# Partial Order in the Normal Category Arising from Normal Bands

C S Preenu
University College Thiruvananthapuram
Research Centre University of Kerala  A R Rajan
Former Professor of Mathematics
University of Kerala  K S Zeenath
Former Professor of Mathematics
School of Distance Education
University of Kerala

###### Abstract

The notion of normal category was introduced by KSS Nambooripad in connection
with the study of the structure of regular semigroups using cross
connections[4]. It is an abstraction of the category of principal left ideals
of a regular semigroup. A normal band is a semigroup $B$ satisfying $a^{2}=a$
and $abca=acba$ for all $a,b,c\in B$. Since the normal bands are regular
semigroups, the category $\mathcal{L}(B)$ of principal left ideals of a normal
band $B$ is a normal category. One of the special properties of this category
is that the morphism sets admit a partial order compatible with the
composition of morphisms. In this article we derive several properties of this
partial order and obtain a new characterization of this partial order.

Keywords: Partial order, Normal categories, Normal bands.
AMS Subject Classification : 20M17

##  1 Preliminaries

In this section we discuss some basic concepts of normal categories, normal
bands and the normal category derived from normal bands. We follow [2, 4] for
other notations and definitions which are not mentioned.

###  1.1 Normal Categories

We begin with the category theory, some basic definitions and notations. In a
category $C$, the class of objects is denoted by $vC$ and $C$ itself is used
to denote the morphism class. The collection of all morphisms from $a$ to $b$
in $C$ is denoted by $C(a,b)$ and $1_{a}$ denote the identity morphism in
$C(a,a)$. A morphism $f\in C$ is said to be a monomorphism(epimorphism) if
$fg=fh$($gf=hf$) implies $g=h$ for any two morphisms $g,h\in C$. A morphism
$f\in C(a,b)$ is said to be right(left) invertible if there exists $g\in
C(b,a)$ such that $fg=1_{a}$($gf=1_{b}$). An isomorphism means a morphism
which is both left and right invertible.

Let $S$ be a regular semigroup. A right translation of $S$ is a function
$\phi$ from $S$ to $S$ such that $(xy)\phi=x((y)\phi)$ for all $x,y\in S$.
Similarly a function $\psi:S\to S$ is called a left translation if
$\psi(xy)=(\psi(x))y$ for all $x,y\in S$. For a fixed $u\in S$, the map
$\rho_{u}:x\mapsto xu$ for $x\in S$ is a right translation. This map can be
viewed as a function from $S$ to the principal left ideal $Su$ of $S$. Since
$S$ is regular, we can find an idempotent $f\in S$ such that $Su=Sf$. In
particular if we choose $u\in eSf$ the restriction of $\rho_{u}$ to $Se$,
denoted by $\rho(e,u,f)$, can be viewed as a function from $Se$ to $Sf$. If
$\rho(e,u,f):Se\to Sf$ and $\rho(g,v,h):Sg\to Sh$ are such that $Sf=Sg$, the
composition of these maps will be $\rho(e,uv,h)$. Now we arrive at a category
structure, $\mathcal{L}(S)$, with objects are the principal left ideals of $S$
and morphisms are of the form $\rho(e,u,f):Se\to Sf$ where $u\in eSf$. The
usual set inclusion of the principal left ideals will induce a partial order
in the objects class of $\mathcal{L}(S)$. Also $Se\subseteq Sf$ implies $ef=e$
and hence the morphism $\rho(e,e,f)$ is an inclusion map from $Se$ to $Sf$ and
is denoted by $j(Se,Sf)$. The category $\mathcal{L}(S)$ become a normal
category whose definition is given below.

A preorder is a category $P$ such that each of the hom sets contain at most
one morphism. We write $a\leq b$ whenever $P(a,b)\neq\emptyset$.

######  Definition 1.1 ([8]).

A category with normal factorization is a small category $C$ with the
following properties.

  1. 1.

The vertex set $vC$ of $C$ is a partially ordered set such that whenever
$a\leq b$ in $vC$, there is a monomorphism $j(a,b):a\to b$ in $C$. This
morphism is called the inclusion from $a$ to $b$.

  2. 2.

$j:(vC,\leq)\to C$ is a functor from the preorder $(vC,\leq)$ to $C$ which
maps $a\leq b$ to $j(a,b)$ for $a,b\in vC$.

  3. 3.

For $a,b\leq c$ in $vC$, if

$j(a,c)=fj(b,c)$

for some $f:a\to b$ then $a\leq b$ and $f=j(a,b)$.

  4. 4.

Every morphism $j(a,b):a\to b$ has a right inverse $q:b\to a$ such that

$j(a,b)q=1_{a}$

Such a morphism $q$ is called a retraction in $C$.

  5. 5.

Every morphism $f$ in $C$ has a factorization

$f=quj$

where $q$ is a retraction, $u$ is an isomorphism and $j$ is an inclusion. Such
a factorization is called normal factorization in $C$.

######  Proposition 1.1 ([4]).

Let $C$ is a category with normal factorization. If
$f=quj=q^{\prime}u^{\prime}j^{\prime}$ are two normal factorizations of $f\in
C$ then $j=j^{\prime}$ and $qu=q^{\prime}u^{\prime}$.

In this case $f^{0}=qu$ is called the epimorphic component of $f$. The
codomain of $f^{\circ}$ is called the image of $f$ and is denoted by
$\text{im}\,f$

Corresponding to a right translation $\rho_{u}$ from $S$ to $Su$, we get
morphisms from each of the principal left ideals of $S$ to $Su$ by restricting
the $\rho_{u}$. This phenomena is similar to the concept of cones in the
category theory[2]. Now we define normal cones and normal categories.

######  Definition 1.2.

[4] Let $C$ be a category with normal factorization. A cone $\gamma$ with
vertex $a\in vC$ is a function from $vC$ to $C$ satisfying the following

  1. 1.

$\gamma(c)\in C(c,a)$ for all $c\in vC$

  2. 2.

If $c_{1}\subseteq c_{2}$ then $\gamma(c_{1})=j(c_{1},c_{2})\gamma(c_{2})$

  3. 3.

There is an object $b\in vC$ such that $\gamma(b)$ is an isomorphism.

######  Definition 1.3 ([6]).

A normal category is a category with normal factorization such that for each
$a\in vC$ there is a normal cone $\gamma$ with $\gamma(a)=1_{a}$.

The cones induced by a right translation $\rho_{u}$ is called a principal cone
and is denoted by $\rho^{u}$. The category $\mathcal{L}(S)$ may contain normal
cones other than principal cones.

###  1.2 Normal Bands

Several studies have been carried out for recent years about the normal
categories arising from the semigroups satisfying some special conditions[8,
3, 5]. We focus on the normal category of principal left ideals of a normal
band. A semigroup $B$ is said to be a normal band if $a^{2}=a$ and $abca=acba$
for all $a,b,c\in B$. The following theorem, by Yamada and Kimura[1], states
that a semigroup is a normal band if and only if it is a strong semilattice of
rectangular bands.

######  Theorem 1.2 ([1, 9]).

The following conditions are equivalent for a semigroup $B$

  1. 1.

$B$ is a normal band

  2. 2.

There exists a semilattice $\Gamma$, a disjoint family of rectangular bands
$\\{E_{\alpha}:\alpha\in\Gamma\\}$ such that
$B=\bigcup\\{E_{\alpha}:\alpha\in\Gamma\\}$ and a family of homomorphisms
$\\{\phi_{\alpha,\beta}:E_{\alpha}\to E_{\beta},\text{ where
}\alpha,\beta\in\Gamma\text{ and }\beta\leq\alpha\\}$ satisfying the following

    1. (a)

$\phi_{\alpha,\alpha}$ is the identity function,

    2. (b)

$\phi_{\alpha,\beta}\phi_{\beta,\gamma}=\phi_{\alpha,\gamma}$ for
$\gamma\leq\beta\leq\alpha$ and

    3. (c)

For $a,b\in B$ with $a\in E_{\alpha}$ and $b\in E_{\beta}$

$ab=(a\phi_{\alpha,\alpha\beta})(b\phi_{\beta,\alpha\beta})$

  3. 3.

$B$ is a band and

$abcd=acbd\text{ for all }a,b,c,d\in B.$

###  1.3 The category $\mathcal{L}(B)$

Let $B$ is a normal band. In addition to that the category $\mathcal{L}(B)$ is
normal, it possess some special qualities which are not common in general.
Several properties of the category $\mathcal{L}(B)$ are described in [5]. Here
the normal factorization is unique and there is only one isomorphism between
any two isomorphic objects. Also the principal cones respect the isomorphic
objects in the sense that

$\gamma(Ba)=\rho(a,ab,b)\gamma(Bb)$

for any principal cone $\gamma$ and isomorphic objects $Ba$ and $Bb$. This
leads to the definition of strong cones by AR Rajan[7].

######  Definition 1.4.

[7] A normal cone $\gamma$ in a normal category $C$ is said to be a strong
cone if the following holds

If $c$ and $d$ are two isomorphic objects, then there is a unique isomorphism
$\alpha$ from $c$ to $d$ such that $\gamma(c)=\alpha\gamma(d)$

Thus every principal cone is a strong cone. That is for every objects $Ba$ of
$\mathcal{L}(B)$ there is a strong cone with vertex $Ba$. The category of
principal left ideals of any normal band is characterized as follows.

######  Theorem 1.3.

[5] A normal category $C$ satisfies following conditions if and only if it is
isomorphic to $\mathcal{L}(B)$ for some normal band $B$

  1. SC1

The isomorphism between any two objects is unique.

  2. SC2

The right inverse of an inclusion is unique.

  3. SC3

For every object $c$ in $C$, there is a strong cone with vertex $c$.

##  2 Partial order in $\mathcal{L}(B)$

Let $B$ is a normal band. Let us denote by $[Ba,Bb]$, the collection of all
morphisms from $Ba$ to $Bb$ in $\mathcal{L}(B)$. Now we show that there is a
partial order on the morphism sets $[Ba,Bb]$ and that $[Ba,Bb]$ is an order
ideal with respect to this partial order. In particular there is a maximum
element in $[Ba,Bb]$. We note that any normal band $B$ has a natural partial
order given by $a\leq b$ if and only if $ab=ba=a$.

######  Theorem 2.1.

[5] Let $B$ be a normal band and $\rho(a,u,b)$, $\rho(a,v,b)\in[Ba,Bb]$ be
morphisms in $\mathcal{L}(B)$. Define

$\rho(a,u,b)\leq\rho(a,v,b)\text{ if }u\leq v$

Then $\leq$ is a partial order on $[Ba,Bb]$.

######  Proposition 2.2.

Every morphism set $[Ba,Bb]$ contains a maximum with respect to the above
partial order.

###### Proof.

Taking $\delta=\alpha\beta$ where $a\in E_{\alpha}$ and $b\in E_{\beta}$ we
see that there is a unique $u\in E_{\alpha\beta}$ such that
$\rho(a,u,b)\in[Ba,Bb]$.

Since $\omega^{l}(b)\cap\omega^{r}(a)=\omega(h)$ for some $h\in
E_{\alpha\beta}$ and $\rho(a,h,b)\in[Ba,Bb]$ we conclude that $u=h$. It
follows that $\rho(a,u,b)$ is the maximum in $[Ba,Bb]$ ∎

The following compatibility properties hold for this partial order.

######  Proposition 2.3.

Let $\rho(a,u,b)$ and $\rho(a,v,b)$ be elements in the morphism set $[Ba,Bb]$
in $\mathcal{L}(B)$ such that $\rho(a,u,b)\leq\rho(a,v,b)$. Then the following
compatibility relations hold.

  1. 1.

If $\rho(b,w,c)\in[Bb,Bc]$ then
$\rho(a,u,b)\rho(b,w,c)\leq\rho(a,v,b)\rho(b,w,c)$.

  2. 2.

If $\rho(d,x,a)\in[Bd,Ba]$ then
$\rho(d,x,a)\rho(a,u,b)\leq\rho(d,x,a)\rho(a,v,b)$.

###### Proof.

Follows from the compatibility of natural partial order on normal bands. ∎

Now we describe some properties of the maximum elements in the morphism sets.

######  Proposition 2.4.

Let $m=m(a,b)$ be the maximum element in the morphism set $[Ba,Bb]$ in
$\mathcal{L}(B)$ and $\alpha=\alpha(b,c)$ be an isomorphism from $Bb$ to $Bc$
in $\mathcal{L}(B)$. Then $m\alpha$ is the maximum element in the morphism set
$[Ba,Bc]$.

###### Proof.

Let $f\in[Ba,Bc]$. Then $f\alpha^{-1}\in[Ba,Bb]$. Since $m$ is maximum element
in $[Ba,Bb]$ we have $f\alpha^{-1}\leq m$. Now by compatibility given by
Proposition 2.3

$f=f\alpha^{-1}\alpha\leq m\alpha$

. So $m\alpha$ is the maximum in $[Ba,Bc]$. ∎

######  Proposition 2.5.

Let $m=m(a,b)$ be the maximum element in the morphism set $[Ba,Bb]$ in
$\mathcal{L}(B)$ and $j=j(a_{0},a)$ be an inclusion in $\mathcal{L}(B)$. Then
$jm$ is the maximum element in the morphism set $[Ba_{0},Bb]$.

###### Proof.

Let $f\in[Ba_{0},Bb]$. Then $f=\rho(a_{0},u,b)$ for some $u\leq k$ such that
$\omega(k)=a_{0}Bb$. Now

$jm=\rho(a_{0},a_{0},a)\rho(a,h,b)=\rho(a_{0},a_{0}h,b)=\rho(a_{0},k,b)$

where $\omega(h)=aBb$. Therefore $f\leq jm$ so that $jm$ is maximum element in
$[Ba_{0},Bb]$. ∎

######  Proposition 2.6.

Let $m=m(a,b)$ be the maximum element in the morphism set $[Ba,Bb]$ in
$\mathcal{L}(B)$ and $q:b\to b_{0}$ be a retraction in $\mathcal{L}(B)$. Then
$mq$ is the maximum element in the morphism set $[Ba,Bb_{0}]$.

###### Proof.

Let $f\in[Ba,Bb_{0}]$. Then $fj(b_{0},b)$ is in $[Ba,Bb]$. Since $m$ is the
maximum element in $[Ba,Bb]$ we have $fj(b_{0},b)\leq m$. Now by compatibility
given by Proposition 2.3

$f=fj(b_{0},b)q\leq mq.$

So $mq$ is the maximum in $[Ba,Bb_{0}]$. ∎

As a consequence of this propositions we can see that $mf^{\circ}$ is always a
maximum if $m$ is a maximum.

######  Proposition 2.7.

Let $m=m(a,b)$ be the maximum element in the morphism set $[Ba,Bb]$ in
$\mathcal{L}(B)$ and $f:b\to c$ be any morphism in $\mathcal{L}(B)$. Then
$mf^{\circ}$ is the maximum element in the morphism set $[Ba,Bc_{0}]$ where
$c_{0}\leq c$ is the codomain of $f^{\circ}$.

###### Proof.

Suppose $f=\varrho uj$, where $\varrho$ is a retraction, $u$ is an isomorphism
and $j$ is an inclusion. Then by the above propositions 2.4 and 2.6 we can say
that $mf^{\circ}=m\varrho u$ is the maximum in $[Ba,Bc_{0}]$ ∎

The following proposition exactly locates the maximum element in each hom set
in $\mathcal{L}(B)$

######  Proposition 2.8.

The maximum element in $[Ba,Bb]$ is $\rho(a,ab,b)$

###### Proof.

Suppose $\rho(a,u,b)\in[Ba,Bb]$. Then $u\in aBb$ and hence $u=aub$. Now

$(ab)(u)=(ab)(aub)=aabub=aubb=aub=u$

Similarly we get

$(u)(ab)=aubab=auab=aub=u$

That is $u\leq ab$ for all $u\in aBb$ and this proves that

$\rho(a,u,b)\leq\rho(a,ab,b)$

for all $\rho(a,u,b)\in[Ba,Bb]$. ∎

Note that $\rho(a,ab,b)$ is the unique isomorphism in $[Ba,Bb]$ whenever $Ba$
and $Bb$ are isomorphic. Also the following theorem shows that these morphisms
are the building blocks of principal cones in $\mathcal{L}(B)$.

######  Theorem 2.9.

The component of the principal cone $\rho^{a}$ at $Bb\in\mathcal{L}(B)$ is the
maximum element in $[Bb,Ba]$.

###### Proof.

We have the principal cone $\rho^{a}$ with vertex $Ba$ is defined as

$\rho^{a}(Bb)=\rho(b,ba,a)\text{ for all }Bb\in v\mathcal{L}(B)$

But $\rho(b,ba,a)$ is the maximum in $[Bb,Ba]$ by proposition 2.8. Hence the
theorem. ∎

Now we give another characterization of the partial order in the hom sets of
$\mathcal{L}(B)$ using the image of the morphisms.

######  Theorem 2.10.

For any two morphisms $\rho(a,u,b),\rho(a,v,h)\in[Ba,Bb]$

$\rho(a,u,b)\leq\rho(a,v,b)\text{ if and only if
}\text{im}\,\rho(a,u,b)\subseteq\text{im},\rho(a,v,b)$

###### Proof.

To prove the theorem it is enough to show the following

$\text{im}\,\rho(a,u,b)\subseteq\text{im},\rho(a,v,b)\text{ if and only if
}u\leq v$

The unique normal factorization[5] of $\rho(a,u,b)$ is given by

$\rho(a,u,b)=\rho(a,g,g)\rho(g,u,h)\rho(h,h,b).$

Where $u\mathscr{R}g\leq a$ and $u\mathscr{L}h\leq b$. Therefore
$\text{im}\,\rho(a,u,b)=Bh=Bu$. Similarly we have $\text{im}\,\rho(a,v,b)=Bv$.

Assume that $u\leq v$. Then $uv=vu=u$. This implies $Bu=Buv\subseteq Bv$. Thus
$\text{im}\,\rho(a,u,b)\subseteq\text{im}\,\rho(a,v,b)$.

Conversely suppose that
$\text{im}\,\rho(a,u,b)\subseteq\text{im},\rho(a,v,b)$. That is $Bu\subseteq
Bv$. Then $uv=v$.

Note that $u,v\in aBb=aB\cap Bb$. Therefore there exists $p,q,r,s\in B$ such
that $u=ar=pb$ and $v=as=qb$. Also $u=u^{2}=arpb$ and $v=v^{2}=asqb$. Then
$uv=u$ implies $arqb=ar$ and $pbqb=pb$. Now we get

$\displaystyle vu$ $\displaystyle=(asqb)(arpb)$ $\displaystyle=(aras)(pbqb)$
By repeted application of normality $\displaystyle=(ar)(pb)$
$\displaystyle=uu$ $\displaystyle=u$

Therefore $\text{im}\,\rho(a,u,b)\subseteq\text{im},\rho(a,v,b)$ implies
$uv=vu=u$ and hence $u\leq v$. Hence the theorem.

∎

## References

  * [1] Naoki Kimura.  The structure of idempotent semigroups. i.  Pacific Journal of Mathematics, 8(2):257–275, 1958.
  * [2] Saunders Mac Lane.  Categories for the working mathematician, volume 5.  Springer Science & Business Media, 2013.
  * [3] PA Azeef Muhammed.  Normal categories from completely simple semigroups.  In Algebra and its Applications, pages 387–396. Springer, 2016\.
  * [4] KSS Nambooripad.  Theory of Cross-connections.  Centre for Mathematical Sciences, 1994.
  * [5] C S Preenu, AR Rajan, and KS Zeenath.  Category of principal left ideals of normal bands.  In Springer Proceedings in Mathematics & Statistics.  2020(to appear).
  * [6] A R Rajan.  Inductive groupoids and normal categories of regular semigroups.  In Proceedings of the International Conference on Algebra and Its Applications, pages 193–200. Aligarh Muslim University, Walter de Gruyter GmbH & Co KG, 2018.
  * [7] AR Rajan.  Normal categories associated with bands.  In Proceedings of the International Conference on Semigroups Algebra and Applications, pages 21–29. Cochin University of Science and Technology, 2015.
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# A note on branching of $V(\rho)$

Santosh Nadimpalli and Santosha Pattanayak

###### Abstract.

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\mathfrak{g}_{0}$
be the sub-algebra fixed by a diagram automorphism of $\mathfrak{g}$. Let $G$
be the complex, simply-connected, simple algebraic group with Lie algebra
$\mathfrak{g}$, and let $G_{0}$ be the connected subgroup of $G$ with Lie
algebra $\mathfrak{g}_{0}$. Let $\rho$ be the half sum of positive roots of
$\mathfrak{g}$. In this article, we give a necessary and sufficient condition
for a highest weight $\mathfrak{g}_{0}$-representation $V_{0}(d\mu)$ to occur
in the representation $\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$, for any
saturation factor $d$ of the pair $(G_{0},G)$.

##  1\. Introduction

Let $\mathfrak{g}$ be a complex simple Lie algebra, and let $\rho$ be the half
sum of positive roots, for some choice of positive roots of the root system of
$\mathfrak{g}$ with respect to a Cartan subalgebra $\mathfrak{h}$ and a Borel
subalgebra $\mathfrak{b}$, with $\mathfrak{h}\subset\mathfrak{b}$. Let $\mu$
be a dominant weight with respect to this chosen set of positive roots, and
let $V(\mu)$ be the irreducible representation of $\mathfrak{g}$ with highest
weight $\mu$. A conjecture of Kostant predicts that the representation
$V(\mu)$ occurs in $V(\rho)\otimes V(\rho)$ for all $\mu\preceq 2\rho$ with
respect to the dominance order. In the article [3], the authors show that
$V(d\mu)$ occurs in $V(d\rho)\otimes V(d\rho)$ for all $\mu\preceq 2\rho$, and
for any saturation factor $d$ associated to the Lie algebra $\mathfrak{g}$
(see [7, Definition 11]). The results of the article [3] are related to the
branching of the $\mathfrak{g}\times\mathfrak{g}$ representation
$V(\rho)\otimes V(\rho)$ to the diagonal embedding of $\mathfrak{g}$ in
$\mathfrak{g}\times\mathfrak{g}$. In this article, we consider a similar
branching problem for symmetric pairs associated to diagram automorphisms.

Let $\Phi$ be the set of roots of $\mathfrak{g}$ with respect to
$\mathfrak{h}$. Let $\\{X_{\alpha}:\alpha\in\Phi\\}$ be a Chevalley basis for
the Lie algebra $\mathfrak{g}$. Let $\theta$ be a non-trivial automorphism
fixing the pinning
$(\mathfrak{g},\mathfrak{b},\mathfrak{h},\\{X_{\alpha}\\})$. Let
$\mathfrak{g}_{0}$ be the fixed point Lie sub-algebra
$\\{X\in\mathfrak{g}:\theta(X)=X\\}$. Let $\mathfrak{h}_{0}$ be the Cartan
sub-algebra $\mathfrak{g}_{0}\cap\mathfrak{h}$ of $\mathfrak{g}_{0}$, and let
$\mathfrak{b}_{0}$ be the Borel sub-algebra $\mathfrak{g}_{0}\cap\mathfrak{b}$
of $\mathfrak{g}_{0}$. Let $\Phi_{0}$ be the set of roots of
$\mathfrak{g}_{0}$ with respect to $\mathfrak{h}_{0}$. Let $\rho_{0}$ be the
half sum of positive roots for the choice of $\mathfrak{b}_{0}$ as a Borel
sub-algebra of $\mathfrak{g}_{0}$. Let $G$ be the complex, simply-connected,
simple algebraic group with Lie algebra $\mathfrak{g}$, and let $G_{0}$ be the
connected subgroup of $G$ with Lie algebra $\mathfrak{g}_{0}$.

Let $d$ be a saturation factor for the pair $(G_{0},G)$ (see [7, Definition
12] or see Definition 2.1). Our main theorem gives a necessary and sufficient
condition for a $\mathfrak{g}_{0}$-representation $V_{0}(d\mu)$ to occur in
the restriction $\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$. Let
$p:\mathfrak{h}^{\ast}\rightarrow\mathfrak{h}_{0}^{\ast}$ be the restriction
map. A subset $\Gamma$ of $\Phi_{0}$ is said to be a root subsystem if
$\Gamma$ is a root system in the Euclidean space spanned by $\Gamma$. We do
not necessarily assume that $\Gamma$ is closed in $\Phi$. For instance, the
set of short roots $\Phi_{0,s}$ of $\Phi_{0}$ is a root subsystem of
$\Phi_{0}$ but it is not necessarily closed. In this note, we show that
$V_{0}(d\mu)$ occurs in the restriction
$\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$ if and only if $\mu\preceq
p(\rho)$ in the dominance order of $\Phi_{0,s}$ and $\mu-(2\rho_{0}-p(\rho))$
is dominant in $\Phi_{0,s}$. Here, the ordering in $\Phi_{0,s}$ is induced by
the Borel sub-algebra $\mathfrak{b}_{0}$ of $\mathfrak{g}_{0}$. The condition
that $\mu\preceq p(\rho)$ is analogous to $\mu\preceq 2\rho$ in the tensor
product case.

Kostant’s conjecture is inspired by the $\mathfrak{g}$-representation on the
exterior algebra $\wedge^{\bullet}\mathfrak{g}$. If $\mathfrak{g}$ has a
diagram automorphism $\theta$ of order two, we set $\mathfrak{g}_{1}$ to be
the $-1$ eigenspace of $\theta$. We then get
$\wedge^{\bullet}\mathfrak{g}=\wedge^{\bullet}\mathfrak{g}_{0}\otimes\wedge^{\bullet}\mathfrak{g}_{1}$.
Based on the work of Kostant ([6, Proposition 20, Corollary 36]) and Panyushev
([8, Theorem 6.5]), we obtain that

$\operatorname{res}_{\mathfrak{g}_{0}}V(\rho)\simeq V_{0}(\rho_{0})\otimes
V_{0}(p(\rho)-\rho_{0})$ (1.1)

as $\mathfrak{g}_{0}$ modules. The results of this article are motivated by
the above identity. Indeed if the saturation factor $d$ of the pair
$(G_{0},G)$ is equal to one, our results describe the decomposition of the
tensor product $V_{0}(\rho_{0})\otimes V_{0}(p(\rho)-\rho_{0})$ of
$\mathfrak{g}_{0}$-representations.

##  2\. A decomposition of $V(\rho)$

Let $B$ and $T$ be the Borel subgroup and the maximal torus of $G$ such that
the Lie algebras of $B$ and $T$ are $\mathfrak{b}$ and $\mathfrak{h}$
respectively. Let $Z$ be the center of $G$. Let $\Phi^{\vee}$ be the set of
co-roots of $\mathfrak{g}$ with respect to $\mathfrak{h}$. Let $\Phi^{+}$
(resp. $(\Phi^{\vee})^{+}$) be the set of positive roots (resp. co-roots) of
$\Phi$ (resp. $\Phi^{\vee}$) for the choice of the Borel subalgebra
$\mathfrak{b}$. Let $\Delta$ and $\Delta^{\vee}$ be a system of simple roots
and simple co-roots in $\Phi$ and $\Phi^{\vee}$ respectively. Let $G_{0}$ be
the connected subgroup of $G$ with Lie algebra $\mathfrak{g}_{0}$. We denote
by $B_{0}$ and $T_{0}$ the connected components containing identity of the
groups $B\cap G_{0}$ and $T\cap G_{0}$ respectively. Let $\Phi_{0}^{+}$ be the
set of positive roots of $\Phi_{0}$ with respect to the Borel subalgebra
$\mathfrak{b}_{0}$, and let $\Delta_{0}$ be the set of simple roots in
$\Phi^{+}_{0}$. Let ${\rm X}^{\ast}(T)$ (resp. ${\rm X}^{\ast}(T_{0})$) be the
character lattice of $T$ (resp. $T_{0}$).

Let $P\subset\mathfrak{h}^{\ast}$ and $P_{0}\subset\mathfrak{h}_{0}^{\ast}$ be
the weight lattices of $\mathfrak{g}$ and $\mathfrak{g}_{0}$ respectively. Let
$P^{+}$ be the set of dominant weights of $\mathfrak{g}$ with respect to the
choices of $\mathfrak{h}$ and $\mathfrak{b}$. For $\lambda\in P^{+}$, let
$V(\lambda)$ be the irreducible representation of $\mathfrak{g}$ with highest
weight $\lambda$. For $\mu\in P$, we denote by $V(\lambda)_{\mu}$ the
$\mu$-weight subspace of $V(\lambda)$. The formal character associated to the
representation $V(\lambda)$, defined by the formal sum $\sum_{\mu\in
P}\dim(V(\lambda)_{\mu})e^{\mu}$, is denoted by $ch_{\lambda}$. Similarly, let
$P_{0}^{+}$ be the set of dominant weights of $\mathfrak{g}_{0}$ with respect
to the choice of $\mathfrak{h}_{0}$ and $\mathfrak{b}_{0}$. Recall that we
have $p:\mathfrak{h}^{\ast}\rightarrow\mathfrak{h}_{0}^{\ast}$, the
restriction map.

Since $\mathfrak{g}_{0}$ is a simple Lie algebra but not simply laced,
$\Phi_{0}$ has two root lengths. Let $\Phi_{0,s}$ (resp. $\Phi_{0,l}$) be the
set of short (resp. long) roots in $\Phi_{0}$. Note that $\Phi_{0,s}$ and
$\Phi_{0,l}$ are root subsystems of $\Phi_{0}$, and the latter subsystem is
closed. We set $\Phi_{0,s}^{+}$ to be the set $\Phi_{0}^{+}\cap\Phi_{0,s}$.
Let $P_{0,s}^{+}$ be the set of dominant weights of the sub-root system
$\Phi_{0,s}$ with respect to the choice of positive roots $\Phi_{0,s}^{+}$.
The set $P_{0,s}^{+}$ is contained in $P_{0}^{+}$. We write
$\mu\preceq_{s}\lambda$, if $\lambda-\mu$ is a non-negative integral linear
combination of positive short roots. Let
$\rho_{s}=\frac{1}{2}\sum_{\beta\in\Phi_{0,s}^{+}}\beta$ and let
$\rho_{l}=\frac{1}{2}\sum_{\beta\in\Phi_{0,l}^{+}}\beta$. Let $\Delta_{0,s}$
be the set $\Delta_{0}\cap\Phi_{0,s}$ and let $\Delta_{0,l}$ be the set of
simple roots $\Delta_{0}\backslash\Delta_{0,s}$. We see that
$s_{\alpha}(\rho_{s})=\rho_{s}-\alpha$, for $\alpha\in\Delta_{0,s}$ and
$s_{\alpha}(\rho_{s})=\rho_{s}$ for $\alpha\in\Delta_{0,l}$, where
$s_{\alpha}$ is the reflection corresponding to the root $\alpha$. We then
have $\rho_{s}=\sum_{\alpha\in\Delta_{0,s}}\varpi_{\alpha}$ and
$\rho_{l}=\rho_{0}-\rho_{s}$. We note that $p(\rho)-\rho_{0}=\rho_{s}$ if
$\theta$ is of order $2$ and $p(\rho)-\rho_{0}=2\rho_{s}$ if $\theta$ has
order $3$. Let $(\mathfrak{g}_{s},\mathfrak{g}_{s}\cap\mathfrak{h})$ be the
simple Lie algebra with its Cartan sub-algebra corresponding to the root
system $\Phi_{0,s}$.

We come to the crucial definition of saturation factor associated to the pair
$(G_{0},G)$.

######  Definition 2.1.

A positive integer $d$ is called a saturation factor for the pair $(G_{0},G)$
if for any $(\mu,\lambda)\in P_{0}^{+}\times P^{+}$ such that
$\mu(z)\lambda(z)=1$, for all $z\in Z\cap G_{0}$, and
$\operatorname{Hom}_{G_{0}}(V_{0}(N\mu),V(N\lambda))\neq 0$ for some positive
integer $N$, then $\operatorname{Hom}_{G_{0}}(V_{0}(d\mu),V(d\lambda))\neq 0$.

Let $C(G_{0},G)$ be the set of all pairs $(\mu,\lambda)\in P_{0}^{+}\times
P^{+}$ such that $\operatorname{Hom}_{G_{0}}(V_{0}(N\mu),V(N\lambda))\neq 0$
for some positive integer $N$. The set $C(G_{0},G)$ is called an eigencone of
the pair $(G_{0},G)$. Note that the finite dimensional representations of
$G_{0}$ are self dual. Hence, we have

$\dim_{\mathbb{C}}\operatorname{Hom}_{G_{0}}(V_{0}(N\mu),V(N\lambda))=\dim_{\mathbb{C}}(V_{0}(N\mu)\otimes
V(N\lambda))^{G_{0}}.$

Our result (Theorem 2.2) generalises the main result of [3] on the components
of the form $V(d\lambda)$ in the tensor product $V(d\rho)\otimes V(d\rho)$,
where $d$ is a saturation factor of the group $G$ and $\lambda\in P^{+}$. We
use Ressayre’s work (see [10]) on saturation problem in the context of
branching to prove the following theorem.

######  Theorem 2.2.

Let $d$ be a saturation factor of the pair $(G_{0},G)$, and let $\mu\in
P_{0}^{+}$. The representation $V_{0}(d\mu)$ occurs in
$\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$ if and only if
$\mu=\rho_{0}+\beta$ where $\beta$ occurs as a weight of the
$\mathfrak{g}_{0}$-representation $V_{0}(p(\rho)-\rho_{0})$.

Before we prove the theorem, we recall some notations. For any
$G_{0}$-dominant one parameter subgroup $\delta$ in $T_{0}$, let
$P_{0}(\delta)$ (resp. $P(\delta)$) be the parabolic subgroup associated to
$\delta$ inside $G_{0}$ (resp. $G$). Let $W_{0,P_{0}(\delta)}$ (resp.
$W_{P(\delta)}$) be the Weyl group of $P_{0}(\delta)$ (resp. $P(\delta)$) and
let $W_{0}^{P_{0}(\delta)}$ (resp. $W^{P(\delta)}$) be the set of minimal
length representatives in the cosets of $W_{0}/W_{0,P_{0}(\delta)}$ (resp.
$W/W_{P(\delta)}$). Let $i:G_{0}/P_{0}(\delta)\rightarrow G/P(\delta)$ be the
natural $G_{0}$-equivariant embedding and let
$i^{*}:H^{*}(G/P(\delta),\mathbb{Z})\rightarrow
H^{*}(G_{0}/P_{0}(\delta),\mathbb{Z})$ be the corresponding morphism of
cohomology rings. For $w^{\prime}\in W_{0}^{P_{0}(\delta)}$ (resp. $w\in
W^{P(\delta)}$), let $[X_{w^{\prime}}]\in
H^{*}(G_{0}/P_{0}(\delta),\mathbb{Z})$ (resp. $[X_{w}]\in
H^{*}(G_{0}/P_{0}(\delta),\mathbb{Z})$) be the cohomology class of the sub-
variety $X_{w^{\prime}}:=\overline{B_{0}w^{\prime}P_{0}(\delta)}\subseteq
G_{0}/P_{0}(\delta)$ (resp. $X_{w}:=\overline{BwP(\delta)}\subseteq
G/P(\delta)$). A one parameter subgroup in $T_{0}$ is said to be admissible if
the hyperplane of ${\rm X}_{\mathbb{R}}^{*}(T_{0}):={\rm
X}^{*}(T_{0})\otimes\mathbb{R}$ which it defines is spanned by the weights of
the representation of $T_{0}$ in $\mathfrak{g}/\mathfrak{g}_{0}$ (see [2,
Définition 4.7]). For any complex algebraic group $H$, and for any one
parameter subgroup $\gamma:{\rm G}_{m}\rightarrow H$, we denote by
$\dot{\gamma}\in{\rm Lie}(H)$, the element $d\gamma(1)$.

###### Proof of Theorem 2.2.

We will use the results proved by Ressayre (see [10, Theorem A]), to show that
for any dominant weight $\mu$ such that $\mu=\rho_{0}+\beta$ with
$V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$, the representation $V_{0}(d\mu)$
occurs in the representation $\operatorname{res}_{G_{0}}V(d\rho)$, where $d$
is any saturation factor for the pair $(G_{0},G)$. We will use the exposition
in Brion’s Bourbaki article [2].

The pair $(\lambda,\rho)\in P_{0}^{+}\times P^{+}$ belongs to $C(G_{0},G)$ if
and only if, for any admissible and dominant one parameter subgroup $\delta$
of $T_{0}$ and for any $(w^{\prime},w)\in W_{0}^{P_{0}(\delta)}\times
W^{P(\delta)}$ such that

  1. (1)

$i^{*}([X_{w}]).[X_{w^{\prime}}]=[X_{e}]\in
H^{*}(G_{0}/P_{0}(\delta),\mathbb{Z})$ and

  2. (2)

$(p(w^{-1}\rho)+p(\rho))(\dot{\delta})=(\rho_{0}-w^{\prime-1}\rho_{0})(\dot{\delta})$,

the following inequalities are satisfied:

$I_{\delta,w^{\prime},w}:(p(w^{-1}\rho)+w^{\prime-1}\lambda)(\dot{\delta})\leq
0.$

For the above result we refer to [2, Théorème 4.9]. We will show that
condition $(2)$ already implies the inequalities $I_{\delta,w^{\prime},w}$,
and we will not explicitly use the cohomological condition $(1)$. Note that
condition $(1)$ implies that

$(p(w^{-1}\rho)+p(\rho))(\dot{\delta})\leq(\rho_{0}-w^{\prime-1}\rho_{0})(\dot{\delta}).$

(see [11, Proposition 2.3] and [7, equations (30)-(31)]). For the above
results one may as well refer to the survey article: [7, Theorem 33].

Let $(w^{\prime},w)\in W_{0}^{P_{0}(\delta)}\times W^{P(\delta)}$ and $\delta$
be as above. Since $(\rho_{0}-w^{\prime-1}\rho_{0})(\dot{\delta})$ is equal to
$(p(w^{-1}\rho)+p(\rho))(\dot{\delta})$, we have

$(p(w^{-1}\rho)+w^{\prime-1}\lambda)(\dot{\delta})=(\rho_{0}-w^{\prime-1}\rho_{0}-p(\rho)+w^{\prime-1}\lambda)(\dot{\delta}).$

We apply these results to the present case, i.e., $\lambda=\mu$, where
$\mu=\rho_{0}+\beta$ and $V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$. Note that
$\mu\in{\rm X}^{\ast}(T_{0})\subseteq P_{0}$. We have,
$w^{\prime-1}\mu=w^{\prime-1}\rho_{0}+w^{\prime-1}\beta$. Since
$V_{0}(p(\rho)-\rho_{0})_{w^{\prime-1}\beta}\neq 0$ we have
$w^{\prime-1}\beta\preceq p(\rho)-\rho_{0}$. Thus,

$\displaystyle(p(w^{-1}\rho)+w^{\prime-1}\mu)(\dot{\delta})=$
$\displaystyle(\rho_{0}-w^{\prime-1}\rho_{0}-p(\rho)+w^{\prime-1}\rho_{0}+w^{\prime-1}\beta)(\dot{\delta})$
$\displaystyle\leq(\rho_{0}-w^{\prime-1}\rho_{0}-p(\rho)+w^{\prime-1}\rho_{0}+p(\rho)-\rho_{0})(\dot{\delta})=0.$

Hence, $(\mu,\rho)$ belongs to the eigencone $C(G_{0},G)$. From the assumption
that $\mu=\rho_{0}+\beta$, where $V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$, we
get that $p(\rho)-\mu$ is contained in the root lattice of $G_{0}$. Thus we
get that $\mu(z)\rho(z)=1$, for all $z\in Z\cap G_{0}$. We conclude that
$V_{0}(d\mu)$ occurs as a sub-representation of
$\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$, for any saturation factor $d$
of the pair $(G_{0},G)$.

Conversely, assume that $V_{0}(d\mu)$ is contained in the representation
$\operatorname{res}_{\mathfrak{g}_{0}}V(d\rho)$. Note that
$\Phi_{0}^{+}=\Phi_{0,s}^{+}\cup\Phi_{0,l}^{+}$. The restriction of the map
$p:\mathfrak{h}^{\ast}\rightarrow(\mathfrak{h}_{0})^{\ast}$, to the set
$\Phi^{+}$ surjects onto $\Phi_{0}^{+}$ and the preimage of a short root has
cardinality equal to the order of $\theta$ and the preimage of a long root has
cardinality $1$. So, from the Weyl Character formula, we have

$p(ch_{d\rho})=\prod_{\beta\in\Phi_{0}^{+}}(e^{d\beta/2}+\cdots+e^{-d\beta/2})^{m(\beta)},$

where $m(\beta)$ is the order of $\theta$ (which is either $2$ or $3$) if
$\beta$ is short, and $m(\beta)=1$ if $\beta$ is long. Hence, we get that

$p(ch_{d\rho})=\prod_{\beta\in\Phi_{0,s}^{+}}(e^{d\beta/2}+\cdots+e^{-d\beta/2})^{m(\beta)-1}ch_{d\rho_{0}}.$

Since $p(ch_{d\rho})/ch_{d\rho_{0}}$ is invariant under the Weyl group,
$W_{0}$, of $G_{0}$, we get that

$\prod_{\beta\in\Phi_{0,s}^{+}}(e^{d\beta/2}+\cdots+e^{-d\beta/2})^{m(\beta)-1}=\sum_{i}n_{i}ch_{\mu_{i}},$

for some $n_{i}\in\mathbb{Z}$, and $\mu_{i}\preceq dp(\rho)-d\rho_{0}$ because
$\rho_{s}=p(\rho)-\rho_{0}$ when $\theta$ has order $2$ and
$2\rho_{s}=p(\rho)-\rho_{0}$ when $\theta$ has order $3$. Therefore, we get
that $V_{0}(d\mu)$ is contained in $V_{0}(\mu_{i})\otimes V(d\rho_{0})$, where
$\mu_{i}\preceq dp(\rho)-d\rho_{0}$. Hence, the weight $d\mu$ is equal to
$d\rho_{0}+\beta$ where $V_{0}(\mu_{i})_{\beta}\neq 0$. Now $d\mu-d\rho_{0}$
belongs to the convex hull of $W_{0}(\mu_{i})$. Then, $d\mu-d\rho_{0}$ belongs
to the convex hull of $W_{0}(dp(\rho)-d\rho_{0})$. Thus we get that
$\mu-\rho_{0}$ belongs to the convex hull of $W_{0}(p(\rho)-\rho_{0})$. Since
$\mu-\rho_{0}$ is a weight, and it belongs to $W_{0}(p(\rho)-\rho_{0})$, we
get that $\mu-\rho_{0}$ is comparable to $p(\rho)-\rho_{0}$ (see [4,
Proposition 8.44]). Thus, $V_{0}(p(\rho)-\rho_{0})_{\mu-\rho_{0}}\neq 0$.
Here, $V_{0}(p(\rho)-\rho_{0})$ is considered as a
$\mathfrak{g}_{0}$-representation. This completes the proof of the theorem. ∎

######  Remark 2.3.

The above theorem holds true for any automorphism $\theta$ of order $2$ with
some modification in the set up. We need to choose a $\theta$-stable Cartan
subalgebra $\mathfrak{h}$ and a $\theta$-stable Borel sub-algebra
$\mathfrak{b}$ of $\mathfrak{g}$ and the rest of the notations are as defined
above. The proof is also the same. However, in what follows, we focus on the
case of diagram automorphisms–in which case we describe the weights $\beta$
occurring in the $\mathfrak{g}_{0}$-representation $V_{0}(p(\rho)-\rho_{0})$
such that $\rho_{0}+\beta$ is dominant.

######  Remark 2.4.

Let us consider the case where $\mathfrak{g}$ is a simple Lie algebra of type
${\rm D}_{4}$, and let $\mathfrak{g}_{0}$ be a sub-algebra of type ${\rm
G}_{2}$, fixed by a triality automorphism. Using the methods as in the above
theorem, we can see that the irreducible sub-representations of
$\operatorname{res}_{\mathfrak{g}_{0}}V(\rho)$ are exactly the sub-
representations of

$V_{0}(\rho_{0})\otimes V_{0}(p(\rho)-\rho_{0}).$

In this small rank case, using Klimyk’s formula, we check that the irreducible
sub-representations which occur in the above tensor product are precisely the
following set:

$\\{V_{0}(\rho_{0}+\beta):\rho_{0}+\beta\in
P_{0}^{+}\,\,\text{and}\,\,V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0\\}.$

In the following Lemma and Proposition, we assume that $\theta$ is of order
$2$. The following lemma gives a characterisation of the weights occurring in
the $\mathfrak{g}_{0}$-representation $V_{0}(p(\rho)-\rho_{0})$. For any
subset $\Psi\subseteq\Phi_{0}$, we denote by ${\rm sum}(\Psi)$ the vector
$\sum_{v\in\Psi}v$.

######  Lemma 2.5.

The space $V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$ if and only if
$\beta=p(\rho)-\rho_{0}-{\rm sum}(\Psi)$, where $\Psi$ is a subset of
$\Phi^{+}_{s,0}$.

###### Proof.

Note that the character of the $\mathfrak{g}_{0}$-representation
$V_{0}(p(\rho)-\rho_{0})$ is given by

$\prod_{\beta\in\Phi_{s,0}^{+}}(e^{\beta/2}+e^{-\beta/2}).$

The above is the formal character of the irreducible
$\mathfrak{g}_{s}$-representation $V(\rho_{s})$ with highest weight
$\rho_{s}$. The lemma now follows from the result [5, Lemma 5.9] of Kostant. ∎

######  Proposition 2.6.

Let $\mu\in P_{0}^{+}$. Then, $V_{0}(p(\rho)-\rho_{0})_{\mu-\rho_{0}}\neq 0$
if and only if $\mu\preceq_{s}p(\rho)$ and $\mu-(2\rho_{0}-p(\rho))\in
P_{0,s}^{+}$.

###### Proof.

Assume that $\mu\preceq_{s}p(\rho)$. Then
$\mu-(2\rho_{0}-p(\rho))\preceq_{s}p(\rho)-(2\rho_{0}-p(\rho))=2\rho_{s}$.
Since $\mu-(2\rho_{0}-p(\rho))\in P_{0,s}^{+}$, using the result [3,
Proposition 9] we get that

$\mu-(2\rho_{0}-p(\rho))=\rho_{s}+\beta$

where $V(\rho_{s})_{\beta}\neq 0$. Here, $V(\rho_{s})$ is the
$\mathfrak{g}_{s}$-representation with highest weight $\rho_{s}$. Since the
weights occurring in the $\mathfrak{g}_{s}$-representation $V(\rho_{s})$ are
the same as the weights occurring in the $\mathfrak{g}_{0}$-representation
$V_{0}(p(\rho)-\rho_{0})$, we have $\mu=\rho_{0}+\beta$ where
$V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$.

Conversely, assume that $\mu=\rho_{0}+\beta$ where
$V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$. By Lemma 2.5, we have
$\beta=p(\rho)-\rho_{0}-{\rm sum}(\Psi)$, where $\Psi$ is a subset of
$\Phi^{+}_{s,0}$. So
$p(\rho)-\mu=p(\rho)-\rho_{0}-\beta=\rho_{s}-\beta=\rho_{s}-p(\rho)+\rho_{0}+{\rm
sum}(\Psi)={\rm sum}(\Psi)$. So, $p(\rho)-\mu$ is a sum of positive short
roots. Hence $\mu\preceq_{s}p(\rho)$.

Write $\rho_{0}=\rho_{s}+\rho_{l}$. Then $\mu=\rho_{s}+\rho_{l}+\beta$ and
$\mu-(2\rho_{0}-p(\rho))=\rho_{s}+\beta$. For $\alpha\in\Delta_{0,s}$, we have
$\langle\rho_{l},\alpha\rangle=0$, and hence,

$\langle\rho_{s}+\beta,\alpha\rangle=\langle\mu-\rho_{l},\alpha\rangle=\langle\mu,\alpha\rangle\geq
0.$

This completes the proof of the proposition. ∎

######  Remark 2.7.

Although we assume that $\mathfrak{g}$ is simple, we may consider the case
where $\mathfrak{g}=\mathfrak{g}_{0}\times\mathfrak{g}_{0}$ and $\theta$ to be
the permutation automorphism of the two simple factors. With the rest of the
notations being similar we get that $p(\rho)=2\rho_{0}$. This situation is
considered in the paper [3, Proposition 9]. The hypothesis of Proposition 2.6
is similar to the hypothesis in [3, Proposition 9]. However, when $\theta$ is
an outer involution of a simple Lie algebra $\mathfrak{g}$, i.e., for the pair
$(\mathfrak{g},\mathfrak{g}_{0})$, where $\mathfrak{g}_{0}$ is the fixed point
sub-algebra of an outer automorphism of a simple Lie algebra $\mathfrak{g}$,
there exist dominant weights $\mu$ such that $p(\rho)-\mu$ is a non-negative
integral linear combination of short positive roots in $\Phi_{0}$ but
$V_{0}(p(\rho)-\rho_{0})_{\mu-\rho_{0}}=0$.

Note that $p(\rho)$ is in the root lattice except in the case where
$\mathfrak{g}$ is of the type ${\rm A}_{2n-1}$ and $\mathfrak{g}_{0}$ is of
the type ${\rm C}_{n}$, for $n$ is odd. When $p(\rho)$ is in the root lattice,
we take $\mu=0$. In the case where the pair $(\mathfrak{g},\mathfrak{g}_{0})$
is of type $(A_{2n-1},C_{n})$, for $n$ is odd, we take $\mu=(2n-1)\varpi_{1}$;
here, $\varpi_{1}$ is the first fundamental weight of $\mathfrak{g}_{0}$. Here
the numbering of fundamental weights is according to the conventions in [1,
PLANCHE III]. Then it is easy to see that $p(\rho)-\mu$ is a non-negative
integral linear combination of short positive roots. But,
$V_{0}(p(\rho)-\rho_{0})_{\mu-\rho_{0}}=0$ since $p(\rho)-\mu$ is not of the
form $p(\rho)-\rho_{0}-{\rm sum}(\Psi)$, for any
$\Psi\subseteq\Phi_{s,0}^{+}$.

######  Remark 2.8.

Let $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g_{1}}$ be the eigen-
decomposition of $\theta$. From Kostant’s description of
$\wedge^{\bullet}\mathfrak{g}$, and Panyushev’s work on the description of
$\wedge^{\bullet}\mathfrak{g}_{1}$, we get that

$\operatorname{res}_{\mathfrak{g}_{0}}V(\rho)\simeq V_{0}(\rho_{0})\otimes
V_{0}(p(\rho)-\rho_{0})$

as $\mathfrak{g}_{0}$ modules. If $V_{0}(\mu)$ occurs in
$\operatorname{res}_{\mathfrak{g}_{0}}V(\rho)$, then $\mu=\rho_{0}+\beta$,
where $V_{0}(p(\rho)-\rho_{0})_{\beta}\neq 0$. If $d=1$, for instance, in the
cases where $(\mathfrak{g},\mathfrak{g}_{0})$ are of the type: $({\rm
D}_{n},{\rm B}_{n-1})$, $({\rm E}_{6},{\rm F}_{4})$ and $({\rm A}_{2n-1},{\rm
C}_{n})$, for $2\leq n\leq 5$, (see [9]) then the main theorem gives the
converse of this statement. In general, using computations on SAGE, we
observed that the representation $V_{0}(\mu)$ occurs in
$\operatorname{res}_{\mathfrak{g}_{0}}V(\rho)$ if and only if
$V_{0}(p(\rho)-\rho_{0})_{\mu-\rho_{0}}\neq 0$.

Acknowledgements: We would like to thank Prof. Michel Brion for the careful
reading of the manuscript and for some helpful comments. We also thank Prof.
Shrawan Kumar for some useful comments.

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Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur,
U.P. India, 208016.
email<EMAIL_ADDRESS><EMAIL_ADDRESS>