$r=a_{k}+\cdots+a_{s_{\alpha}^{\pm}},\ \
s=(k-1)a_{k}+\cdots+(s_{\alpha}^{\pm}-1)a_{s_{\alpha}^{\pm}}.$

We have the following relations in $M^{\prime}_{\mu,\mathbf{p}}$:

${}_{k-1}^{t}\mathbf{x}^{-}_{\alpha}(r,s)v=0\ \ (\text{if $\alpha\in
R^{+}(\mu)$}),\ \ \ {}_{k}\mathbf{x}^{+}_{\alpha}(r,r+s)v=0\ \ (\text{if
$\alpha\in R^{-}(\mu)$}).$

###### Proof.

The statement follows immediately from Proposition Proposition and Proposition
Proposition. ∎

######  Theorem 2.

For $\mu\in P$ and $\mathbf{p}$ as in Proposition Proposition we have

$M^{\prime\prime}_{\mu,\mathbf{p}}\twoheadrightarrow
M^{\prime}_{\mu,\mathbf{p}}\twoheadrightarrow M_{\mu,\mathbf{p}}.$

Moreover, the first map is an isomorphism if
${}_{\pm}\xi^{\alpha}_{1}=\cdots=_{\pm}\xi^{\alpha}_{s_{\alpha}^{\pm}-1}$ and
the second map is an isomorphism if
${}_{\pm}\xi^{\alpha}_{1}\neq_{\pm}\xi^{\alpha}_{2}$.

###### Proof.

The first surjective map follows immediately from Corollary Corollary if we
set $a_{k}=p_{\alpha}^{-}(k)+1$ and the remaining integers to be zero. Then,
in the notation of the above Corollary, we have

${}_{k-1}^{t}\mathbf{x}^{-}_{\alpha}(r,s)v=(x_{\alpha}^{-}\otimes
t^{k})^{p_{\alpha}^{-}(k)+1}v=0\ \ (\text{if $\alpha\in R^{+}(\mu)$})$
${}_{k}\mathbf{x}^{+}_{\alpha}(r,r+s)v=(x_{\alpha}^{+}\otimes
t^{k})^{p_{\alpha}^{+}(k)+1}v=0\ \ (\text{if $\alpha\in R^{-}(\mu)$}).$

The second surjectivity holds by definition. If
${}_{\pm}\xi^{\alpha}_{1}=\cdots=_{\pm}\xi^{\alpha}_{s_{\alpha}^{\pm}-1}$, it
is clear that all relations in $M_{\mu,\mathbf{p}}^{\prime}$ also hold in
$M_{\mu,\mathbf{p}}^{\prime\prime}$. Now assume that
${}_{\pm}\xi^{\alpha}_{1}\neq_{\pm}\xi^{\alpha}_{2}$. To see that the second
map is an isomorphism, it will be enough to show that the monomial relations
from Definition Definition(3) (for fixed $i$) hold in
$M_{\mu,\mathbf{p}}^{\prime}$ when ${}_{\pm}\xi_{i}^{\alpha}=\
_{\pm}\xi_{i+1}^{\alpha}$, i.e.

$2p_{\alpha}^{\pm}(i)=p_{\alpha}^{\pm}(i-1)+p_{\alpha}^{\pm}(i+1).$

We will show this by downward induction on
$i\in\\{1,\dots,s_{\alpha}^{\pm}-1\\}$. If $i=s_{\alpha}^{\pm}-1$, then the
answer is immediate and the relation already holds in
$M_{\mu,\mathbf{p}}^{\prime\prime}$. Otherwise, we can assume by induction
that

$a_{i}+a_{i+1}+\cdots+a_{s_{\alpha}^{\pm}}>p_{\alpha}^{\pm}(i)-p_{\alpha}^{\pm}(i+1)$

and hence setting $a_{i-1}=0$ we get

$a_{i-1}+2a_{i}+3a_{i+1}+\cdots+(s_{\alpha}^{\pm}-i+1)a_{s_{\alpha}^{\pm}}\geq
1+2p_{\alpha}^{\pm}(i)-p_{\alpha}^{\pm}(i+1)=1+p_{\alpha}^{\pm}(i-1).$

So either ${}_{\pm}\xi_{i-1}^{\alpha}=\ _{\pm}\xi_{i}^{\alpha}$ or the
relation holds in $M_{\mu,\mathbf{p}}^{\prime}$. Continuing in this way we
obtain the desired relation, since the process has to stop by the reason of
${}_{\pm}\xi^{\alpha}_{1}\neq_{\pm}\xi^{\alpha}_{2}$. ∎

######  Remark.

For general $\mu$ and $\mathbf{p}$ it is quite hard to figure out the
connection between these three modules. If the aforementioned conditions do
not hold, than these maps do not need to be isomorphisms. However, the
surjectivity will be enough to simplify the presentation of all Demazure
modules.

##  4\. Simplified presentations and applications: embeddings

### 4.1.

We first discuss the trivial consequence of Theorem 2 and extend the results
of [13] from $\mathfrak{g}$-stable Demazure modules to the general setting.
For $\mu\in P$ and $\alpha\in R^{\mp}(\mu)$ we denote by
$s^{\pm}_{\alpha},m^{\pm}_{\alpha}\in\mathbb{Z}_{+}$ the unique integers so
that

$\mp\mu(h_{\alpha})=(s^{\pm}_{\alpha}-1)d_{\alpha}k+m^{\pm}_{\alpha},\ \ \
0<m^{\pm}_{\alpha}\leq d_{\alpha}k$ (4.1)

It follows that the function $\mathbf{p}$ from Example Example(2) turns into

$p_{\alpha}^{\pm}(i)=\mathrm{max}\\{0,(s^{\pm}_{\alpha}-i-1)d_{\alpha}k+m^{\pm}_{\alpha}\\},\
\ i\in\mathbb{Z}_{+}$

In particular, from Lemma Lemma we obtain that
${}_{\pm}\xi^{\alpha}_{i}=_{\pm}\xi^{\alpha}_{i+1}$ for all $1\leq i\leq
s_{\alpha}^{\pm}-2$ and
${}_{\pm}\xi^{\alpha}_{s_{\alpha}^{\pm}-1}=_{\pm}\xi^{\alpha}_{s_{\alpha}^{\pm}}$
provided that $m^{\pm}_{\alpha}=d_{\alpha}k$. So in this case, the module
$M^{\prime}_{\mu,\mathbf{p}}$ is generated by $v$ subject to the relations in
Definition Definition(1)-(2) and for $\alpha\in R^{\mp}(\mu)$

$(x_{\alpha}^{\pm}\otimes t^{s_{\alpha}^{\pm}})v=0$ $(x_{\alpha}^{\pm}\otimes
t^{s^{\pm}_{\alpha}-1})^{m^{\pm}_{\alpha}+1}v=0,\ \text{ if
$m^{\pm}_{\alpha}<d_{\alpha}k$}$

######  Corollary.

Let $\mu\in P$ and $k\in\mathbb{N}$. Then $\mathbf{D}_{\mu}^{k}$ is a cyclic
$\mathbf{U}(\mathfrak{I}[t])$–module generated by a non-zero vector $v$ with
the following relations:

$\displaystyle(h\otimes t^{s})v=\delta_{s,0}\cdot\mu(h)v,\text{ for all
}h\in\mathfrak{h}$

and for $\alpha\in R^{\mp}(\mu)$ we have

$\big{(}x_{\alpha}^{\pm}\otimes
t^{s^{\pm}_{\alpha}-1}\big{)}^{m^{\pm}_{\alpha}+1}v=0,\ \text{if
$m^{\pm}_{\alpha}<d_{\alpha}k$};\ \ \big{(}x_{\alpha}^{\pm}\otimes
t^{s^{\pm}_{\alpha}}\big{)}v=0,$ (4.2)
$\left(x_{\alpha}^{+}\otimes\mathbb{C}[t]\right)v=0,\
\left(x_{\alpha}^{-}\otimes t\right)^{\mathrm{max}\\{0,\
\mu(h_{\alpha})-d_{\alpha}k\\}+1}v=0,\ \text{ if $\alpha\in R^{+}(\mu)$}$
(4.3) $\left(x_{\alpha}^{-}\otimes t\mathbb{C}[t]\right)v=0,\
\left(x_{\alpha}^{+}\otimes 1\right)^{-\mu(h_{\alpha})+1}v=0,\ \text{ if
$\alpha\in R^{-}(\mu)$}$ (4.4)

###### Proof.

Let us denote the module generated by $v$ with the simplified relations listed
above by $\mathbf{N}_{\mu}^{k}$. The presentation of Demazure modules stated
in Theorem 1 gives a surjective map
$\mathbf{N}_{\mu}^{k}\twoheadrightarrow\mathbf{D}_{\mu}^{k}$. In order to
prove that the map is an isomorphism, we only have to show that the defining
relations of $\mathbf{D}_{\mu}^{k}$ hold in $\mathbf{N}_{\mu}^{k}$. But
$\mathbf{N}_{\mu}^{k}$ is isomorphic to a quotient of
$M^{\prime}_{\mu,\mathbf{p}}\cong M^{\prime\prime}_{\mu,\mathbf{p}}$ and hence
all relations also hold in $\mathbf{D}_{\mu}^{k}$. ∎

######  Remark.

In the special case $k=1$, a similar observation as in [13, Proposition 3.4]
shows that some of the relations in (4.2) are still redundant. To be more
precise, the second relation in (4.2) is only needed if $d_{\alpha}>1$ and the
first relation in (4.2) is only needed when $d_{\alpha}=3=m^{\mp}_{\alpha}+2$.
For example, this is one of the key observations why $\mathbf{D}_{\mu}^{1}$
for anti-dominant weights can be identified with local Weyl modules in the
simply-laced case.

So given a module $L$ by generators and relations, let $L^{\mathrm{simp}}$ a
module obtained by erasing certain relations in the presentation. In
particular we have $L^{\mathrm{simp}}\rightarrow L\rightarrow 0$. The key idea
above for showing that $L^{\mathrm{simp}}\cong L$ was to identify $\mu$ and
$\mathbf{p}$ (the induced tuple in (3.1) has to be a partition) such that
there exists surjections

$M_{\mu,\mathbf{p}}^{\prime}\rightarrow L^{\mathrm{simp}}\rightarrow 0\ \text{
and }\ L\rightarrow M_{\mu,\mathbf{p}}^{\prime\prime}\rightarrow 0.$

This strategy can be used to simplify the presentation of certain more modules
$V(\xi)$ introduced in [13]. On can read the isomorphism
$L^{\mathrm{simp}}\cong L$ also conversely, namely that we have more relations
in $L^{\mathrm{simp}}$. This can be used for example to show that certain
relations hold in the generalized Weyl module [15].

### 4.2.

The simplified presentation of Demazure modules allow us to embed
$\mathbf{D}_{\mu}^{rk}$ into a $k$-fold tensor product of Demazure modules of
the form $\mathbf{D}_{\mu^{\prime}}^{r}$ (see Theorem 3). In the limit
$k\rightarrow\infty$ this will become for anti-dominant weights the well-known
embedding of irreducible $\mathfrak{g}$-modules into a tensor product of
fundamental irreducible $\mathfrak{g}$-modules. The interest of this embedding
is twofold.

On the one hand we hope that this embedding can be used to come up with
explicit combinatorial models for the Demazure crystal of
$\mathbf{D}_{\mu}^{k}$ viewing it as a “connected component” inside the tensor
product of suitable lower level crystals.

On the other hand, the motivation comes from quantum affine algebras and
graded limits of irreducible finite-dimensional representations. It is
sometimes important to prove surjective maps from the graded limit of a
suitable representation to a higher level Demazure module as pointed out, for
example, for level two in the article [6] for the graded limit of a prime
irreducible object in the Hernandez-Leclerc category. To achieve such a map,
the authors proved in type $A$ and $k=2$ (see [6, Theorem 4]) an embedding of
Demazure modules. The aim of this section is to generalize their result.

We state the main theorem of this section. For $k\in\mathbb{N}$ and $\mu\in P$
we define

$\mathcal{P}(\mu,k)=\\{(\mu_{1},\dots,\mu_{k})\in
P^{\times{k}}:\mu_{1}+\cdots+\mu_{k}=\mu\\}.$

We say that an element
$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{k})\in\mathcal{P}(\mu,k)$ is pre-
admissible if

$\alpha\in R^{\pm}(\mu)\Rightarrow\alpha\in R^{\pm}(\mu_{i}),\ \ 1\leq i\leq
k$ (4.5)

and for such an element $\boldsymbol{\mu}$ we define non-negative integers
$t_{\alpha}$ and $p^{\alpha}_{j}(\boldsymbol{\mu}),\ 0\leq j\leq t_{\alpha}$,
$\alpha\in R^{\mp}(\mu)$ by the following equation

$\big{(}\mp\mu_{i_{1}}(h_{\alpha}),\dots,\mp\mu_{i_{k}}(h_{\alpha})\big{)}=\big{(}x,\dots,x,(x-1),\dots,(x-1),\dots,(x-t_{\alpha}),\dots,(x-t_{\alpha})\big{)}$
(4.6)

where $p_{j}^{\alpha}(\boldsymbol{\mu})$ encodes how often $(x-j)$ appears in
the above tuple. By convention we assume that we have
$p_{0}^{\alpha}(\boldsymbol{\mu})>0$. Given $r\in\mathbb{N}$ we define also an
element $0<m^{\pm}_{\alpha}(\boldsymbol{\mu},r)\leq d_{\alpha}r$ determined by

$x=\mp\mu_{i_{1}}(h_{\alpha})=m^{\pm}_{\alpha}(\boldsymbol{\mu},r)\mod
d_{\alpha}r$

We say that a pre-admissible element $\boldsymbol{\mu}$ is r-admissible if it
satisfies

$m^{\pm}_{\alpha}(\boldsymbol{\mu},r)\cdot k>\sum_{j=0}^{t_{\alpha}}j\cdot
p^{\alpha}_{j}(\boldsymbol{\mu})\ \ \forall\alpha\in R^{\mp}(\mu)$ (4.7)

and

$\mu(h_{\alpha})>kd_{\alpha}r\Rightarrow
x=\mu_{i_{1}}(h_{\alpha})\geq(t_{\alpha}+d_{\alpha}r),\ \ \forall\alpha\in
R^{+}(\mu)$ (4.8)

We emphasize that (4.8) is only needed for roots in $R^{+}(\mu)$. It is not
hard to see that each tuple $(\mu_{1},\dots,\mu_{k})$ is $r$-admissible for
some $r>>0$. So it is natural to ask for the minimal $r$ such that a tuple is
$r$-admissible. We will discuss this question in the next subsection and
conjecture that there exists always a 1-admissible element. The main theorem
of this section is the following.

######  Theorem 3.

Given $\mu\in P$ and $r,k\in\mathbb{N}$, let
$\boldsymbol{\mu}=(\mu_{1},\dots,\mu_{k})\in\mathcal{P}(\mu,k)$ be
$r$-admissible. Then we have an injective map of graded
$\mathfrak{I}[t]$-modules

$\mathbf{D}_{\mu}^{rk}\hookrightarrow\mathbf{D}_{\mu_{1}}^{r}\otimes\cdots\otimes\mathbf{D}_{\mu_{k}}^{r},\
\ v_{\mu}\mapsto v_{\mu_{1}}\otimes\cdots\otimes v_{\mu_{k}}.$

The proof of this theorem occupies the rest of this section.

### 4.3.

The fact that $\mathbf{D}_{\mu}^{k}$ appears as a subquotient follows from the
next result which can be derived from [30].

######  Proposition.

Let $r\in\mathbb{N}$ and choose positive integers $k_{1},\dots,k_{r}$ such
that $k_{1}+\cdots+k_{r}=k$ and
$(\mu_{1},\dots,\mu_{r})\in\mathcal{P}(\mu,r)$. Then we have a surjective map

$\mathbf{D}_{\mu_{1}}^{k_{1}}\otimes\cdots\otimes\mathbf{D}_{\mu_{k}}^{k_{r}}\supseteq\mathbf{U}(\mathfrak{I}[t])(v_{\mu_{1}}\otimes\cdots\otimes
v_{\mu_{r}})\rightarrow\mathbf{D}_{\mu}^{k}\rightarrow 0$

∎

The previous proposition implies that it is enough to show that for an
$r$-admissible element $(\mu_{1},\dots,\mu_{k})\in\mathcal{P}(\mu,k)$ the
weight vector $(v_{\mu_{1}}\otimes\cdots\otimes v_{\mu_{k}})$ satisfies the
defining relations of $\mathbf{D}_{\mu}^{rk}$ which were simplified in
Corollary Corollary. Without this simplification it was not possible for us to
show that the required relations hold. The following lemma is trivial but
useful:

######  Lemma.

Let $x,j\in\mathbb{Z}_{+}$ such that $x\geq j$ and $\ell\in\mathbb{N}$. We
write $x=(s-1)\ell+m$, $x-j=(s^{\prime}-1)\ell+m^{\prime}$ and
$j=(q-1)\ell+m^{\prime\prime}$, $0<m,m^{\prime},m^{\prime\prime}\leq\ell$.
Then we have

$(s,m)=\begin{cases}(s^{\prime}+q-1,m^{\prime}+m^{\prime\prime})&\ \text{if
$m^{\prime}+m^{\prime\prime}\leq\ell$}\\\
(s^{\prime}+q,m^{\prime}+m^{\prime\prime}-\ell)&\ \text{if
$m^{\prime}+m^{\prime\prime}>\ell$}\\\ \end{cases}$

∎

###  4.4. Proof of Theorem 3

Fix an element $\alpha\in R^{\mp}(\mu)$. As usual we define the numbers
$s_{\alpha}^{\pm},m_{\alpha}^{\pm}$ for $\mu$ as in (4.1) (with respect to
$d_{\alpha}kr$) and denote the corresponding elements for $\mu_{i}$ by
$(s^{i}_{\alpha})^{\pm},(m^{i}_{\alpha})^{\pm}$ (with respect to
$d_{\alpha}r$). Recall that $\boldsymbol{\mu}$ satisfies property (4.5). In
the rest of this section we abbreviate $v_{j}$ for the cyclic generator of
$\boldsymbol{D}_{\mu_{j}}^{r}$, $1\leq j\leq k$ and let $v$ be the cyclic
generator of $\boldsymbol{D}_{\mu}^{rk}$. We abbreviate further
$p_{j}=p_{j}^{\alpha}(\boldsymbol{\mu})$ and $t=t_{\alpha}$ and recall that
$p_{0}+\cdots+p_{t}=k.$ We have

$\displaystyle(s_{\alpha}^{\pm}-1)d_{\alpha}kr+m_{\alpha}^{\pm}$
$\displaystyle=\mp\mu(h_{\alpha})=p_{0}x+p_{1}(x-1)+\cdots+p_{t}(x-t)$
$\displaystyle=kx-(p_{1}+2p_{2}+\cdots+tp_{t})$
$\displaystyle=\big{(}(s^{i_{1}}_{\alpha})^{\pm}-1\big{)}d_{\alpha}kr+\big{(}km^{\pm}_{\alpha}(\boldsymbol{\mu},r)-(p_{1}+2p_{2}+\cdots+tp_{t})\big{)}.$
(4.9)

From (4.7) and $0<m^{\pm}_{\alpha}(\boldsymbol{\mu},r)\leq d_{\alpha}r$ we
know that

$0<km^{\pm}_{\alpha}(\boldsymbol{\mu},r)-(p_{1}+2p_{2}+\cdots+tp_{t})\leq
d_{\alpha}r$

and hence $s_{\alpha}^{\pm}=(s^{i_{1}}_{\alpha})^{\pm}$. Now using Lemma Lemma
we also get that

$s_{\alpha}^{\pm}=(s^{i_{1}}_{\alpha})^{\pm}\geq(s^{j}_{\alpha})^{\pm},\text{
for all $1\leq j\leq k$}$

and hence we have proved the connection

$s_{\alpha}^{\pm}=\max\\{(s^{i}_{\alpha})^{\pm}:1\leq i\leq k\\}.$ (4.10)

Relations in (4.2): The relation $(x_{\alpha}^{\pm}\otimes
t^{s^{\pm}_{\alpha}})(v_{1}\otimes\cdots\otimes v_{k})=0$ is immediate from
(4.10) and Corollary Corollary. So we argue in the remaining part why the
following relation holds:

$(x_{\alpha}^{\pm}\otimes
t^{s^{\pm}_{\alpha}-1})^{m_{\alpha}^{\pm}+1}(v_{1}\otimes\cdots\otimes
v_{k})=0.$

So assume by contradiction that the above relation does not hold. Write
$Y=(x_{\alpha}^{\pm}\otimes t^{s^{\pm}_{\alpha}-1})$ and
$m=m_{\alpha}^{\pm}+1$, then

$Y^{m}(v_{1}\otimes\cdots\otimes
v_{k})=\sum\limits_{j_{1}+\cdots+j_{k}=m}Y^{j_{1}}v_{1}\otimes\cdots\otimes
Y^{j_{k}}v_{k}.$

Note that we either have $Yv_{i}=0$ (which is the case if
$s^{\pm}_{\alpha}\geq(s^{i}_{\alpha})^{\pm}+1$) or
$s^{\pm}_{\alpha}\leq(s^{i}_{\alpha})^{\pm}$ and hence by (4.10) we must have
equality $s^{\pm}_{\alpha}=(s^{i}_{\alpha})^{\pm}$. In the latter case we have
$Y^{s}v_{i}=0$ if $s\geq(m^{i}_{\alpha})^{\pm}+1$. So there must exist a tuple
$(j_{1},\dots,j_{k})$ in the above sum which has the following conditions:

  1. (1)

$j_{1}+\cdots+j_{k}=m$

  2. (2)

$j_{i}=0$ if $s^{\pm}_{\alpha}\neq(s^{i}_{\alpha})^{\pm}$

  3. (3)

$j_{i}\leq(m^{i}_{\alpha})^{\pm}$ if $s^{\pm}_{\alpha}=(s^{i}_{\alpha})^{\pm}$

Let $A=\\{1\leq i\leq k:s^{\pm}_{\alpha}=(s^{i}_{\alpha})^{\pm}\\}$ and set
$I_{j}=\big{[}p_{0}+\cdots+p_{j-1}+1,p_{0}+\cdots+p_{j}\big{]}$. From Lemma
Lemma we see that an element $i\in I_{j}$ lies in $A$ if and only if

$0\leq j\leq d_{\alpha}r\ \ \text{and}\ \ (m_{\alpha}^{i})^{\pm}+j\leq
d_{\alpha}r.$

In particular, if $i\in I_{j}\cap A$ we have the equality
$(m^{i_{1}}_{\alpha})^{\pm}=(m^{i}_{\alpha})^{\pm}+j$ and therefore

$m=\sum\limits_{i\in A}j_{i}=\sum_{j=0}^{t}\ \sum_{i\in A\cap
I_{j}}j_{i}\leq\sum_{j=0}^{t}|A\cap
I_{j}|\big{(}(m^{i_{1}}_{\alpha})^{\pm}-j\big{)}$ (4.11)

On the other hand from (4.9) we get

$m=k(m^{i_{1}}_{\alpha})^{\pm}+1-(p_{1}+2p_{2}+\cdots+tp_{t}).$

So if we substitute this into inequality (4.11) we end up in a contradiction
since $|A\cap I_{j}|\leq p_{j}$:

$p_{0}(m^{i_{1}}_{\alpha})^{\pm}+p_{1}\big{(}(m^{i_{1}}_{\alpha})^{\pm}-1\big{)}+\cdots+p_{t}\big{(}(m^{i_{1}}_{\alpha})^{\pm}-t\big{)}<\sum_{j=0}^{t}|A\cap
I_{j}|\big{(}(m^{i_{1}}_{\alpha})^{\pm}-j\big{)}.$

Relations in (4.3): Now suppose that $\alpha\in R^{+}(\mu)$. The only non-
trivial relation is given by

$(x_{\alpha}^{-}\otimes
t)^{\mathrm{max}\\{0,\mu(h_{\alpha})-d_{\alpha}kr\\}+1}(v_{1}\otimes\cdots\otimes
v_{k})=0.$ (4.12)

Suppose that $\mu(h_{\alpha})\leq kd_{\alpha}r$ then $s_{\alpha}^{-}\leq 1$
which implies together with (4.10) that $(s^{i}_{\alpha})^{-}\leq 1$ for each
$1\leq i\leq k$. Hence the above relation is immediate in this case. Suppose
from now on that $\mu(h_{\alpha})>kd_{\alpha}r$ and note that
$\mu_{i}(h_{\alpha})\geq d_{\alpha}r$ for all $1\leq i\leq k$ by (4.8). Again
we write $Y=(x_{\alpha}^{-}\otimes t)$ and consider a typical summand on the
left hand side of (4.12)

$Y^{j_{1}}v_{1}\otimes\dots\otimes Y^{j_{k}}v_{k}.$

So if for all $i\in\\{1,\dots,n\\}$ the inequality
$j_{i}\leq\max\\{0,\mu_{i}(h_{\alpha})-d_{\alpha}r\\}$ holds we would get a
contradiction togehter with (4.8):

$\mu(h_{\alpha})-kd_{\alpha}r+1=j_{1}+\dots+j_{k}\leq\sum_{i=1}^{k}\max\\{0,\mu_{i}(h_{\alpha})-d_{\alpha}r\\}\leq\mu(h_{\alpha})-kd_{\alpha}r$

Relations in (4.4): The first one follows directly from property (4.5) and the
second relation is standard. This completes the proof.

### 4.5.

Given $\mu\in P$ and $k\in\mathbb{N}$, there are a few natural candidates for
$\boldsymbol{\mu}$ being $1$-admissible arising from [16] which we will
explain now. Let $\sigma\in W$ such that $\sigma(\mu)\in P^{+}$ and consider

$\mathcal{P}^{+}(\sigma(\mu),k)=\\{(\lambda_{1},\dots,\lambda_{k})\in(P^{+})^{\times{k}}:\lambda_{1}+\cdots+\lambda_{k}=\sigma(\mu)\\}.$

This set is partially ordered and has a unique maximal element
$\boldsymbol{\lambda}^{\mathbf{max}}$ maximizing the dimension of tensor
products of the corresponding irreducible representations [16]. This unique
element can be explicitly defined (see [16, Section 3.1]) and gives at least a
pre-admissible tuple

$\boldsymbol{\mu}=(\sigma^{-1}(\lambda_{1}),\dots,\sigma^{-1}(\lambda_{k}))\in\mathcal{P}(\mu,k),\
\ \boldsymbol{\lambda}^{\mathbf{max}}=(\lambda_{1},\dots,\lambda_{k}).$ (4.13)

However this element is generically only $1$-admissible in type A which we
illustrate in the next example.

######  Example.

  1. (1)

Let $\mathfrak{g}$ be of type A and $\mu\in P$, $\sigma\in W$ and
$\boldsymbol{\lambda}^{\mathbf{max}}\in\mathcal{P}^{+}(\sigma(\mu),k)$ as
above. From the explicit description (see [16, Section 3.1]) we see that
$\boldsymbol{\lambda}^{\mathbf{max}}$ satisfies

$|\lambda_{i}(h_{\alpha})-\lambda_{j}(h_{\alpha})|\leq 1,\ \ \forall\alpha\in
R^{+}.$ (4.14)

Hence the element
$\boldsymbol{\mu}=(\sigma^{-1}(\mu_{1}),\dots,\sigma^{-1}(\mu_{k}))$ is
$r$-admissible for all $r\geq 1$ (we have $t_{\alpha}\leq 1$ in this case).

  2. (2)

Let $\mathfrak{g}$ be of type $C_{2}$, $k=2$, and
$\mu=2\varpi_{1}+\varpi_{2}$. The unique maximal element from [16] is given by
$(\varpi_{1}+\varpi_{2},\varpi_{1})$, however (4.7) is violated for $r=1$ and
$\alpha=\alpha_{1}+\alpha_{2}$. So it is not 1-admissible, but 2-admissible.
Nevertheless, $(2\varpi_{1},\varpi_{2})$ provides a 1-admissible element which
is also 2-admissible.

So the previous example shows that in type A the unique maximal element from
[16] gives rise to a 1-admissible element in $\mathcal{P}(\mu,k)$. We
conjecture that there is always a $1$-admissible element. For now, we can only
prove the following.

######  Lemma.

Let $\boldsymbol{\mu}$ be the pre-admissible element from (4.13) and let
$\mathfrak{g}$ be of non-exceptional type. Then $\boldsymbol{\mu}$ is
$1$-admissible or there exists $\alpha\in R^{\mp}(\mu)$ such that
$t_{\alpha}=2=(m_{\alpha}^{i_{1}})^{\pm}+1$. If additionally

$\mp\mu_{i_{1}}(h_{\alpha})\equiv(d_{\alpha}+1)\mod 2d_{\alpha}$

for those $\alpha$, then $\boldsymbol{\mu}$ is $2$-admissible.

###### Proof.

For $\mathfrak{g}$ of non-exceptional type we always have $t_{\alpha}\leq 2$
since (4.14) holds for all roots $\alpha\in R^{+}$ such that
$\varpi_{j}(h_{\alpha})\leq 1$ for all $j\in I$ (see [16, Section 3.1]) and
every coroot can be written as a sum of two such coroots. Assume that
$\boldsymbol{\mu}$ is not $1$-admissible. Since $t_{\alpha}\leq 1$ for all
$\alpha\in R^{+}$ would imply $1$-admissibility, we deduce that there exists a
root $\alpha\in R^{+}$ satisfying $t_{\alpha}=2$. Among those $\alpha$ with
$t_{\alpha}=2$ there must exist at least one which satisfies
$(m_{\alpha}^{i_{1}})^{\pm}=1$; otherwise we would again get
$1$-admissibility. This show the first part of the statement. Now let
$\alpha\in R^{\mp}(\mu)$ such that $t_{\alpha}=2=(m_{\alpha}^{i_{1}})^{\pm}+1$
and write

$\mp\mu_{i_{1}}(h_{\alpha})=q_{\alpha}d_{\alpha}+(m_{\alpha}^{i_{1}})^{\pm}=q^{\prime}_{\alpha}2d_{\alpha}+(d_{\alpha}+1)$

where the second equality follows from the additional assumption. So the new
remainder is $(d_{\alpha}+1)$ with respect to $2d_{\alpha}$ and hence greater
or equal to $2$. Now if we have an $\alpha\in R^{\mp}(\mu)$ with
$t_{\alpha}=2$ and $(m_{\alpha}^{i_{1}})^{\pm}\geq 2$, the new remainder can
be calculated as follows. We write

$\mp\mu_{i_{1}}(h_{\alpha})=q_{\alpha}d_{\alpha}+(m_{\alpha}^{i_{1}})^{\pm}$

as above. If $q_{\alpha}\equiv 0\mod 2$, the remainder with respect to
$2d_{\alpha}$ does not change or $q_{\alpha}\equiv 1\mod 2$ and the new
remainder is $(m_{\alpha}^{i_{1}})^{\pm}+d_{\alpha}\leq 2d_{\alpha}.$ In both
case we increase the remainder and we have proved the following. For all
$\alpha\in R^{\mp}(\mu)$ we have $t_{\alpha}\leq 1$ or $t_{\alpha}=2$ in which
case the remainder with respect to $2d_{\alpha}$ is greater or equal to $2$.
This shows that (4.7) and (4.8) hold for $r=2$ and $\boldsymbol{\mu}$ is
$2$-admissible. ∎

### 4.6.

We assume in this subsection that $\mathfrak{g}$ is of simply-laced type. As
mentioned earlier, we can give a lower bound for the dimension of certain
representations of the quantum affine algebra
$\widehat{\mathbf{U}}_{q}(\mathfrak{g})$. We denote by $\mathcal{P}^{+}$ the
monoid generated by $\boldsymbol{\varpi}_{i,a}$ where $i\in I$ and
$a\in\mathbb{C}(q)^{\times}$ and let $\mathcal{P}_{\mathbb{Z}}^{+}$ the
submonoid generated by the elements $\boldsymbol{\varpi}_{i,a}$ with $a\in
q^{\mathbb{Z}}$. It is well-known [10, 11] that the isomorphism classes of
irreducible finite-dimensional representations of
$\widehat{\mathbf{U}}_{q}(\mathfrak{g})$ are indexed by elements in
$\mathcal{P}^{+}$. For $\boldsymbol{\pi}\in\mathcal{P}_{\mathbb{Z}}^{+}$ we
denote the corresponding finite-dimensional irreducible module by
$V(\boldsymbol{\pi})$ and its graded limit by $L(\boldsymbol{\pi})$ (for a
precise definition and comments on the existence see [6, Section 1.8] for
example). Note that we have a weight map
$\mathrm{wt}:\mathcal{P}^{+}\rightarrow P^{+}$ given by extending the
assignment $\mathrm{wt}(\boldsymbol{\varpi}_{i,a})=\varpi_{i}$ to a morphism
of monoids.

######  Corollary.

Let $\mathfrak{g}$ be of simply-laced type. Assume that
$\boldsymbol{\pi}^{1},\dots,\boldsymbol{\pi}^{k}\in\mathcal{P}_{\mathbb{Z}}^{+}$
are such that

  1. (1)

There exists a map of $\widehat{\mathbf{U}}_{q}(\mathfrak{g})$-modules

$V(\boldsymbol{\pi})\rightarrow V(\boldsymbol{\pi}^{1})\otimes\cdots\otimes
V(\boldsymbol{\pi}^{k}),\ \
\boldsymbol{\pi}=\boldsymbol{\pi}^{1}\cdots\boldsymbol{\pi}^{k}$

  2. (2)

$V(\boldsymbol{\pi}^{i})$ is a quantum Weyl module for all $1\leq i\leq k$

  3. (3)

The element
$(w_{0}\mathrm{wt}(\boldsymbol{\pi}^{1}),\dots,w_{0}\mathrm{wt}(\boldsymbol{\pi}^{k}))$
is $1$-admissible.

Then $\dim
V(\boldsymbol{\pi})\geq\dim\mathbf{D}_{w_{0}\mathrm{wt}(\boldsymbol{\pi})}^{k}$.

###### Proof.

From property (1) and [35, Lemma 2.20] we know that there exists a map
$L(\boldsymbol{\pi})\rightarrow L(\boldsymbol{\pi}^{1})\otimes\cdots\otimes
L(\boldsymbol{\pi}^{k})$ on the level of graded limits which is surjective
onto the image. By property (2) we obtain an isomorphism between
$L(\boldsymbol{\pi}^{i})$ and
$\mathbf{D}_{w_{0}\mathrm{wt}(\boldsymbol{\pi}^{i})}^{1}$. Hence the image is
isomorphic to $\mathbf{D}_{w_{0}\mathrm{wt}(\boldsymbol{\pi})}^{k}$ by
property (3) and Theorem 3. The rest follows from the fact that the graded
limit preserves dimensions. ∎

Although the conditions of the above corollary might look strong, there are
sufficient conditions worked out for property (1) and (2) to hold (see for
example [8, 12]). Using the formula for the denominator of the normalized
$R$-matrix (see [20]) and Hernandez reduction theorem [22] there are even
necessary and sufficient conditions for property (2).

### 4.7.

It is still an open problem to determine the classical decompositions of
Demazure modules with respect to a maximal semi-simple Lie subalgebra
$\mathfrak{g}_{0}\subseteq\mathfrak{g}$. The problem in the
$\mathfrak{g}$-stable case has some solutions and is easier to understand for
rectangular anti-dominant weights (i.e. $\mu\in-k\cdot P^{+}$); see for
example [7, 17]. However, the question seems difficult for prime Demazure
modules (see [1, 41] for some progress). In the $\mathfrak{g}$-stable case
there is another way to attack this problem using tensor products of Kirillov-
Reshetikhin crystals.

If $B^{r,s}$ denotes the KR crystal of classical highest weight $s\varpi_{r}$
and $B=\bigotimes_{j=1}^{N}B^{r_{j},s_{j}}$ is a tensor product of KR crystals
bounded by $k$, i.e. $\lceil\frac{s_{j}}{c_{r_{j}}}\rceil\leq k$ for all
$1\leq j\leq N$ (see [40] for the table defining the integers $c_{r}$), then
the proof of [33, Theorem 3.8] shows for nonexceptional types that the
Demazure crystal of $\mathbf{D}_{\mu}^{k}$ for anti-dominant
$\mu=\sum_{j=1}^{N}r_{j}w_{0}\varpi_{j}$ can be obtained by the following
steps.

$\bullet$ Remove all edges in $B$ which are contained in the length $k$ head
of a $0$-string; following [33] we call this subcrystal by $\tilde{D}_{k}(B)$.

$\bullet$ Then, up to certain $0$-arrows, the Demazure crystal of
$\mathbf{D}_{\mu}^{k}$ is the connected component $D_{k}(B)$ of
$\tilde{D}_{k}(B)$ containing the unique element of weight $\mu$.

Removing all $0$-arrows of $D_{k}(B)$ in a third step, provides the
$\mathfrak{g}$-decomposition of $\mathbf{D}_{\mu}^{k}$. Finding very explicit
combinatorial models for these subcrystals is another problem.

Now if $\mathfrak{g}$ and $\mu$ are arbitrary such that
$R^{-}(\mu)\neq\emptyset$, we consider the maximal semi-simple Lie subalgebra
$\mathfrak{g}_{0}\subseteq\mathfrak{g}$ such that the module
$\mathbf{D}_{\mu}^{k}$ is $\mathfrak{g}_{0}$-stable. An interesting problem is
to describe explicitly the Demazure crystal of $\mathbf{D}_{\mu}^{k}$ as the
connected component in the corresponding tensor product of Demazure crystals
containing the unique element of weight $\mu$. We emphasize here that the way
of splitting $\mu$ is very important. We will address this problem and precise
statements elsewhere and discuss only an example.

######  Example.

Let $\mathfrak{g}$ be of type $A_{2}$ and $\mu=\varpi_{1}-2\varpi_{2}$. Then

$V_{s_{2}s_{1}}(\Lambda_{1}+\Lambda_{2})=\mathbf{D}^{2}_{\mu}\hookrightarrow\mathbf{D}^{1}_{-\varpi_{2}}\otimes\mathbf{D}^{1}_{\varpi_{1}-\varpi_{2}}=V_{s_{2}s_{1}}(\Lambda_{1})\otimes
V_{s_{2}}(\Lambda_{2}).$

We know from [27] that the crystal base $B_{w}(\Lambda)$ of $V_{w}(\Lambda)$
is the full subgraph of the crystal base of $V(\Lambda)$ whose vertices
consist precisely of those elements that are reachable by raising operators
from the unique element of weight $w\Lambda$. In this particular case we have,
as sets,

$B_{s_{2}s_{1}}(\Lambda_{1})=\big{\\{}v_{\Lambda_{1}},\tilde{f}_{1}v_{\Lambda_{1}},\tilde{f}_{2}\tilde{f}_{1}v_{\Lambda_{1}}\big{\\}}\
\
B_{s_{2}}(\Lambda_{2})=\\{v_{\Lambda_{2}},\tilde{f}_{2}v_{\Lambda_{2}}\big{\\}}$

Now, tensoring the corresponding crystals

$\bullet$$\bullet$$\bullet$$\otimes$$\bullet$$\bullet$$1$$2$$2$

and considering the connected component containing the element of weight $\mu$
gives the graph

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$1$$2$$2$$1$$2$

Deleting all arrows whose label is not $2$ gives the $\mathfrak{sl}_{2}$
decomposition (corresponding to the simple root $\alpha_{2}$) of
$\mathbf{D}^{2}_{\mu}$. In particular, $\dim(\mathbf{D}^{2}_{\mu})=5$.

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Each odd instanton bundle ${\mathcal{E}}$ on ${\mathbb{P}^{3}}$ with quantum
number $k$ is the cohomology of a monad ${\mathcal{C}}^{\bullet}$ of the form

$0\longrightarrow(H\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}(h))^{\vee}\mathbin{\smash{\mathop{\longrightarrow}\limits^{u\psi^{\vee}(-h)}}}(K\otimes\Omega_{{\mathbb{P}^{3}}}(2h))^{\vee}\oplus
K\otimes\Omega_{{\mathbb{P}^{3}}}(h)\mathbin{\smash{\mathop{\longrightarrow}\limits^{\psi}}}H\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}\longrightarrow
0,$ (9.1)

where $H$ and $K$ are vector spaces with

$\dim(H)=\frac{3k}{2}-1,\qquad\dim(K)=\frac{k}{2},$

and

$u\colon(K\otimes\Omega_{{\mathbb{P}^{3}}}(2h)\oplus(K\otimes\Omega_{{\mathbb{P}^{3}}}(h))^{\vee})(-h)\to(K\otimes\Omega_{{\mathbb{P}^{3}}}(2h))^{\vee}\oplus
K\otimes\Omega_{{\mathbb{P}^{3}}}(h)$

is skew–symmetric.

Conversely, if the cohomology ${\mathcal{E}}$ of the monad
${\mathcal{C}}^{\bullet}$ is a vector bundle, then ${\mathcal{E}}$ is an odd
instanton bundle with $c_{2}({\mathcal{E}})=k$ on ${\mathbb{P}^{3}}$.

###### Proof.

If ${\mathcal{E}}$ is an odd instanton bundle on ${\mathbb{P}^{3}}$, then it
is the cohomology in degree $0$ of a complex ${{\mathcal{C}}}^{\bullet}$ with
$i^{th}$–module

${{\mathcal{C}}}^{i}:=\bigoplus_{p+q=i}H^{q}\big{(}{\mathcal{E}}(ph)\big{)}\otimes\Omega^{-p}_{{\mathbb{P}^{3}}}(-ph)$

where, as usual,
$\Omega^{t}_{{\mathbb{P}^{3}}}:=\wedge^{t}\Omega_{{\mathbb{P}^{3}}}$, thanks
to [2, Beilinson’s Theorem (Strong form)]. Our first task is to prove that
$h^{q}\big{(}{\mathcal{E}}(ph)\big{)}$ is the number in position $(p,q)$ in
table 2.

| 0 | 0 | 0 | 0 | $q=3$
---|---|---|---|---|---
| $\frac{3k}{2}-1$ | $\frac{k}{2}$ | 0 | 0 | $q=2$
| 0 | 0 | $\frac{k}{2}$ | $\frac{3k}{2}-1$ | $q=1$
| 0 | 0 | 0 | 0 | $q=0$
| $p=-3$ | $p=-2$ | $p=-1$ | $p=0$ |
Table 2. The values of $h^{q}\big{(}{\mathcal{E}}(ph)\big{)}$ for
${\mathbb{P}^{3}}$

Indeed, thanks to Proposition 2.5 the cohomology of ${\mathcal{E}}$ is natural
in degree $0$, hence $h^{q}\big{(}{\mathcal{E}}\big{)}=0$ when $q\neq 1$.
Moreover,
$h^{2}\big{(}{\mathcal{E}}(-h)\big{)}=h^{1}\big{(}{\mathcal{E}}(-2h)\big{)}=0$
by definition. The vanishing $h^{0}\big{(}{\mathcal{E}}\big{)}=0$ implies
$h^{0}\big{(}{\mathcal{E}}(ph)\big{)}=0$ for $-3\leq p\leq 0$. Equality (2.4)
then returns $h^{3}\big{(}{\mathcal{E}}(-ph)\big{)}=0$ in the same range. For
the same reason
$h^{q}\big{(}{\mathcal{E}}(ph)\big{)}=h^{3-q}\big{(}{\mathcal{E}}((-p-3)h)\big{)}$
for $p=0,-1$ and $q=1,2$. In order to compute
$h^{1}\big{(}{\mathcal{E}}\big{)}=h^{2}\big{(}{\mathcal{E}}(-3h)\big{)}$ and
$h^{1}\big{(}{\mathcal{E}}(-h)\big{)}=h^{2}\big{(}{\mathcal{E}}(-2h)\big{)}$
we apply Equality (2.6) to ${\mathcal{E}}$ and ${\mathcal{E}}(-h)$
respectively.

Let $H:=H^{1}\big{(}{\mathcal{E}}\big{)}$ and
$K:=H^{1}\big{(}{\mathcal{E}}(-h)\big{)}$. Equality (2.4) implies

$H^{2}\big{(}{\mathcal{E}}(-3h)\big{)}\cong
H^{1}\big{(}{\mathcal{E}}\big{)}^{\vee},\qquad
H^{2}\big{(}{\mathcal{E}}(-2h)\big{)}\cong
H^{1}\big{(}{\mathcal{E}}(-h)\big{)}^{\vee},$

hence the existence of a monad ${\mathcal{C}}^{\bullet}$ of the form

$0\longrightarrow
H^{\vee}\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)\mathbin{\smash{\mathop{\longrightarrow}\limits^{\eta}}}(K\otimes\Omega_{{\mathbb{P}^{3}}}(2h))^{\vee}\oplus
K\otimes\Omega_{{\mathbb{P}^{3}}}(h)\mathbin{\smash{\mathop{\longrightarrow}\limits^{\psi}}}H\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}\longrightarrow
0,$

follows directly from the isomorphism
$\Omega^{t}_{{\mathbb{P}^{3}}}(4h)\cong(\Omega^{3-t}_{{\mathbb{P}^{3}}})^{\vee}$
and Table 2 above.

The monads ${\mathcal{C}}^{\bullet}$ and $({\mathcal{C}}^{\bullet})^{\vee}$
satisfy the hypothesis of [46, Lemma II.4.1.3], hence the skew–symmetric
isomorphism ${\mathcal{E}}^{\vee}(-h)\cong{\mathcal{E}}$ induces an
isomorphism of the monads
$({\mathcal{C}}^{\bullet})^{\vee}(-h)\cong{\mathcal{C}}^{\bullet}$ such that
the map $u\colon{\mathcal{C}}_{0}^{\vee}(-h)\to{\mathcal{C}}_{0}$ is
skew–symmetric. The same argument described after the proof of [46, Corollary
2 of Lemma II.4.1.3] finally yields Monad (9.1).

Conversely, if the cohomology ${\mathcal{E}}$ of Monad (9.1) is a vector
bundle, then it is easy to check that $\mathrm{rk}({\mathcal{E}})=2$,
$c_{1}({\mathcal{E}})=-1$, $c_{2}({\mathcal{E}})=k$. We have to check that
$h^{0}\big{(}{\mathcal{E}}\big{)}=h^{1}\big{(}{\mathcal{E}}(-2h)\big{)}=0$. To
this purpose consider the two short exact sequences

$\begin{gathered}0\longrightarrow\mathcal{K}\longrightarrow(K\otimes\Omega_{{\mathbb{P}^{3}}}(2h))^{\vee}\oplus
K\otimes\Omega_{{\mathbb{P}^{3}}}(h)\longrightarrow
H^{\vee}\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}\longrightarrow 0,\\\
0\longrightarrow
H\otimes{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)\longrightarrow\mathcal{K}\longrightarrow{\mathcal{E}}\longrightarrow
0.\end{gathered}$ (9.2)

The cohomology of Sequences (9.2) easily yields

$\displaystyle 2h^{0}\big{(}{\mathcal{E}}\big{)}\leq
kh^{0}\big{(}\Omega^{\vee}_{{\mathbb{P}^{3}}}(-2h)\big{)}+kh^{0}\big{(}\Omega_{{\mathbb{P}^{3}}}(h)\big{)},$
$\displaystyle 2h^{1}\big{(}{\mathcal{E}}(-2h)\big{)}\leq
kh^{1}\big{(}\Omega^{\vee}_{{\mathbb{P}^{3}}}(-4h)\big{)}+kh^{1}\big{(}\Omega_{{\mathbb{P}^{3}}}(-h)\big{)}.$

The statement then follows by computing the cohomology of suitable twists of
the Euler exact sequence and its dual. ∎

######  Remark 9.2.

As pointed out in [23, Section 5.1], if ${\mathcal{E}}$ is an odd instanton
bundle on ${\mathbb{P}^{3}}$ with quantum number $k$, then it is the
cohomology of a monad of the form

$\displaystyle 0\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{3}}}(-2h)^{\oplus
k}\oplus{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)^{\oplus t_{0}}$
$\displaystyle\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)^{\oplus
k+t_{0}+1}\oplus{\mathcal{O}}_{{\mathbb{P}^{3}}}^{\oplus k+t_{0}+1}$
$\displaystyle\longrightarrow{\mathcal{O}}_{{\mathbb{P}^{3}}}^{\oplus
t_{0}}\oplus{\mathcal{O}}_{{\mathbb{P}^{3}}}(h)^{\oplus k}\longrightarrow 0$

where $t_{0}$ is the number of minimal generators of
$\bigoplus_{t\in\mathbb{Z}}H^{1}\big{(}{\mathcal{O}}_{{\mathbb{P}^{3}}}(t)\big{)}$
lying in $H^{1}\big{(}{\mathcal{O}}_{{\mathbb{P}^{3}}}\big{)}$.

We now turn the attention to even instanton bundles on $Q$. If ${\mathcal{E}}$
is even, then it is $\mu$–stable. Thus, we have the following well–known
result.

######  Proposition 9.3.

Each even instanton bundle ${\mathcal{E}}$ on $Q$ with
$c_{2}({\mathcal{E}})=k\ell$ is the cohomology of a monad
${\mathcal{C}}^{\bullet}$ of the form

$0\longrightarrow
H^{\vee}\otimes{\mathcal{O}}_{Q}(-h)\mathbin{\smash{\mathop{\longrightarrow}\limits^{u\psi^{\vee}}}}K\otimes{\mathcal{O}}_{Q}\mathbin{\smash{\mathop{\longrightarrow}\limits^{\psi}}}H\otimes{\mathcal{O}}_{Q}(h)\longrightarrow
0,$ (9.3)

where $H$ and $K$ are vector spaces with

$\dim(H)=\frac{k}{2},\qquad\dim(K)=k+2$

and

$u\colon K\otimes{\mathcal{O}}_{Q}\to K^{\vee}\otimes{\mathcal{O}}_{Q}$

is skew–symmetric.

Conversely, if the cohomology ${\mathcal{E}}$ of the monad
${\mathcal{C}}^{\bullet}$ is a vector bundle, then ${\mathcal{E}}$ is an even
instanton bundle with $c_{2}({\mathcal{E}})=k\ell$ on $Q$.

###### Proof.

For the existence of a monad see [48, Proposition 1.1]. We can deduce its
self–duality as in the proof of Theorem 9.1. In the same way we prove the
converse part of the statement. ∎

######  Remark 9.4.

As pointed out in [22, Theorem 3.5] a rank $2$ bundle ${\mathcal{E}}$ on $Q$
is the cohomology of Monad (9.3) if and only if there is at most one $i\geq 0$
such that $h^{i}\big{(}{\mathcal{E}}(-th)\big{)}\neq 0$ where $-2\leq t\leq
0$, at most one $i\geq 0$ such that
$h^{i}\big{(}{\mathcal{E}}\otimes{\mathcal{S}}(-2h)\big{)}\neq 0$ and
$h^{i}\big{(}{\mathcal{E}}\otimes{\mathcal{S}}(-h)\big{)}=0$ for all $i\geq
0$.

Thus Theorem 9.3 guarantees that all such sets of vanishings are automatically
satisfied by an even instanton bundle on $Q$. Notice that the first set of
vanishings above is exactly Proposition 2.5.

Thanks to [46, Corollary 2 of Theorem II.2.1.4] and [20, Proposition 5.3]
every instanton bundle ${\mathcal{E}}$ on either ${\mathbb{P}^{3}}$ or $Q$ is
always generically trivial. We conclude the section by dealing with the locus
of lines where the splitting of ${\mathcal{E}}$ is not balanced.

######  Definition 9.5.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$.

If ${\mathcal{E}}$ is an instanton bundle on $X$ with
$c_{1}({\mathcal{E}})=-\varepsilon h$, then a line $L\subseteq X$ is called
jumping line of ${\mathcal{E}}$ if
$h^{1}\big{(}{\mathcal{E}}((i_{X}+\varepsilon-2q_{X}^{\varepsilon}-1)h)\otimes{\mathcal{O}}_{L}\big{)}\neq
0$. The locus of jumping lines of ${\mathcal{E}}$ inside $\Lambda(X)$ will be
denoted by $J_{\mathcal{E}}$.

We say that ${\mathcal{E}}$ has expected splitting type if the codimension of
all the components of $J_{\mathcal{E}}$ inside $\Lambda(X)$ is
$1+\varepsilon$.

######  Example 9.6.

If $i_{X}\geq 3$ and ${\mathcal{E}}$ is an even instanton bundle, then [46,
Corollary 2 of Theorem II.2.1.4] and [20, Proposition 5.3] imply that
${\mathcal{E}}$ has expected splitting type. This is no longer true if
$i_{X}\leq 2$. Indeed in this case the projection $\chi\colon\mathscr{L}\to X$
can have disconnected fibres as pointed out in [38, Section 3.7], hence the
standard approach of [46, 20] does not work.

When $X={\mathbb{P}^{3}}$ and ${\mathcal{E}}$ is odd with quantum number $2$,
then ${\mathcal{E}}$ has expected splitting type, thanks to [31, Section 4].
As far as we know, there are no other general results even when $i_{X}\geq 3$:
indeed one cannot say anything even about the analogous problem for bundles on
${\mathbb{P}^{2}}$: see [32, Example 10.7.2].

Let $\widehat{\lambda}$ and $\widehat{\chi}$ the natural projections
$\Lambda(X)\times X$ on the first and second factor respectively. We will
denote by $\lambda$ and $\chi$ their restrictions to the universal line
$\mathscr{L}\subseteq\Lambda(X)\times X$. The following lemma follows
immediately by the above definition.

######  Lemma 9.7.

Let $X$ be a Fano threefold with very ample ${\mathcal{O}}_{X}(h)$.

If ${\mathcal{E}}$ is an instanton bundle on $X$ with
$c_{1}({\mathcal{E}})=-\varepsilon h$, then the locus
$J_{\mathcal{E}}\subseteq\Lambda(X)$ is the support of the sheaf
$R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}((i_{X}+\varepsilon-2q_{X}^{\varepsilon}-1)h)$.

###### Proof.

The open locus $\Lambda(X)\setminus J_{\mathcal{E}}$ is by definition 9.5 the
set of $L\in\Lambda(X)$ such that
$h^{1}\big{(}{\mathcal{E}}((i_{X}+\varepsilon-2q_{X}^{\varepsilon}-1)h)\otimes{\mathcal{O}}_{L}\big{)}=0$.
Thanks to [28, Corollary III.12.9], it is thus the locus where
$R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}((i_{X}+\varepsilon-2q_{X}^{\varepsilon}-1)h))=0$.
∎

We first deal with instanton bundles on ${\mathbb{P}^{3}}$. Recall that
$\Lambda({\mathbb{P}^{3}})=G(2,4)$ is the Grassmannian of lines in
${\mathbb{P}^{3}}$ which is a quadric in ${\mathbb{P}^{5}}$ endowed with the
induced natural polarization ${\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1)$.
In this case $J_{\mathcal{E}}$ is the support of the sheaf
$R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}$.

The quadric $\Lambda({\mathbb{P}^{3}})$ contains two families of planes which
are classically known as $\alpha$–planes (the lines through a fixed point) and
$\beta$–planes (the lines contained in a fixed plane). Both these families of
planes correspond to the non–zero sections of the two rank $2$ spinor bundles
on $\Lambda({\mathbb{P}^{3}})$ (see [47]).

As pointed out in [26, Section 1.5], $H^{*}(\Lambda({\mathbb{P}^{3}}))$ is
generated by the Schubert cycles. More precisely,
$H^{2}(\Lambda({\mathbb{P}^{3}}))$ is generated by $\sigma_{1}$ (representing
the set of lines in $\Lambda({\mathbb{P}^{3}})$ intersecting a fixed one),
$A^{2}(\Lambda({\mathbb{P}^{3}}))$ by $\sigma_{2}$ and $\sigma_{1,1}$
(representing respectively the classes of the $\alpha$–planes and
$\beta$–planes in $\Lambda({\mathbb{P}^{3}})$). Moreover, the relation
$\sigma_{1}^{2}=\sigma_{2}+\sigma_{1,1}$ holds inside
$\Lambda({\mathbb{P}^{3}})$.

Let ${\mathcal{S}}$ be the spinor bundle corresponding to the $\alpha$–planes.

######  Proposition 9.8.

Let ${\mathcal{E}}$ be an odd instanton bundle on ${\mathbb{P}^{3}}$ with
quantum number $k$.

If ${\mathcal{E}}$ has expected splitting type, then
$J_{{\mathcal{E}}}\subseteq\Lambda({\mathbb{P}^{3}})$ is endowed with a sheaf
$\mathscr{G}$ fitting into

$\displaystyle 0\longrightarrow{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-k)$
$\displaystyle\longrightarrow(K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\oplus(K\otimes{\mathcal{S}})$
(9.4)
$\displaystyle\mathbin{\smash{\mathop{\longrightarrow}\limits^{\Psi}}}H\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}\longrightarrow\mathscr{G}\longrightarrow
0.$

Moreover, the class of $J_{\mathcal{E}}$ in $H^{*}(\Lambda({\mathbb{P}^{3}}))$
is

$\frac{k^{2}}{2}\sigma_{2}+\frac{k(k-1)}{2}\sigma_{1,1}.$

###### Proof.

Splitting Monad (9.1) we obtain the exact sequences

$\begin{gathered}{\begin{aligned}
0\longrightarrow\chi^{*}\mathcal{K}\longrightarrow(K\otimes\chi^{*}\Omega_{{\mathbb{P}^{3}}}(2h))^{\vee}&\oplus(K\otimes\chi^{*}\Omega_{{\mathbb{P}^{3}}}(h))\\\
&\longrightarrow
H\otimes\chi^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}\longrightarrow
0,\end{aligned}}\\\ 0\longrightarrow
H^{\vee}\otimes\chi^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)\longrightarrow\chi^{*}\mathcal{K}\longrightarrow\chi^{*}{\mathcal{E}}\longrightarrow
0.\end{gathered}$ (9.5)

As in the previous proof we apply the functor $\lambda_{*}$ to the two above
sequences. In particular we have to identify for all $i\geq 0$ the sheaves
$R^{i}\lambda_{*}\chi^{*}\Omega^{\vee}_{{\mathbb{P}^{3}}}(-2h)$,
$R^{i}\lambda_{*}\chi^{*}\Omega_{{\mathbb{P}^{3}}}(h)$,
$R^{i}\lambda_{*}\chi^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}(-th)$ where $t=0,1$.

The structure sheaf of the universal line fits into an exact sequence of the
form

$\displaystyle
0\longrightarrow{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-1)\boxtimes{\mathcal{O}}_{{\mathbb{P}^{3}}}(-2h)$
$\displaystyle\longrightarrow{\mathcal{S}}\boxtimes{\mathcal{O}}_{{\mathbb{P}^{3}}}(-h)$
(9.6)
$\displaystyle\longrightarrow{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})\times{{\mathbb{P}^{3}}}}\longrightarrow{\mathcal{O}}_{\mathscr{L}}\longrightarrow
0.$

We will compute
$R^{i}\lambda_{*}\chi^{*}\Omega^{\vee}_{{\mathbb{P}^{3}}}(-2h)$,
$R^{i}\lambda_{*}\chi^{*}\Omega_{{\mathbb{P}^{3}}}(h)$,
$R^{i}\lambda_{*}\chi^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}(-th)$ where $t=0,1$
by applying the functor $\widehat{\lambda}_{*}$ to Sequence (9.6) tensored by
$\widehat{\chi}^{*}\Omega^{\vee}_{{\mathbb{P}^{3}}}(-2h)$,
$\widehat{\chi}^{*}\Omega_{{\mathbb{P}^{3}}}(h)$,
$\widehat{\chi}^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}(-th)$ where $t=0,1$,
making use of the projection formula [28, Exercise III.8.3] and taking into
account that
$\widehat{\lambda}_{*}\widehat{\mathcal{F}}\cong{\lambda}_{*}{\mathcal{F}}$
for each sheaf supported on $\mathscr{L}$.

Let ${\mathcal{G}}$ be a sheaf on ${\mathbb{P}^{3}}$. For each
$L\in\Lambda({\mathbb{P}^{3}})$ we denote by $k(L)$ the residue field of
${\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}$ at $L$. Trivially
$h^{i}\big{(}\widehat{\chi}^{*}{\mathcal{G}}\otimes
k(L)\big{)}=h^{i}\big{(}{\mathcal{G}}\big{)}$, hence the function $L\mapsto
h^{i}\big{(}\widehat{\chi}^{*}{\mathcal{G}}\otimes k(L)\big{)}$ is constant on
$\Lambda({\mathbb{P}^{3}})$. Thus [28, Corollary III.12.9] implies that

$R^{i}\widehat{\lambda}_{*}({\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-s)\boxtimes{\mathcal{G}})\cong{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-s)\otimes
R^{i}\widehat{\lambda}_{*}\widehat{\chi}^{*}{\mathcal{G}}\cong{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-s)\otimes
H^{i}\big{(}{\mathcal{G}}\big{)}$

In this way we deduce that

$R^{0}{\lambda}_{*}{\chi}^{*}{\mathcal{O}}_{{\mathbb{P}^{3}}}\cong{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})},\qquad
R^{0}{\lambda}_{*}{\chi}^{*}\Omega_{{\mathbb{P}^{3}}}^{\vee}(-2h)\cong{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-1),\qquad
R^{0}{\lambda}_{*}{\chi}^{*}\Omega_{{\mathbb{P}^{3}}}(h)\cong{\mathcal{S}},$

all the other $R^{i}$’s being zero. Replacing the above isomorphisms in the
first of Sequence (9.5) and taking into account that the second of such
sequences gives $R^{i}\lambda_{*}\chi^{*}\mathcal{K}\cong
R^{i}\lambda_{*}\chi^{*}{\mathcal{E}}$ we finally obtain the exact sequence

$\displaystyle 0\longrightarrow\lambda_{*}\chi^{*}{\mathcal{E}}$
$\displaystyle\longrightarrow(K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\oplus(K\otimes{\mathcal{S}})$
(9.7)
$\displaystyle\mathbin{\smash{\mathop{\longrightarrow}\limits^{\Psi}}}H\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}\longrightarrow
R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}\longrightarrow 0,$

where the map $\Psi$ is induced by $\psi^{\vee}(-h)$. In particular
$R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}$ has rank $1$. Since ${\mathcal{E}}$
has expected splitting type, then all the components of $J_{\mathcal{E}}$ have
codimension $2$. In this case
$c_{1}(R^{1}\lambda_{*}\chi^{*}{\mathcal{E}})=0$, hence computing it from
Sequence (9.7) we obtain
$\lambda_{*}\chi^{*}{\mathcal{E}}\cong{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(-k)$.
Taking into account such a latter isomorphism, Sequence (9.4) is exactly
Sequence (9.7).

In order to complete the proof, notice that the class of $J_{\mathcal{E}}$
inside $H^{*}(\Lambda({\mathbb{P}^{3}}))$ is

$\displaystyle-c_{2}({\mathcal{G}})$
$\displaystyle=c_{2}\left((K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\oplus(K\otimes{\mathcal{S}})\right)$
$\displaystyle=c_{2}\left((K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\right)+c_{2}\left(K\otimes{\mathcal{S}}\right)+c_{1}\left((K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\right)c_{1}\left(K\otimes{\mathcal{S}}\right).$

By direct computation one checks that

$\displaystyle
c_{1}\left((K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\right)=c_{1}\left(K\otimes{\mathcal{S}}\right)=-\frac{k}{2}\sigma_{1},$
$\displaystyle
c_{2}\left((K\otimes{\mathcal{O}}_{\Lambda({\mathbb{P}^{3}})}(1))^{\vee}\right)=\frac{k(k-2)}{8}\sigma_{1}^{2},\qquad
c_{2}\left(K\otimes{\mathcal{S}}\right)=\frac{k(k-2)}{8}\sigma_{1}^{2}+\frac{k}{2}\sigma_{2}.$

The statement then follows by combining the above equalities ∎

Now we turn our attention to the quadric $Q\subseteq{\mathbb{P}^{4}}$. Recall
that $\Lambda(Q)\cong{\mathbb{P}^{3}}$. In this case $J_{\mathcal{E}}$ is the
support of the sheaf $R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}(-h)$ and it is a
divisor (see Example 9.6).

######  Proposition 9.9.

Let ${\mathcal{E}}$ be an even instanton bundle on $Q$ with quantum number
$k$.

Then $J_{{\mathcal{E}}}\subseteq\Lambda(Q)$ is a divisor of degree $k$ endowed
with a sheaf $\mathscr{G}$ fitting into

$0\longrightarrow
H^{\vee}\otimes{\mathcal{O}}_{\Lambda(Q)}(-2)\mathbin{\smash{\mathop{\longrightarrow}\limits^{\Psi}}}H\otimes{\mathcal{O}}_{\Lambda(Q)}\longrightarrow\mathscr{G}\longrightarrow
0.$ (9.8)

Moreover, $J_{\mathcal{E}}$ is a hypersurface of degree $k$.

###### Proof.

Splitting Monad (9.3) we obtain the exact sequences

$\begin{gathered}0\longrightarrow\chi^{*}\mathcal{K}(-h)\longrightarrow
K\otimes\chi^{*}{\mathcal{O}}_{Q}(-h)\longrightarrow
H\otimes\chi^{*}{\mathcal{O}}_{Q}\longrightarrow 0,\\\ 0\longrightarrow
H^{\vee}\otimes\chi^{*}{\mathcal{O}}_{Q}(-2h)\longrightarrow\chi^{*}\mathcal{K}(-h)\longrightarrow\chi^{*}{\mathcal{E}}(-h)\longrightarrow
0.\end{gathered}$ (9.9)

As in the previous proof we apply the functor $\lambda_{*}$ to the two above
sequences. In particular we have to identify the sheaves
$R^{i}\lambda_{*}\chi^{*}{\mathcal{O}}_{Q}(-th)$ for $t=0,1,2$ and $i\geq 0$.

Recall that $\Lambda(Q)\cong{\mathbb{P}^{3}}$ is endowed with the standard
polarization ${\mathcal{O}}_{{\mathbb{P}^{3}}}(1)$ and
${\mathcal{O}}_{\mathscr{L}}$ fits into the exact sequence

$\displaystyle
0\longrightarrow{\mathcal{O}}_{\Lambda(Q)}(-2)\boxtimes{\mathcal{O}}_{Q}(-h)$
$\displaystyle\longrightarrow{\mathcal{O}}_{\Lambda(Q)}(-1)\boxtimes{\mathcal{S}}$
(9.10)
$\displaystyle\longrightarrow{\mathcal{O}}_{\Lambda(Q)\times{Q}}\longrightarrow{\mathcal{O}}_{\mathscr{L}}\longrightarrow
0$

where ${\mathcal{S}}$ is the unique spinor bundle on $Q$, whose non–zero
sections vanish exactly on the lines inside $Q$. We then compute
$R^{i}\lambda_{*}\chi^{*}{\mathcal{O}}_{Q}(-th)$ by applying the functor
$\widehat{\lambda}_{*}$ to Sequence (9.10) tensored by
$\widehat{\chi}^{*}{\mathcal{O}}_{Q}(-th)$. We then deduce

$R^{1}{\lambda}_{*}{\chi}^{*}{\mathcal{O}}_{Q}(-2h)\cong{\mathcal{O}}_{\Lambda(Q)}(-2),\qquad
R^{0}{\lambda}_{*}{\chi}^{*}{\mathcal{O}}_{Q}\cong{\mathcal{O}}_{\Lambda(Q)},$

all the other $R^{i}$’s being zero. Replacing the above isomorphisms in
Sequences (9.9) we finally obtain Sequence (9.8) where $\mathscr{G}\cong
R^{1}\lambda_{*}\chi^{*}{\mathcal{E}}(-h)$ and $\Psi$ is induced by the map
$\psi\colon H^{\vee}\otimes\chi^{*}{\mathcal{O}}_{Q}(-2h)\to
K\otimes\chi^{*}{\mathcal{O}}_{Q}(-h)$.

Finally observe that ${\mathcal{G}}$ is supported on the vanishing locus of
$\det(\Psi)$. The statement then follows by noticing that $\Psi$ is
represented by a ${k}/{2}$ square matrix of degree two homogenous forms (see
Sequence (9.8)). ∎

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Vincenzo Antonelli,
Dipartimento di Scienze Matematiche, Politecnico di Torino,
c.so Duca degli Abruzzi 24,
10129 Torino, Italy
e-mail<EMAIL_ADDRESS>

Gianfranco Casnati,
Dipartimento di Scienze Matematiche, Politecnico di Torino,
c.so Duca degli Abruzzi 24,
10129 Torino, Italy
e-mail<EMAIL_ADDRESS>

Ozhan Genc,
Faculty of Mathematics and Computer Science, Jagiellonian University,
ul. Łojasiewicza 6,
30-348 Kraków, Poland
e-mail<EMAIL_ADDRESS>