---
abstract: 'If $G$ is a complex simply connected semisimple algebraic group and if ${\lambda}$ is a dominant weight, we consider the compactification $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is normal; as well, we give necessary and sufficient conditions on the support of ${\lambda}$ so that $X_{\lambda}$ is smooth.'
author:
- 'Paolo Bravi, Jacopo Gandini, Andrea Maffei, Alessandro Ruzzi'
title: 'Normality and non-normality of group compactifications in simple projective spaces'
---

Introduction {#introduction .unnumbered}
============

Consider a semisimple simply connected algebraic group $G$ over an algebraically closed field ${\Bbbk}$ of characteristic zero. If ${\lambda}$ is a dominant weight (with respect to a fixed maximal torus $T$ and a fixed Borel subgroup $B\supset T$) and if ${V({{\lambda}})}$ is the simple $G$-module of highest weight ${\lambda}$, then $\operatorname{End}\big({V({{\lambda}})}\big)$ is a simple $G\times G$-module. Let $I_{\lambda}\in \operatorname{End}\big({V({{\lambda}})}\big)$ be the identity map and consider the variety $X_{\lambda}\subset {\mathbb P}\big(\operatorname{End}({V({{\lambda}})})\big)$ given by the closure of the $G\times G$-orbit of $[I_{\lambda}]$. In [@Ka], S. Kannan studied for which ${\lambda}$ this variety is projectively normal, and this happens precisely when ${\lambda}$ is minuscule. In [@Ti], D. Timashev studied the more general situation of a sum of irreducible representations, giving necessary and sufficient conditions for the normality and smoothness of these compactifications; however the conditions for normality are not completely explicit. In this paper we give an explicit characterization of the normality of $X_{\lambda}$, which allows to simplify the conditions for the smoothness as well.

To explain our results we need some notation. Let ${\Delta}$ be the set of simple roots (w.r.t. $T\subset B$) and identify ${\Delta}$ with the vertices of the Dynkin diagram. Define the support of ${\lambda}$ as the set $\operatorname{Supp}({\lambda})=\{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0
\}$.

\[see Theorem \[teo:normalita\]\] The variety $X_{\lambda}$ is normal if and only if ${\lambda}$ satisfies the following property:

-   For every non-simply laced connected component $\Delta'$ of $\Delta$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, then it contains also the short root which is adjacent to a long simple root.

In particular, if the Dynkin diagram of $G$ is simply laced then $X_{\lambda}$ is normal, for all ${\lambda}$. In the paper we will prove the theorem in a more general form, for simple (i.e. with a unique closed orbit) linear projective compactifications of an adjoint group (see section \[ssez:XSigma\]). We will make use of the wonderful compactification of $G_\mathrm{ad}$, the adjoint group of $G$, and of the results on projective normality of these compactifications proved by S. Kannan in [@Ka]. These results hold in the more general case of a symmetric variety; however our method does not apply to this more general situation (see section \[ssez:simmetrichenormalita\]).

\[see Theorem \[smooth Xsigma\]\] The variety $X_{\lambda}$ is smooth if and only if $\lambda$ satisfies property $(\star)$ of Theorem A together with the following properties:

-   For every connected component $\Delta'$ of $\Delta$, $\operatorname{Supp}(\lambda)\cap \Delta'$ is connected and, in case it contains a unique element, then this element is an extreme of ${\Delta}'$;

-   $\operatorname{Supp}(\lambda)$ contains every simple root which is adjacent to three other simple roots and at least two of the latter;

-   Every connected component of ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ is of type ${\mathsf A}$.

Theorem B can be generalized to any simple and normal adjoint symmetric variety. Following a criterion of ${\mathbb Q}$-factoriality for spherical varieties given by M. Brion in [@Br2], properties i) and ii) characterize the $\mathbb{Q}$-factoriality of the normalization of $X_{\lambda}$ (see Proposition \[Q-fattorialita\]), while property iii) arises from a criterion of smoothness given by D. Timashev in [@Ti] in the case of a linear projective compactification of a reductive group.

As a corollary of Theorem B, we get that $X_{\lambda}$ is smooth if and only if its normalization is smooth.

The paper is organized as follows. In the first section we introduce the wonderful compactification of $G_\mathrm{ad}$ and the normalization of the variety $X_{\lambda}$. In the second section we prove Theorem A, and in the third section Theorem B. In the last section we discuss some possible generalizations of our results.

Preliminaries {#sez:preliminari}
=============

Notation {#ssez:notazioni}
--------

Recall that $G$ is semisimple and simply connected. Fix a Borel subgroup $B\subset G$, a maximal torus $T\subset B$ and let $U$ denote the unipotent radical of $B$. Lie algebras of groups denoted by upper-case latin letters ($G,U,L,\ldots$) will be denoted by the corresponding lower-case german letter (${\mathfrak g}, \mathfrak u, \mathfrak l,\ldots$). Let $\Phi$ denote the set of roots of $G$ relatively to $T$ and ${\Delta}\subset \Phi$ the basis associated to the choice of $B$. For all ${\alpha}\in {\Delta}$ let $e_{\alpha}, {\alpha}^{{\vee}},f_{\alpha}$ be an $\mathfrak{sl}(2)$-triple of $T$-weights ${\alpha},0,-{\alpha}$. Let ${\Lambda}$ denote the weight lattice of $T$ and ${\Lambda}^+$ the subset of dominant weights. For all ${\alpha}\in{\Delta}$, denote by $\omega_{\alpha}$ the corresponding fundamental weight.

If ${\lambda}\in {\Lambda}$, recall the definition of its *support*: $$\operatorname{Supp}({\lambda}) = \{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0 \}.$$

If $I \subset {\Delta}$, define its *border* $\partial{I}$, its *interior* $I^\circ$ and its *closure* ${\overline}{I}$ as follows: $$\partial{I} = 
\{ {\alpha}\in {\Delta}{\smallsetminus}I {\, : \,}{\exists\,}{\beta}\in I \mathrm{ \; such \,  that \; } 
\langle {\beta},{\alpha}^\vee \rangle \neq 0\};$$ $$I^\circ = I {\smallsetminus}\partial{({\Delta}{\smallsetminus}I)};$$ $$\overline{I} = I \cup \partial{I}.$$

For ${\lambda}\in \Lambda$, denote by ${\mathcal L}_{\lambda}$ the line bundle on $G/B$ whose $T$-weight in the point fixed by $B$ is $-{\lambda}$. For ${\lambda}$ dominant, ${V({{\lambda}})} = {\Gamma}(G/B,{\mathcal L}_{\lambda})^*$ is an irreducible $G$-module of highest weight ${\lambda}$; when we deal with different groups we will use the notation ${V_G({{\lambda}})}$.

Denote by ${\Pi({{\lambda}})}$ the set of weights occurring in ${V({{\lambda}})}$ and set ${\Pi^+({{\lambda}})}={\Pi({{\lambda}})}\cap {\Lambda}^+$. Let ${\lambda}\mapsto {\lambda}^*$ be the linear involution of ${\Lambda}$ defined by $({V({{\lambda}})})^*{\simeq}{V({{\lambda}^*})}$, for any dominant weight ${\lambda}$.

The weight lattice ${\Lambda}$ is endowed with the dominance order ${\leqslant}$ defined as follows: $\mu {\leqslant}{\lambda}$ if and only if ${\lambda}- \mu \in
{\mathbb N}{\Delta}$. If ${\beta}= \sum_{{\alpha}\in {\Delta}} n_{\alpha}{\alpha}\in {\mathbb Z}{\Delta}$, define its *support over* ${\Delta}$ (not to be confused with the previous one) as follows: $$\operatorname{Supp}_{\Delta}({\beta}) = \{ {\alpha}\in {\Delta}{\, : \,}n_{\alpha}\neq 0 \}.$$

We introduce also some notations about the multiplication of sections. Notice that, for all ${\lambda},\mu \in\Lambda$, ${\mathcal L}_{\lambda}\otimes {\mathcal L}_\mu ={\mathcal L}_{{\lambda}+\mu}$. Therefore, if ${\lambda},\mu$ are dominant weights and $n\in {\mathbb N}$, the multiplication of sections defines maps as follows: $$m_{{\lambda},\mu}:{V({{\lambda}})}\times {V({\mu})} {\rightarrow}{V({{\lambda}+\mu})} \; \text{ and }
m_{\lambda}^n : {V({{\lambda}})} {\rightarrow}{V({n{\lambda}})}.$$ We will also write $uv$ for $m_{{\lambda},\mu}(u,v)$ and $u^n$ for $m^n_{\lambda}(u)$. Since $G/B$ is irreducible, $m_{{\lambda},\mu}$ and $m^n_{\lambda}$ induce the following maps at the level of projective spaces: $$\psi_{{\lambda},\mu}: {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) {\rightarrow}{\mathbb P}({V({{\lambda}+ \mu})})
\;{\text{ and }}\; \psi^n_{\lambda}: {\mathbb P}({V({{\lambda}})}) {\rightarrow}{\mathbb P}({V({n{\lambda}})}).$$

The following lemma is certainly well known; however we do not know any reference.

\[lem:immersioni\] Let ${\lambda},\mu$ be dominant weights.

1.  If $\operatorname{Supp}({\lambda}) \cap \operatorname{Supp}(\mu) = {\varnothing}$, then the map $\psi_{{\lambda},\mu}\colon {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) \to {\mathbb P}({V({{\lambda}+
      \mu})})$ is a closed embedding.

2.  For any $n>0$, the map $\psi_{\lambda}^n: {\mathbb P}({V({{\lambda}})})  {\rightarrow}{\mathbb P}({V({n{\lambda}})})$ is a closed embedding.

$i)$. Fix highest weight vectors $v_{\lambda}\in {V({{\lambda}})}$, $v_\mu \in {V({\mu})}$ and $v_{{\lambda}+\mu}= v_{\lambda}v_\mu \in {V({{\lambda}+\mu})}$.

If $V$ is irreducible, then ${\mathbb P}(V)$ has a unique closed orbit, namely the orbit of the highest weight vector. Consequently, since ${\mathbb P}(V({\lambda})) \times {\mathbb P}(V(\mu))$ has a unique closed orbit, in order to prove the claim it suffices to prove that $\psi_{{\lambda},\mu}$ is smooth in $x=([v_{\lambda}],[v_\mu])$ and that the inverse image of $[v_{{\lambda}+\mu}]$ is $x$. The second claim is clear for weight reasons.

In order to prove that $\psi_{{\lambda},\mu}$ is smooth in $x$, consider $T$-stable complements $U \subset {V({{\lambda}})}$, $V \subset {V({\mu})}$ and $W\subset
  {V({{\lambda}+\mu})}$ of ${\Bbbk}\,v_{\lambda}$, ${\Bbbk}\,v_\mu$ and ${\Bbbk}\,v_{{\lambda}+\mu}$. So in a neighbourhood of $x$ the map $\psi_{{\lambda},\mu}$ can be described as $$\psi:U\times V {\longrightarrow}W \;\text{ where } \psi(u,v)= u v_\mu + v_{\lambda}v + u v.$$ The differential of $\psi_{{\lambda},\mu}$ in $x$ is then given by the differential of $\psi$ in $(0,0)$, thus it is described as follows: $$d\psi_x(u,v)=uv_\mu+v_{\lambda}v.$$ Suppose that $d\psi_x$ is not injective. Since it is $T$-equivariant, consider a maximal weight $\eta \in {\Pi({{\lambda}+\mu})}{\smallsetminus}\{{\lambda}+\mu\}$ such that there exists a couple of non-zero $T$-eigenvectors $(u,v)\in \ker d
  \psi_x$ with weights respectively $\eta - \mu$ and $\eta - {\lambda}$. Suppose that $\eta - \mu \in {\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ is not maximal and take ${\alpha}\in {\Delta}$ such that $\eta - \mu + {\alpha}\in
  {\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ and $e_{\alpha}u\neq 0$: then $$(e_{\alpha}u)v_\mu + v_{\lambda}(e_{\alpha}v) = e_{\alpha}(u v_\mu + v_{\lambda}v) = 0$$ and $\eta + {\alpha}\in {\Pi({{\lambda}+ \mu})}{\smallsetminus}\{{\lambda}+\mu\}$, against the maximality of $\eta$. Thus $\eta - \mu$ is maximal in ${\Pi({{\lambda}})}
{\smallsetminus}\{{\lambda}\}$ and similarly $\eta - {\lambda}$ is maximal in ${\Pi({\mu})}
{\smallsetminus}\{\mu\}$. Therefore, on one hand it must be $$\eta - \mu = {\lambda}- {\alpha}$$ with ${\alpha}\in \operatorname{Supp}({\lambda})$, while on the other hand it must be $$\eta - {\lambda}= \mu - {\beta}$$ with ${\beta}\in \operatorname{Supp}(\mu)$. Since $\operatorname{Supp}({\lambda}) \cap \operatorname{Supp}(\mu) =
{\varnothing}$, this is impossible and shows that, if $(u, v) \in \ker d
\psi_x$, then it must be $u = 0$ or $v = 0$. Suppose now that $(u,0)
\in \ker d\psi_x$: then $u v_\mu = 0$ and by the irreducibility of $G/B$ also $u=0$. A similar argument applies if $v=0$.

$ii).$ Suppose that $v,w \in V({\lambda})$ are such that $v^{n} = w^{n}$: then $v=tw$ for some $t\in {\Bbbk}$. Thus $\psi_{\lambda}^n$ is injective. Let us show now that $\psi_{\lambda}^n$ is smooth; it is enough to show it in $x = [v_{\lambda}]$ where $v_{\lambda}\in V({\lambda})$ is a highest weight vector. Let $V \subset V({\lambda})$ be the $T$-stable complement of ${\Bbbk}v_{\lambda}$, identified with the tangent space $T_{x} {\mathbb P}(V({\lambda}))$. If $v
\in V$, the differential $d (\psi_{\lambda}^n)_{x}$ is described as follows $$d (\psi^n_{\lambda})_{x} (v) = n v_{\lambda}^{n-1} v.$$ Thus $d (\psi^n_{\lambda})_{x}$ is injective and $\psi_{\lambda}^n$ is smooth.

The variety $X_{\lambda}$ {#ssez:Xlambda}
-------------------------

If ${\lambda}$ is a dominant weight, denote by ${E({{\lambda}})}$ the $G\times
G$-module $\operatorname{End}({V({{\lambda}})})$ and set $X_{\lambda}$ the closure of the $G\times G$-orbit of $[I_{\lambda}] \in {\mathbb P}({E({{\lambda}})})$. More generally if ${\lambda}_1,\dots,{\lambda}_m$ are dominant weights we define $$X_{{\lambda}_1, \ldots, {\lambda}_m}
= {\overline}{G\times G([I_{{\lambda}_1}], \ldots, [I_{{\lambda}_m}])} \subset
{\mathbb P}({E({{\lambda}_1})}) \times \cdots \times {\mathbb P}({E({{\lambda}_m})}).$$ Since ${E({{\lambda}})}$ is an irreducible $G\times G$-module of highest weight $({\lambda},{\lambda}^*)$, as a consequence of Lemma \[lem:immersioni\] we get that if ${\lambda}$ and $\mu$ have non-intersecting supports and if $n\in {\mathbb N}$ then $$X_{{\lambda}+\mu}\simeq X_{{\lambda},\mu} \qquad \mathrm{ and } \qquad X_{n{\lambda}}{\simeq}X_{\lambda}.$$ As a consequence we get the following proposition:

\[prp:supporto\]Let ${\lambda},\mu$ be dominant weights. Then $X_{\lambda}\simeq  X_\mu$ as $G\times G$-varieties if and only if ${\lambda}$ and $\mu$ have the same support. Moreover, if $\operatorname{Supp}({\lambda})=\{{\alpha}_1,\dots,{\alpha}_m\}$ then $$X_{\lambda}{\simeq}X_{\omega_{{\alpha}_1},\dots,\omega_{{\alpha}_m}}.$$

By the discussion above we have to prove only that the condition is necessary. This follows by noticing that if $X_{\lambda}$ and $X_\mu$ are $G\times G$-isomorphic then also their closed $G\times G$-orbits are isomorphic, which is equivalent to the fact that ${\lambda}$ and $\mu$ have the same support.

The wonderful compactification of $G_{\mathrm{ad}}$ and the normalization of $X_{\lambda}$ {#ssez:meravigliosa}
------------------------------------------------------------------------------------------

When ${\lambda}$ is a regular weight (i.e. $\operatorname{Supp}({\lambda})={\Delta}$) the variety $X_{\lambda}$ is called the wonderful compactification of $G_\mathrm{ad}$ and it has been studied by C. De Concini and C. Procesi in [@CP]. We will denote this variety by $M$: it is smooth and the complement of its open orbit is the union of smooth prime divisors with normal crossings whose intersection is the closed orbit. The closed orbit of $M$ is isomorphic to $G/B \times G/B$ and the restriction of line bundles determines an embedding of $\operatorname{Pic}(M)$ into $\operatorname{Pic}(G/B\times G/B)$, that we identify with ${\Lambda}\times {\Lambda}$ as before; the image of this map is the set of weights of the form $({\lambda},{\lambda}^*)$. Therefore $\operatorname{Pic}(M)$ is identified with ${\Lambda}$ and we denote by ${\mathcal M}_{{\lambda}}$ a line bundle on $M$ whose restriction to $G/B \times G/B$ is isomorphic to ${\mathcal L}_{\lambda}\boxtimes {\mathcal L}_{{\lambda}^*}$. If $D\subset M$ is a $G\times G$-stable prime divisor then the line bundle defined by $D$ is of the form ${\mathcal M}_{{\alpha}_D}$, where ${\alpha}_D$ is a simple root. The map $D\mapsto {\alpha}_D$ defines a bijection between the set of $G\times G$-stable prime divisors and ${\Delta}$, and we denote by $M_{\alpha}$ the prime divisor which corresponds to a simple root ${\alpha}$. We denote by $s_{\alpha}$ a section of ${\mathcal M}_{\alpha}$ whose associated divisor is $M_{\alpha}$; notice that such a section is $G\times G$-invariant. More generally if $\nu = \sum_{{\alpha}\in {\Delta}} n_{{\alpha}}{\alpha}\in {\mathbb N}{\Delta}$, set $s^\nu = \prod_{{\alpha}\in {\Delta}} s_{{\alpha}} ^{n_{{\alpha}}} \in
{\Gamma}(M,{\mathcal M}_\nu)$. Then, given any ${\lambda}\in {\Lambda}$, the multiplication by $s^\nu$ injects ${\Gamma}(M,{\mathcal M}_{{\lambda}- \nu})$ in ${\Gamma}(M,{\mathcal M}_{\lambda})$.

If ${\lambda}$ is a dominant weight, the map $G_\mathrm{ad}{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ extends to a map $q_{\lambda}:M{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ (see [@CP]) whose image is $X_{\lambda}$ and such that ${\mathcal M}_{\lambda}=q_{\lambda}^*({\mathcal O}_{{\mathbb P}({E({{\lambda}})})}(1))$. If we pull back the homogeneous coordinates of ${\mathbb P}({E({{\lambda}})})$ to $M$, we get then a submodule of ${\Gamma}(M,{\mathcal M}_{\lambda})$ which is isomorphic to ${E({{\lambda}})}^*$; by abuse of notation we will denote this submodule by ${E({{\lambda}})}^*$.

If ${\lambda}\in {\Lambda}$, in [@CP Theorem 8.3] the following decomposition of ${\Gamma}(M,{\mathcal M}_{\lambda})$ is given: $${\Gamma}(M,{\mathcal M}_{\lambda}) = \bigoplus_{\mu\in {\Lambda}^+ {\, : \,}\mu {\leqslant}{\lambda}} s^{{\lambda}-\mu}{E({\mu})}^*.$$

Consider the graded algebra $A({\lambda})=\bigoplus_{n=0}^\infty A_n({\lambda})$, where $A_n({\lambda})= {\Gamma}(M,{\mathcal M}_{n{\lambda}})$, and set $\widetilde{X}_{\lambda}=
\operatorname{Proj}A({\lambda})$. We have then a commutative diagram as follows: $$\xymatrix{M \ar@{->>}^{p_{\lambda}}[r] \ar@{->>}_{q_{\lambda}}[dr]
& \widetilde{X}_{\lambda}\ar@{->>}^{r_{\lambda}}[d]\\  & X_{\lambda}}$$

In [@Ka], it has been shown that $A({\lambda})$ is generated in degree $1$ and in [@DC] that $r=r_{\lambda}$ is the normalization of $X_{\lambda}$. Notice that the projective coordinate ring of $X_{\lambda}\subset
{\mathbb P}({E({{\lambda}})})$ is given by the graded subalgebra $B({\lambda})=\bigoplus_{n=0}^\infty
 B_n({\lambda})$ of $A({\lambda})$ generated by ${E({{\lambda}})}^* \subset {\Gamma}(M,{\mathcal M}_{\lambda})$.

The variety $X_\Sigma$ {#ssez:XSigma}
----------------------

We consider now a generalization of the variety $X_{\lambda}$. Let $\Sigma$ be a finite set of dominant weights and denote ${E({\Sigma})} =
\bigoplus_{\mu \in \Sigma} {E({\mu})}$; let $x_\Sigma=[(I_\mu)_{\mu\in\Sigma}]\in {\mathbb P}({E({\Sigma})})$ and define $X_\Sigma$ as the closure of the $G\times G$-orbit of $x_\Sigma$ in ${\mathbb P}({E({\Sigma})})$. If $\Sigma = \{{\lambda}\}$, then we get the variety $X_{\lambda}$, while if ${\Sigma}= {\Pi^+({{\lambda}})}$ we get its normalization $\widetilde X _{\lambda}$. Notice that the diagonal action of $G$ fixes the point $x_\Sigma$ so we have a $G\times G$ equivariant map $G{\longrightarrow}X_\Sigma$ given by $g \longmapsto (g,1)x_\Sigma$. This map induces a map from $G_{\mathrm{ad}}$ to $X_\Sigma$ if and only if the action of the center of $G\times G$ on $E({\lambda})$ is the same for all ${\lambda}\in
\Sigma$ or equivalently if ${\Sigma}$ is contained in a coset of ${\Lambda}$ modulo ${\mathbb Z}{\Delta}$. In this case we say that $X_\Sigma$ is a *semi-compactification* of $G_{ad}$. If $G_{\mathrm{ad}}$ is a simple group and and $\Sigma \neq \{ 0 \}$ then $X_\Sigma$ is a compactification of $G_{\mathrm{ad}}$, while if $G_{\mathrm{ad}}$ is not simple we can only say that is a compactification of a group which is a quotient of $G_{\mathrm{ad}}$.

We say that $\Sigma$ is *simple* if there exists ${\lambda}\in\Sigma$ such that $\Sigma \subset {\Pi^+({{\lambda}})}$ or equivalently if ${\Sigma}$ contains a unique maximal element with respect to the dominance order ${\leqslant}$. Notice also that if ${\lambda}\in \Sigma$ is such that for all $\mu \in \Sigma$ different from ${\lambda}$ the vector $\mu - {\lambda}$ is not in ${\mathbb Q}_{{\geqslant}0}[{\Delta}]$ then is easy to construct a cocharacter $\chi
: {\Bbbk}^*{\longrightarrow}G\times G$ such that $\lim_{t\to 0}\chi(t)x_\Sigma $ is the highest weight line in ${\mathbb P}(E({\lambda}))$. In particular $X_\Sigma$ is a simple $G\times G$ semi-compactification of $G_{\mathrm{ad}}$ if and only if $\Sigma$ is simple.

By the description of the normalization of $X_{\lambda}$ is $\Sigma$ is simple and ${\lambda}\in {\Sigma}$ is the maximal element, then we get $$\xymatrix{\widetilde X_{\lambda}\ar[r]^{r} & X_\Sigma \ar[r] & X_{\lambda}}$$ In particular, it follows that $r=r_\Sigma:\widetilde X_{\lambda}{\longrightarrow}X_\Sigma$ is the normalization of $X_\Sigma$.

If ${\Sigma}$ is simple, denote $B(\Sigma)=\bigoplus_{n=0}^\infty
B_n(\Sigma)$ the projective coordinate ring of $X_\Sigma\subset
{\mathbb P}({E({\Sigma})})$: it is the subalgebra of $A({\lambda})$ generated by ${E({\Sigma})}^*\subset {\Gamma}(M,{\mathcal M}_{\lambda})$.

The discussion above and the fact that in ${\mathbb P}(E({\lambda}))$ there is only one point fixed by the diagonal action of $G$ (the line of scalar matrices) proves that any $G\times G$ linear projective compactification of $G_{\mathrm{ad}}$ is of the form $X_\Sigma$. A projective $G\times G$-variety $X$ is said to be *linear* if there exists an equivariant embedding $X\subset {\mathbb P}(V)$ where $V$ is a finite dimensional rational $G\times G$-module. In particular as a consequence of Sumihiro’s Theorem (see for example [@KKLV Corollary 2.6]) all normal projective compactifications are linear. In this paper we study only linear compactifications.

Normality {#sez:normalita}
=========

In this section we determine for which simple $\Sigma$ the variety $X_\Sigma$ is normal, proving in particular Theorem A. In the following, by $\lambda$ we will always denote the maximal element of $\Sigma$.

Let ${\varphi}_{\lambda}\in {E({{\lambda}})}^*$ be a highest weight vector and set $X_\Sigma^\circ \subset X_\Sigma$ the open affine subset defined by the non-vanishing of ${\varphi}_{\lambda}$. In particular, we set ${\widetilde}X_{\lambda}=
X_{{\Pi^+({{\lambda}})}}$ and notice that ${\widetilde}X ^\circ_{\lambda}=
r^{-1}(X_\Sigma^\circ)$. Notice that $X^\circ_\Sigma$ is $B\times B$-stable and, since it intersects the closed orbit, it intersects every orbit: therefore $X_\Sigma$ is normal if and only if $X_\Sigma^\circ$ is normal if and only if the restriction $r{\bigr|}_{{\widetilde}X
^\circ_{\lambda}} : {\widetilde}X^\circ_{\lambda}\to X^\circ_\Sigma$ is an isomorphism. Denote by $\bar B(\Sigma)$ the coordinate ring of $X^\circ_\Sigma$ and by $\bar A({\lambda})$ the coordinate ring of ${\widetilde}X^\circ_{\lambda}$; then we have $$\bar A({\lambda})= \{\frac{{\varphi}}{{\varphi}_{\lambda}^n}{\, : \,}{\varphi}\in A_n({\lambda})\}
\supset \{\frac{{\varphi}}{{\varphi}_{\lambda}^n}{\, : \,}{\varphi}\in B_n(\Sigma)\} = \bar B(\Sigma)$$ and $X_\Sigma$ is normal if and only if $\bar A({\lambda})=\bar B
(\Sigma)$. The rings $\bar A({\lambda})$ and $\bar B(\Sigma)$ are not $G\times G$-modules, however since $X^\circ_\Sigma$ is an open subset of $X_\Sigma$ we still have an action of the Lie algebra ${\mathfrak g}\oplus{\mathfrak g}$ on them.

By [@Ka], $\bar A({\lambda})$ is generated by the elements of the form ${\varphi}/{\varphi}_{\lambda}$ with ${\varphi}\in A_1({\lambda})$. In particular we have the following lemma.

\[lem:general-normality\] The variety $X_\Sigma$ is normal if and only if for all $\mu\in{\Lambda}^+$ such that $\mu{\leqslant}{\lambda}$ there exists $n>0$ such that $$s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*
\subset B_n(\Sigma).$$

Let ${\varphi}_\mu \in s^{{\lambda}-\mu}{E({\mu})}^*$ be a highest weight vector and suppose that $X_\Sigma$ is normal. Then, by the descriptions of $\bar A({\lambda})$ and $\bar B(\Sigma)$, for every dominant weight $\mu{\leqslant}{\lambda}$ there exist $n>0$ and $\varphi\in B_n(\Sigma)$ such that ${{\varphi}}/{{\varphi}_{\lambda}^n} = {{\varphi}_\mu}/{{\varphi}_{\lambda}}$ or equivalently ${\varphi}={\varphi}_\mu {\varphi}_{\lambda}^{n-1}\in B_n(\Sigma)$. Since ${\varphi}$ is a highest weight vector of the module $s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*$ the claim follows.

Conversely assume that for every dominant weight $\mu{\leqslant}{\lambda}$ there exists $n$ such that $$s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*\subset B_n(\Sigma);$$ in particular ${\varphi}={\varphi}_\mu {\varphi}_{\lambda}^{n-1}\in
B_n(\Sigma)$. Let’s prove that ${\varphi}/{\varphi}_{\lambda}\in \bar B({\Sigma})$ for every ${\varphi}\in s^{{\lambda}-\mu}{E({\mu})}^*$; this implies the thesis since $\bar A({\lambda})$ is generated in degree one. If ${\varphi}={\varphi}_\mu$ this is clear. Using the action of the Lie algebra ${\mathfrak g}\oplus{\mathfrak g}$ on $\bar
B(\Sigma)$, let’s show that if ${\varphi}/{\varphi}_{\lambda}\in \bar B(\Sigma)$ then $f_{\alpha}({\varphi})/{\varphi}_{\lambda}\in \bar B(\Sigma)$: indeed we have $$\frac{f_{\alpha}({\varphi})}{{\varphi}_{\lambda}} = f_{\alpha}(\frac{{\varphi}}{{\varphi}_{\lambda}}) +
\frac{{\varphi}}{{\varphi}_{\lambda}} \cdot \frac{f_{\alpha}({\varphi}_{\lambda})}{{\varphi}_{\lambda}}$$ and the claim follows since $f_{\alpha}({\varphi}_{\lambda})\in {E({{\lambda}})}^*\subset B_1(\Sigma)$.

We can describe the set $B_n(\Sigma)$ more explicitly. Indeed, as in [@DC] or in [@Ka], it is possible to identify sections of a line bundle on $M$ with functions on $G$ and use the description of the multiplication of matrix coefficients. Recall that as a $G\times G$-module we have ${\Bbbk}[G]=\bigoplus_{{\lambda}\in {\Lambda}^+}{E({{\lambda}})}^* {\simeq}\bigoplus_{{\lambda}\in
  {\Lambda}^+}{V({{\lambda}})}^*\otimes {V({{\lambda}})}$. More explicitly if $V$ is a representation of $G$, define $c_V:V^*\otimes V {\longrightarrow}{\Bbbk}[G]$ as usual by $c_V(\psi \otimes v)(g)= \langle \psi, gv\rangle$. If we multiply functions in ${\Bbbk}[G]$ of this type then we get $$c_{V}( \psi \otimes v) \cdot c_{W}(\chi \otimes w) =
c_{V\otimes W} \big((\psi\otimes\chi)\otimes(v\otimes w)\big):$$ in particular we get that the image of the multiplication ${E({{\lambda}})}^*
\otimes {E({\mu})}^* {\longrightarrow}{\Bbbk}[G]$ is the sum of all ${E({\nu})}^*$ with ${V({\nu})}\subset {V({{\lambda}})}\otimes {V({\mu})}$.

As a consequence we obtain the following Lemma:

\[lem:coefficientimatriciali\] Let $\nu, \nu'$ be dominant weights, then the image of ${E({\nu})}^*
\otimes {E({\nu'})}^*$ in ${\Gamma}(M, {\mathcal M}_{\nu+\nu'})$ via the multiplication map is $$\bigoplus_{{V({\mu})}\subset {V({\nu})}\otimes {V({\nu'})}}\!\!\!\!s^{\nu+\nu'-\mu}{E({\mu})}^*.$$

Indeed let $ \pi : G {\rightarrow}M$ be the map induced by the inclusion $G_\mathrm{ad}\subset M$. Then any line bundle on $G$ can be trivialized so that the image of $\pi^*:{E({{\lambda}})}^*\subset{\Gamma}(M,{\mathcal M}_\nu){\longrightarrow}{\Bbbk}[G]$ is the image of $c_{{V({{\lambda}})}}$ and the claim follows from previous remarks.

Together with Lemma \[lem:general-normality\], this gives the following

\[prp:normalita\] The variety $X_\Sigma$ is normal if and only if, for every $\mu \in {\Lambda}^+$ such that $\mu {\leqslant}{\lambda}$, there exist $n>0$ and ${\lambda}_1,\dots,{\lambda}_n\in \Sigma$ such that $${V({\mu + (n-1){\lambda}})} \subset {V({{\lambda}_1})}\otimes \cdots \otimes {V({{\lambda}_n})} .$$

Remarks on tensor products {#ssez:prodottitensore}
--------------------------

By Proposition \[prp:normalita\], in order to establish the normality (or the non-normality) of $X_\Sigma$, we need some results on tensor product decomposition.

\[lem:riduzionelevi\] Let ${\lambda}, \mu, \nu$ be dominant weights and let ${\Delta}'\subset {\Delta}$ be such that $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$; let $L \subset G$ be the standard Levi subgroup associated to ${\Delta}'$. If $\pi \in {\Lambda}^+$, denote by ${V_L({\pi})}$ the simple $L$-module of highest weight $\pi$. Then $${V({\nu})} \subset {V({{\lambda}})} \otimes {V({\mu})} \iff {V_L({\nu})} \subset {V_L({{\lambda}})} \otimes {V_L({\mu})}.$$

If $\mathfrak a$ is any Lie algebra, denote ${\mathfrak U}(\mathfrak a)$ the corresponding universal enveloping algebra.

Suppose that ${V_L({\nu})} \subset {V_L({{\lambda}})} \otimes {V_L({\mu})}$; fix maximal vectors $v_{\lambda}\in {V_L({{\lambda}})}$ and $v_\mu \in {V_L({\mu})}$ for the Borel subgroup $B\cap L \subset L$ and fix $p \in {\mathfrak U}(\mathfrak
l\cap\mathfrak u^-) \otimes {\mathfrak U}(\mathfrak l\cap\mathfrak u^-)$ such that $p\,(v_{\lambda}\otimes v_\mu) \in {V_L({{\lambda}})} \otimes {V_L({\mu})}$ is a maximal vector of weight $\nu$. Since ${V_L({{\lambda}})} \otimes {V_L({\mu})}
\subset {V({{\lambda}})} \otimes {V({\mu})}$, we only need to prove that $p\,(v_{\lambda}\otimes v_\mu)$ is a maximal vector for $B$ too. If ${\alpha}\in {\Delta}'$ then we have $e_{\alpha}p\,(v_{\lambda}\otimes v_\mu) = 0$ by hypothesis. On the other hand, if ${\alpha}\in {\Delta}{\smallsetminus}{\Delta}'$, notice that $e_{\alpha}$ commutes with $p$, since by its definition $p$ is supported only on the $f_{\alpha}$’s with ${\alpha}\in {\Delta}'$. Since $v_{\lambda}\otimes
v_\mu$ is a maximal vector for $B$, then we get $$e_{\alpha}p\,(v_{\lambda}\otimes v_\mu) = p\, e_{\alpha}(v_{\lambda}\otimes v_\mu) = 0;$$ thus $p\,(v_{\lambda}\otimes v_\mu)$ generates a simple $G$-module of highest weight $\nu$.

Assume conversely that ${V({\nu})} \subset {V({{\lambda}})} \otimes {V({\mu})}$ and fix $p
\in {\mathfrak U}(\mathfrak u^-) \otimes {\mathfrak U}(\mathfrak u^-)$ such that $p\,(v_{\lambda}\otimes v_\mu) \in {V({{\lambda}})} \otimes {V({\mu})}$ is a maximal vector of weight $\nu$. Since $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$, we may assume that the only $f_{\alpha}$’s appearing in $p$ are those with ${\alpha}\in {\Delta}'$; therefore $p\,(v_{\lambda}\otimes v_\mu) \in {V_L({{\lambda}})}
\otimes {V_L({\mu})}$ and it generates a simple $L$-module of highest weight $\nu$.

\[lem:traslazione\] Fix ${\lambda}, \mu, \nu \in {\Lambda}^+$ such that ${V({\nu})} \subset {V({{\lambda}})}
\otimes {V({\mu})}$. Then, for any $\nu' \in {\Lambda}^+$, it also holds $${V({\nu + \nu'})} \subset {V({{\lambda}+ \nu'})} \otimes {V({\mu})}.$$

Fix a maximal vector $v_{\nu'} \in {V({\nu'})}$ and consider the $U$-equivariant map $$\begin{array}{cccc}
\phi: & {V({{\lambda}})} \otimes {V({\mu})} & {\longrightarrow}& {V({{\lambda}+ \nu'})}\otimes {V({\mu})}\\
& w_1 \otimes w_2 & \longmapsto & m_{{\lambda},\nu'}(w_1,v_{\nu'}) \otimes w_2
\end{array}$$ The claim follows since, if $v_\nu \in {V({{\lambda}})}\otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu$, then $\phi(v_\nu) \in {V({{\lambda}+
\nu'})} \otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu +
\nu'$.

We now describe some more explicit results. When we deal with explicit irreducible root systems, unless otherwise stated, we always use the numbering of simple roots and fundamental weights of Bourbaki [@Bo].

In order to describe the simple subsets ${\Sigma}\subset {\Lambda}^+$ which give rise to a non-normal variety $X_{\Sigma}$, we will make use of following lemma.

\[lem:BGno\]

1.  Let $G$ be of type ${\sf B}_r$. Then, for any $n$, ${V({(n-1){\omega}_1})} \not \subset {V({{\omega}_1})}^{\otimes n}$.

2.  Let $G$ be of type ${\sf G}_2$. Then, for any $n$, ${V({{\omega}_1+(n-1){\omega}_2})} \not \subset {V({{\omega}_2})}^{\otimes n}$.

We consider only the first case, the second is similar. Fix a highest weight vector $v_1 \in {V({{\omega}_1})}$. If ${\alpha}$ is any simple root and if $1 {\leqslant}s {\leqslant}r$, notice that $f_{\alpha}$ acts non-trivially on $f_{{\alpha}_{s-1}}\cdots f_{{\alpha}_1}v_1$ if and only if ${\alpha}=
{\alpha}_s$. The $T$-eigenspace of weight 0 in ${V({{\omega}_1})}$ is spanned by $v_0=f_{{\alpha}_r}\cdots f_{{\alpha}_1}v_1$, and similarly the $T$-eigenspace of weight $(n-1){\omega}_1$ in ${V({{\omega}_1})}^{\otimes n}$ is spanned by $v_1^{\otimes i-1} \otimes v_0 \otimes v_1^{\otimes n-i}$, where $1 {\leqslant}i {\leqslant}n$. Since the vectors $$e_{{\alpha}_r} (v_1^{\otimes i-1} \otimes v_0 \otimes v_1^{\otimes
      n-i}) = v_1^{\otimes i-1} \otimes (e_{{\alpha}_r} v_0) \otimes
    v_1^{\otimes n-i}$$ are linearly independent, there exists no maximal vector of weight $(n-1){\omega}_1$ in ${V({{\omega}_1})}^{\otimes n}$.

Dual results will be needed to describe the subsets ${\Sigma}$ which give rise to a normal variety $X_{\Sigma}$, but before we need to introduce some further notation.

If $\Phi$ is an irreducible root system and ${\Delta}$ is a basis for $\Phi$ we will denote by $\eta$ the highest root if $\Phi$ is simply laced or the highest short root if $\Phi$ is not simply laced. For the convenience of the reader we list the highest short root of every irreducible root system in Table \[tab:hsr\].

   type of $\Phi$                                                highest short root
  ----------------- ------------------------------------------------------------------------------------------------------------
   ${\mathsf A}_r$                                ${\alpha}_1+\cdots+{\alpha}_r=\omega_1+\omega_r$
   ${\mathsf B}_r$                                    ${\alpha}_1+\cdots+{\alpha}_r=\omega_1$
   ${\mathsf C}_r$                      ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-1})+{\alpha}_r=\omega_2$
   ${\mathsf D}_r$              ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-2})+{\alpha}_{r-1}+{\alpha}_r=\omega_2$
   ${\mathsf E}_6$                ${\alpha}_1+2{\alpha}_2+2{\alpha}_3+3{\alpha}_4+2{\alpha}_5+{\alpha}_6=\omega_2$
   ${\mathsf E}_7$         $2{\alpha}_1+2{\alpha}_2+3{\alpha}_3+4{\alpha}_4+3{\alpha}_5+2{\alpha}_6+{\alpha}_7=\omega_1$
   ${\mathsf E}_8$   $2{\alpha}_1+3{\alpha}_2+4{\alpha}_3+6{\alpha}_4+5{\alpha}_5+4{\alpha}_6+3{\alpha}_7+2{\alpha}_8=\omega_8$
   ${\mathsf F}_4$                           ${\alpha}_1+2{\alpha}_2+3{\alpha}_3+2{\alpha}_4=\omega_4$
   ${\mathsf G}_2$                                       $2{\alpha}_1+{\alpha}_2=\omega_1$

  : []{data-label="tab:hsr"}

Recall the condition $(\star)$ defined in the introduction: a dominant weight ${\lambda}$ satisfies $(\star)$ if, for every non-simply laced connected component ${\Delta}'\subset {\Delta}$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root then it contains also the short root which is adjacent to a long simple root.

\[twin\] If ${\Delta}' \subset {\Delta}$ is a non-simply laced connected component, order the simple roots in ${\Delta}'= \{ {\alpha}_1, \ldots,
{\alpha}_r\}$ starting from the extreme of the Dynkin diagram of ${\Delta}'$ which contains a long root and denote ${\alpha}_q$ the first short root in ${\Delta}'$. If ${\lambda}$ is a dominant weight such that ${\alpha}_q\not \in \operatorname{Supp}({\lambda})$ and such that $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, denote ${\alpha}_p$ the last long root which occurs in $\operatorname{Supp}({\lambda})\cap {\Delta}'$; for instance, if ${\Delta}'$ is not of type ${\mathsf G}_2$, then the numbering is as follows: $$\begin{picture}(9000,1800)(2000,-900)
           \put(0,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
           \put(3600,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
           \put(7200,0){\multiput(0,0)(1800,0){2}{\circle*{150}}\thicklines\multiput(0,-60)(0,150){2}{\line(1,0){1800}}\multiput(1050,0)(-25,25){10}{\circle*{50}}\multiput(1050,0)(-25,-25){10}{\circle*{50}}}
           \put(9000,0){\multiput(0,0)(3600,0){2}{\circle*{150}}\thicklines\multiput(0,0)(2500,0){2}{\line(1,0){1100}}\multiput(1300,0)(400,0){3}{\line(1,0){200}}}
           \put(-150,-700){\tiny $\alpha_1$}
           \put(3450,-700){\tiny $\alpha_p$}
           \put(8850,-700){\tiny $\alpha_q$}
           \put(12450,-700){\tiny $\alpha_r$}
\end{picture}$$ The *little brother* of ${\lambda}$ with respect to ${\Delta}'$ is the dominant weight $${\lambda}_{{\Delta}'}^{\mathrm{lb}}= {\lambda}- \sum_{i=p}^q {\alpha}_i =
\left\{ \begin{array}{ll}
        {\lambda}-\omega_1+\omega_2 & \textrm{ if $G$ is of type $\sf{G}_2$} \\
        {\lambda}+ {\omega}_{p-1} - {\omega}_{p}  + {\omega}_{q+1} & \textrm{ otherwise}
\end{array} \right.$$ where ${\omega}_i$ is the fundamental weight associated to ${\alpha}_i$ if $1{\leqslant}i {\leqslant}r$, while ${\omega}_0 = {\omega}_{r+1} = 0$. The set of the little brothers of ${\lambda}$ will be denoted by ${\mathrm{LB}}({\lambda})$; notice that ${\mathrm{LB}}({\lambda})$ is empty if and only if ${\lambda}$ satisfies condition $(\star)$ of Theorem A. For convenience, define $\overline {\mathrm{LB}}({\lambda})={\mathrm{LB}}({\lambda})\cup\{{\lambda}\}$, while if ${\Delta}$ is connected and non-simply laced set ${\lambda}^{\mathrm{lb}}= {\lambda}_{\Delta}^{\mathrm{lb}}$.

\[lem:eta\] Assume $G$ to be simple and let ${\lambda}\in {\Lambda}^+{\smallsetminus}\{0\}$. Denote $\eta$ the highest root of $\Phi$ if the latter is simply laced or the highest short root otherwise.

1.  If ${\lambda}$ satisfies the condition $(\star)$ then $${V({{\lambda}})} \subset {V({\eta})} \otimes {V({{\lambda}})}.$$

2.  If ${\lambda}$ does not satisfy the condition $(\star)$ and if ${\lambda}^{\mathrm{lb}}$ is the little brother of ${\lambda}$ then $${V({{\lambda}})} \subset {V({\eta})} \otimes {V({{\lambda}^{\mathrm{lb}}})}.$$

If ${\Delta}$ is simply laced, then ${V({\eta})}\simeq{\mathfrak g}$ is the adjoint representation: in this case the claim follows straightforward by considering the map ${\mathfrak g}\otimes {V({{\lambda}})} \to {V({{\lambda}})}$ induced by the ${\mathfrak g}$-module structure on ${V({{\lambda}})}$, which is non-zero since ${\lambda}$ is non-zero.

Suppose now that ${\Delta}$ is not simply laced. If ${\lambda}$ satisfies condition $(\star)$, then by Lemma \[lem:traslazione\] it is enough to study the case ${\lambda}= \omega_{\alpha}$ where ${\alpha}$ is a short simple root:

*Type ${\mathsf B}_r$*: ${V({\omega_r})}\subset{V({\omega_1})}\otimes {V({\omega_r})}$.

*Type ${\mathsf C}_r$*: ${V({\omega_i})}\subset{V({\omega_2})}\otimes {V({\omega_i})}$, with $i<r$.

*Type ${\mathsf F}_4$*: ${V({\omega_3})}\subset{V({\omega_4})}\otimes {V({\omega_3})}$ and ${V({\omega_4})}\subset{V({\omega_4})}\otimes {V({\omega_4})}$.

*Type ${\mathsf G}_2$*: ${V({\omega_1})}\subset{V({\omega_1})}\otimes {V({\omega_1})}$.

If ${\lambda}$ does not satisfy condition $(\star)$, by Lemma \[lem:traslazione\] we can assume that ${\lambda}={\omega}_{\alpha}$ with ${\alpha}$ a long root:

*Type ${\mathsf B}_r$*: ${V({\omega_i})}\subset{V({\omega_1})}\otimes {V({\omega_{i-1}})}$, if $1<i<r$, and ${V({\omega_1})}\subset{V({\omega_1})}\otimes {V({0})}$.

*Type ${\mathsf C}_r$*: ${V({\omega_r})}\subset{V({\omega_2})}\otimes {V({\omega_{r-2}})}$.

*Type ${\mathsf F}_4$*: ${V({\omega_1})}\subset{V({\omega_4})}\otimes {V({\omega_4})}$ and ${V({\omega_2})}\subset{V({\omega_4})}\otimes {V({\omega_1+\omega_4})}$.

*Type ${\mathsf G}_2$*: ${V({\omega_2})}\subset{V({\omega_1})}\otimes {V({\omega_1})}$.

The above mentioned inclusion relations for tensor products are essentially known: let us treat the case of type ${\mathsf C}_r$ with ${\lambda}=\omega_i$ and $i<r$, the other cases are easier or can be checked directly.

Let $v_0$ be a highest weight vector of ${V({\omega_2})}$ and $w_0$ be a highest weight vector of ${V({\omega_i})}$. Let $f$ be the following product (in the universal enveloping algebra $\mathfrak U(\mathfrak u^-)$) $$f=f_{{\alpha}_i}\cdots f_{{\alpha}_1}\cdot f_{{\alpha}_{i+1}}\cdots f_{{\alpha}_{r-1}}\cdot f_{{\alpha}_r}\cdots f_{{\alpha}_2},$$ and consider all the factorizations $f = p\cdot q$ such that $p,q \in\mathfrak U(\mathfrak u^-)$. If ${\beta}_1,\ldots,{\beta}_j\in{\Delta}$, set $$\,^\mathrm r(f_{{\beta}_1}\cdots f_{{\beta}_j})=(-1)^j2^\delta f_{{\beta}_j}\cdots f_{{\beta}_1},$$ where $\delta$ equals 0 (resp. 1) if $\alpha_i$ occurs an even (resp. odd) number of times in $\{{\beta}_1,\ldots,{\beta}_j\}$. Then it is easy to check that the vector $$\sum_{p\cdot q=f} p.v_0 \otimes \,^\mathrm r\!q.w_0$$ is a $U$-invariant vector in ${V({\omega_2})}\otimes{V({\omega_i})}$ of $T$-weight $\omega_i$.

If the Dynkin diagram of $G$ is not simply laced we will need some further properties of tensor products.

If ${\Delta}$ is connected but not simply laced, we will denote by $\alpha_S$ the short simple root that is adjacent to a long simple root $\alpha_L$; moreover, we will denote the associated fundamental weights by $\omega_S$ and $\omega_L$. Finally, define $\zeta$ as the sum of all simple roots and notice that $\omega_S + \zeta$ is dominant.

\[lem:zeta\] Let ${\lambda}$ be a non-zero dominant weight.

1.  If $G$ is of type ${\mathsf F}_4$ or ${\mathsf C}_r$ ($r{\geqslant}3$) and if $\operatorname{Supp}({\lambda})$ contains a long root then $${V({{\lambda}+\omega_S})} \subset {V({\zeta + \omega_S})} \otimes {V({{\lambda}})}.$$

2.  If $G$ is of type ${\mathsf G}_2$ and if ${\lambda}$ does not satisfy $(\star)$ then $${V({{\lambda}+\omega_1})} \subset {V({\omega_2})} \otimes {V({{\lambda}^{\mathrm{lb}}})}.$$

3.  If $G$ is of type ${\mathsf G}_2$ and if ${\alpha}_S \in \operatorname{Supp}({\lambda})$ then $${V({{\lambda}+\omega_1})} \subset {V({\omega_2})} \otimes {V({{\lambda}})}.$$

By Lemma \[lem:traslazione\] it is enough to check the statements for ${\lambda}=\omega_{\alpha}$ with ${\alpha}$ a long root in the first two cases and ${\alpha}={\alpha}_S$ in the last case.

*Type ${\mathsf C}_r$*: by Lemma \[lem:traslazione\] it is enough to check that ${V({\omega_{r-1}})}\subset {V({\omega_1})} \otimes {V({\omega_r})}$.

*Type ${\mathsf F}_4$*: we have ${\lambda}=\omega_1$ or ${\lambda}=\omega_2$ and $\omega_S+\zeta=\omega_1+\omega_4$.

*Type ${\mathsf G}_2$*: we have ${\lambda}=\omega_2$ and ${\lambda}^{\mathrm{lb}}=\omega_1$ in point (2) and ${\lambda}=\omega_1$ in point (3).

Normality and non-normality of $X_\Sigma$ {#ssez:normalita}
-----------------------------------------

We are now able to state the main theorem.

\[teo:normalita\]Let $\Sigma$ be a simple set of dominant weights and let ${\lambda}$ be its maximal element. The variety $X_\Sigma$ is normal if and only if $\Sigma \supset {\mathrm{LB}}({\lambda})$.

Theorem A stated in the introduction follows immediately by considering the case $\Sigma=\{{\lambda}\}$. The remaining part of this section will be devoted to the proof of Theorem \[teo:normalita\]. The general strategy will be based on Proposition \[prp:normalita\] and will proceed by induction on the dominance order of weights. The ingredients of this induction will be the results proved in section \[ssez:prodottitensore\] together with the description of the dominance order given by J. Stembridge in [@St]: the dominance order between dominant weights is generated by pairs which differ by the highest short root for a subsystem of the root system.

If $K$ is a subset of ${\Delta}$, denote $\Phi_K\subset \Phi$ the associated root subsystem and, in case $K$ is connected, denote by $\eta_K$ the corresponding highest short root. Moreover, if ${\beta}=
\sum_{{\alpha}\in{\Delta}}n_{\alpha}{\alpha}$, set ${\beta}|_K=\sum_{{\alpha}\in
  K}n_{\alpha}{\alpha}$. The result of [@St] that we will use is the following.

\[lem:stembridge\] Let ${\lambda},\mu$ be two dominant weights with ${\lambda}>\mu$; set $I =
\operatorname{Supp}_{\Delta}({\lambda}-\mu)$. Let $\Phi_K$ be an irreducible subsystem of $\Phi_I$ (where $K\subset I$).

1.  If $\langle ({\lambda}-\mu){\bigr|}_K, {\alpha}^\vee \rangle {\geqslant}0$ for all ${\alpha}\in K \cap \operatorname{Supp}(\mu)$, then $\mu+\eta_K {\leqslant}{\lambda}$.

2.  If in addition $\langle \mu + \eta_K, {\alpha}^\vee
          \rangle {\geqslant}0$ for all ${\alpha}\in I {\smallsetminus}K$, then $\mu +
          \eta_K \in {\Lambda}^+$.

The next two lemmas are the main steps of our induction.

\[lem:costruzioneK\] Suppose that $\Phi$ is irreducible; let ${\lambda},\mu\in{\Lambda}^+$ such that ${\lambda}>\mu$ and $\operatorname{Supp}_{\Delta}({\lambda}-\mu)={\Delta}$. Assume that either $\Phi$ is simply laced, or there exists a short root ${\alpha}\in \operatorname{Supp}({\lambda})$ such that $\langle  {\lambda}-\mu,{\alpha}^{\vee}\rangle {\geqslant}0$, or ${\alpha}_S \notin \operatorname{Supp}(\mu)$. Then there exists a connected subset $K$ of ${\Delta}$ such that

1.  $\mu+\eta_K{\leqslant}{\lambda}$;

2.  $\mu+\eta_K\in {\Lambda}^+$;

3.  $K\cap\operatorname{Supp}({\lambda})\neq {\varnothing}$.

Set $K_1= \{{\alpha}\in {\Delta}{\, : \,}\langle{\lambda}-\mu,{\alpha}^{\vee}\rangle{\geqslant}0\}$. Since ${\lambda}>\mu$ we have that $K_1\cap\operatorname{Supp}({\lambda})$ is non-empty. Notice also that $\operatorname{Supp}(\mu)\supset {\Delta}{\smallsetminus}K_1$. Define $K$ as follows:

-   If $\Phi$ is simply laced, let $K$ be a connected component of $K_1$ which intersects $\operatorname{Supp}({\lambda})$.

-   If ${\alpha}\in\operatorname{Supp}({\lambda})$ is a short root such that $\langle {\lambda}-\mu, {\alpha}^{\vee}\rangle {\geqslant}0$ let $K$ be the connected component of $K_1$ containing ${\alpha}$.

-   If $\Phi$ is not simply laced and there does not exist a short root ${\alpha}$ as in b), let $K$ be a connected component of $K_1$ which intersects $\operatorname{Supp}({\lambda})$.

Properties $i)$ and $iii)$ are then easily verified by Lemma \[lem:stembridge\](a) and by construction.

To prove $ii)$ notice that, if $\Phi$ is not simply laced, by the construction of $K$ it follows that if ${\alpha}_L \in K$ then ${\alpha}_S \in K$ as well: indeed, $K$ is a connected component of $K_1$ and if there is no short root ${\alpha}$ as in b) then ${\alpha}_S \not \in \operatorname{Supp}(\mu)$ implies ${\alpha}_S \in K_1$. By the description of highest short roots in Table \[tab:hsr\] we deduce that, if ${\alpha}\in K {\smallsetminus}K^{\circ}$, then the respective coefficient in $\eta_K$ is $1$: hence $\langle \eta_K, {\alpha}^\vee \rangle = -1$ for all ${\alpha}\in \partial K$ and, since $\operatorname{Supp}(\mu)\supset {\Delta}{\smallsetminus}K_1 \supset \partial{K}$, we get $\mu + \eta_K \in {\Lambda}^+$.

In order to proceed with the induction, in the next lemma we will need to consider the condition $(\star)$ also for a Levi subgroup of $G$. If $K\subset {\Delta}$ let $L_K$ be the associated standard Levi subgroup; we say that ${\lambda}\in{\Lambda}^+$ satisfies condition $(\star_K)$ if, for every non-simply laced connected component $K'$ of $K$ such that $\operatorname{Supp}({\lambda})\cap K'$ contains a long root, $\operatorname{Supp}({\lambda})\cap K'$ contains also the short root adjacent to a long root. Notice that if ${\lambda}$ satisfies $(\star)$ then it also satisfies $(\star_K)$ for all $K\subset {\Delta}$.

Similarly we can also define the little brother of a dominant weight w.r.t. the Levi subgroup $L_K$: if $K'$ is a connected component of $K$ such that ${\lambda}$ does not satisfy $(\star_{K'})$, define the little brother ${\lambda}_{K'}^{\mathrm{lb}}$ w.r.t. $K'$ as in Definition \[twin\] and denote by ${\mathrm{LB}}_K({\lambda})$ the set of little brothers of ${\lambda}$ constructed in this way. Notice that if $K'$ is a connected component of $K$ such that ${\lambda}$ does not satisfy $(\star_{K'})$ and if ${\Delta}'$ is the connected component of ${\Delta}$ containing $K'$, then ${\lambda}$ does not satisfy $(\star_{{\Delta}'})$ as well and ${\lambda}_{K'}^{\mathrm{lb}}={\lambda}_{{\Delta}'}^{\mathrm{lb}}$. In particular ${\mathrm{LB}}_K({\lambda})\subset
{\mathrm{LB}}({\lambda})$.

\[lem:induzione\] Assume $G$ to be simple and let ${\lambda},\mu$ be two dominant weights such that ${\lambda}>\mu$ and $\operatorname{Supp}_{\Delta}({\lambda}-\mu)={\Delta}$. Then there exist $\mu'\in{\Lambda}^+$ and ${\lambda}'\in \overline {\mathrm{LB}}({\lambda})$ such that $\mu<\mu'{\leqslant}{\lambda}$ and $${V({\mu+{\lambda}})}\subset
{V({\mu'})}\otimes {V({{\lambda}'})}.$$

Suppose first that either $\Phi$ is simply laced or ${\alpha}_S \notin \operatorname{Supp}(\mu)$ or there exists a short root ${\alpha}$ in $\operatorname{Supp}({\lambda})$ such that $\langle
 {\lambda}-\mu, {\alpha}^{\vee}\rangle {\geqslant}0$. Take $K$ as in Lemma \[lem:costruzioneK\] and set $\mu'=\mu+\eta_K$: then by Lemma \[lem:eta\] together with Lemma \[lem:traslazione\] we get $${V_{L_K}({\mu + {\lambda}})} \subset {V_{L_K}({\mu'})} \otimes {V_{L_K}({{\lambda}'})}$$ with ${\lambda}'\in\overline {\mathrm{LB}}_K({\lambda})$. The claim follows by Lemma \[lem:riduzionelevi\] together with the inclusion ${\mathrm{LB}}_K({\lambda})\subset {\mathrm{LB}}({\lambda})$.

Suppose now that $\Phi$ is not simply laced, that ${\alpha}_S \in \operatorname{Supp}(\mu)$ and that there is no short root ${\alpha}\in \operatorname{Supp}({\lambda})$ such that $\langle
  {\lambda}-\mu, {\alpha}^{\vee}\rangle {\geqslant}0$. Since ${\lambda}>\mu$ there exists ${\alpha}\in \operatorname{Supp}({\lambda})$ such that $\langle {\lambda}-\mu , {\alpha}^\vee\rangle {\geqslant}0$: therefore, $\operatorname{Supp}({\lambda})$ contains at least a long root. Set $\mu'=\mu + \zeta$; notice that $\mu'{\leqslant}{\lambda}$ and that $\mu'$ is dominant. The claim follows then by Lemma \[lem:eta\] and by Lemma \[lem:traslazione\] if $\Phi$ is of type ${\mathsf B}$, while if $\Phi$ is of type ${\mathsf C}$, ${\mathsf F}_4$ or ${\mathsf G}_2$ it follows by Lemma \[lem:zeta\] and by Lemma \[lem:traslazione\].

We prove first that the condition is necessary. Assume that there exists a little brother $\mu = {\lambda}_{{\Delta}'}^{\mathrm{lb}}$ of ${\lambda}$ which is not in $\Sigma$. We prove that for every positive $n$ and for every choice of weights ${\lambda}_1,\dots,{\lambda}_n \in \Sigma$ the module ${V({\mu+(n-1){\lambda}})}$ is not contained in ${V({{\lambda}_1})}\otimes \cdots \otimes {V({{\lambda}_n})}$.

We proceed by contradiction. Assume there exist weights ${\lambda}_1,\dots,{\lambda}_n$ as above and notice that any of them satisfies $\mu{\leqslant}{\lambda}_i{\leqslant}{\lambda}$: indeed, ${\lambda}-\mu=n{\lambda}-(\mu+(n-1){\lambda}){\geqslant}n{\lambda}-\sum{\lambda}_{i}{\geqslant}{\lambda}-{\lambda}_{i}$ for every $i$. Therefore $\operatorname{Supp}_\Delta (\sum {\lambda}_i -(\mu+(n-1){\lambda}) ) \subset \operatorname{Supp}_\Delta ({\lambda}- \mu)$. By Definition \[twin\] together with Lemma \[lem:riduzionelevi\], it is enough to analyse the case $G$ of type ${\mathsf B}_r$ and $\operatorname{Supp}({\lambda}) = \{{\alpha}_1\}$ or $G$ of type ${\mathsf G}_2$ and $\operatorname{Supp}({\lambda}) = \{{\alpha}_2\}$. We analyse these two cases separately.

*Type ${\mathsf B}_r$*: we have ${\lambda}= a \omega_1$, $\mu=(a-1)\omega_1$ and $\mu+(n-1){\lambda}=(na-1)\omega_1$. If $a=1$ we notice that there are no dominant weights between ${\lambda}$ and $\mu$. So the only possibility is ${\lambda}_i={\lambda}=\omega_1$ for all $i$ and this is in contradiction with Lemma \[lem:BGno\]. If $a>1$, notice that there is only one dominant weight between ${\lambda}$ and $\mu$, namely $\nu={\lambda}-{\alpha}_1=(a-2)\omega_1+\omega_2$; hence for all $i$ it must be ${\lambda}_i={\lambda}$ or ${\lambda}_i=\nu$. Since $\sum {\lambda}_i {\geqslant}\mu+(n-1){\lambda}$, at most one ${\lambda}_i$ can be equal to $\nu$; therefore ${V({\mu+(n-1){\lambda}})}\subset {V({{\lambda}})}^{\otimes n}$ or ${V({\mu+(n-1){\lambda}})}\subset {V({\nu})} \otimes {V({{\lambda}})}^{\otimes (n-1)}$. In the first case we obtain $${V({(na-1)\omega_1})}={V({\mu+(n-1){\lambda}})}\subset {V({{\lambda}})}^{\otimes
  n}\subset {V({\omega_1})}^{\otimes na},$$ against Lemma \[lem:BGno\]. In the second case we notice that ${V({\omega_2})} = {\Lambda}^2 {V({\omega_1})} \subset {V({\omega_1})}^{\otimes 2}$, hence ${V({\nu})}\subset {V({(a-2)\omega_1})}\otimes {V({\omega_2})} \subset {V({\omega_1})}^{\otimes a}$ and we can conclude as in the first case.

*Type ${\mathsf G}_2$*: we have ${\lambda}= a \omega_2$, $\mu=\omega_1+(a-1)\omega_2$ and we proceed as in the previous case.

We now prove that the condition is sufficient, showing that for every dominant weight $\mu {\leqslant}{\lambda}$ there exist $n>0$ and weights ${\lambda}_1,\dots,{\lambda}_n \in \overline {\mathrm{LB}}({\lambda})$ such that ${V({\mu+(n-1){\lambda}})}\subset{V({{\lambda}_1})}\otimes \cdots\otimes
{V({{\lambda}_n})}$. To do this, we proceed by decreasing induction with respect to the dominance order.

If $\mu = {\lambda}$ then the claim is clear, so we assume $\mu < {\lambda}$. Let ${\lambda}- \mu = {\beta}_1+\dots+{\beta}_m$ where $\operatorname{Supp}_{\Delta}({\beta}_i)$ are the connected components of $\operatorname{Supp}_{\Delta}({\lambda}-\mu)$. Set $K= \operatorname{Supp}_{\Delta}({\beta}_1)$ and ${\beta}'={\beta}_2+\dots+{\beta}_m$. Notice that $\mu+{\beta}_1$ is dominant: indeed if ${\alpha}\not \in {\overline}{K}$ then $\langle \mu + \beta_1 , {\alpha}^\vee \rangle =  \langle \mu , {\alpha}^\vee \rangle {\geqslant}0$, while if ${\alpha}\in {\overline}{K}$ then $\langle \mu + \beta_1 , {\alpha}^\vee \rangle = \langle {\lambda}- {\beta}',
{\alpha}^\vee \rangle {\geqslant}\langle {\lambda}, {\alpha}^\vee \rangle {\geqslant}0$. Notice moreover that, if $\nu\in \overline {\mathrm{LB}}_K(\mu+{\beta}_1)$, then $\nu+{\beta}' \in \overline
{\mathrm{LB}}({\lambda})$. By Lemma \[lem:induzione\] applied to the semisimple part of the Levi $L=L_K$ associated to $K$, there exists a weight $\mu'$ which is dominant with respect to $K$ such that $\mu < \mu'{\leqslant}\mu+{\beta}_1$ and there exists $\nu \in \overline {\mathrm{LB}}_K(\mu+{\beta}_1)$ which satisfy $${V_L({\mu+{\beta}_1+\mu})}\subset {V_L({\mu'})} \otimes {V_L({\nu})}.$$ By tensoring with ${V_L({{\beta}'})}$, which is a one dimensional representation, we get ${V_L({\mu+{\lambda}})}\subset {V_L({\mu'})} \otimes {V_L({{\lambda}'})}$ with ${\lambda}'=\nu+{\beta}'\in \overline {\mathrm{LB}}({\lambda})$. Since $\langle \mu',{\alpha}^\vee \rangle {\geqslant}\langle \mu+{\beta}_1,{\alpha}^\vee \rangle$ for every ${\alpha}\not \in K$, $\mu'$ is a dominant weight; by Lemma \[lem:riduzionelevi\] we get then ${V({\mu+{\lambda}})}\subset {V({\mu'})} \otimes {V({{\lambda}'})}$ and we may apply the induction on $\mu'{\leqslant}{\lambda}$. Therefore there exist weights ${\lambda}_1,\dots,{\lambda}_n \in \overline {\mathrm{LB}}({\lambda})$ such that ${V({\mu'+(n-1){\lambda}})}\subset{V({{\lambda}_1})}\otimes \cdots\otimes
{V({{\lambda}_n})}$. Finally by Lemma \[lem:traslazione\] we conclude $${V({\mu+n{\lambda}})}\subset {V({\mu'+(n-1){\lambda}})}\otimes {V({{\lambda}'})} \subset
{V({{\lambda}_1})}\otimes \cdots\otimes {V({{\lambda}_n})} \otimes {V({{\lambda}'})}.$$

Smoothness
==========

In this section we will study the variety ${\widetilde}{X}_{\lambda}$; in particular we will give necessary and sufficient conditions on $\operatorname{Supp}({\lambda})$ for its $\mathbb{Q}$-factoriality and for its smoothness.

Thanks to Lemma \[lem:immersioni\], we may assume that $G$ is a simple group. Indeed suppose ${\Delta}= \cup_{i=1}^n {\Delta}_i$ is the decomposition in connected components and write ${\lambda}= {\lambda}_1 +
\ldots + {\lambda}_n$ with $\operatorname{Supp}({\lambda}_i)\subset {\Delta}_i$: correspondingly we get a decomposition $X_{\lambda}= X_{{\lambda}_1} \times \ldots \times
X_{{\lambda}_n}$, and every $X_{{\lambda}_i}$ is an embedding of the corresponding simple factor of $G_\mathrm{ad}$ if ${\lambda}_i \neq 0$ or a point if ${\lambda}_i = 0$. From now on, we will therefore assume that $\Phi$ is an irreducible root system.

By the Bruhat decomposition, the group $G_\mathrm{ad}$ has an open $B\times B^-$-orbit; therefore it is a spherical $G\times
G$-homogeneous space. Following the general theory of spherical embeddings (see [@Kn]), its simple normal embeddings are classified by combinatorial data called the *colored cones*. Here we will skip an overview of such theory, and we will simply recall the definition of the colored cone in the particular case of a simple normal embedding of $G_\mathrm{ad}$.

Recall that a normal variety $X$ is said ${\mathbb Q}$-*factorial* if, given any Weil divisor $D$ of $X$, there exists an integer $n\neq 0$ such that $nD$ is a Cartier divisor. In subsection \[sez smooth-notaz\], we will explicitly describe the colored cone of ${\widetilde}{X}_{\lambda}$; then in subsection \[sez Q-factoriality\] we will study $\mathbb{Q}$-factoriality of ${\widetilde}{X}_{\lambda}$ following [@Br2]. Finally, in subsection \[sezione smoothness\], we will use Theorem \[teo:normalita\] together with the description of the colored cone of ${\widetilde}{X}_{\lambda}$ to make more explicit the criterion of smoothness given in [@Ti] in the case of a linear projective compactification of a reductive group.

The colored cone of ${\widetilde}{X}_{\lambda}$ {#sez smooth-notaz}
-----------------------------------------------

Let $X$ be a simple normal compactification of $G_\mathrm{ad}$, call $Y$ its unique closed orbit. Set ${\mathcal D}(G_\mathrm{ad})$ the set of $B\times B^{-}$-stable prime divisors of $G_\mathrm{ad}$ and ${\mathcal D}(X)\subset {\mathcal D}(G_\mathrm{ad})$ the set of divisors whose closure in $X$ contains $Y$. Let ${\mathcal N}(X)$ be the set of $G\times G$-stable prime divisors of $X$, so that the set of $B\times B^{-}$-stable prime divisors of $X$ is identified with ${\mathcal D}(G_\mathrm{ad})\cup {\mathcal N}(X)$.

Let $T_\mathrm{ad}\subset G_\mathrm{ad}$ be the image of $T$; then the character group ${\mathcal X}(T_\mathrm{ad})$ coincides with the root lattice $\mathbb{Z}{\Delta}$, while the cocharacter group ${\mathcal X}^\vee(T_\mathrm{ad})$ coincides with the coweight lattice ${\Lambda}^\vee$. If $V$ is a simple $G\times G$-module denote by $V^{(B\times B^{-})}$ the subset of $B\times B^{-}$-eigenvectors. Notice that ${\Bbbk}(G_\mathrm{ad})^{(B\times B^{-})}/{\Bbbk}^{*} \simeq \mathbb{Z}{\Delta}$ and define a natural map $\rho : {\mathcal D}(G_\mathrm{ad})\cup {\mathcal N}(X) \to {\Lambda}^\vee$ by associating to a $B\times B^-$-stable prime divisor of $X$ the cocharacter associated to the rational discrete valuation induced by $D$. If $D\in {\mathcal N}(X)$, then $\rho(D)$ is the opposite of a fundamental coweight, while if $D\in {\mathcal D}(G_\mathrm{ad})$, then $\rho(D)$ is a simple coroot; moreover, $\rho$ is injective and $\rho({\mathcal D}(G_\mathrm{ad}))=\Delta^\vee$ (see [@Ti § 7]).

Let ${\mathcal C}(X)$ be the convex cone in ${\Lambda}^\vee_{\mathbb{Q}}$ generated by $\rho\big({\mathcal D}(X)\cup {\mathcal N}(X)\big)$; by the general theory of spherical embeddings we have that ${\mathcal C}(X)$ is generated by $\rho({\mathcal D}(X))$ together with the negative Weyl chamber of $\Phi$. The *colored cone* of $X$ is then the couple $\big({\mathcal C}(X),{\mathcal D}(X)\big)$: up to equivariant isomorphisms, it uniquely determines $X$ as a $G\times G$-compactification of $G_\mathrm{ad}$.

In the case of the compactification $\widetilde{X}_{{\lambda}}$, then $\rho({\mathcal D}(X)) = {\Delta}^\vee {\smallsetminus}\operatorname{Supp}(\lambda)^\vee$ (see [@Ti Theorem 7]).

$\mathbb{Q}$-factoriality {#sez Q-factoriality}
-------------------------

In order to give a necessary and sufficient condition for the $\mathbb{Q}$-factoriality of ${\widetilde}{X}_{\lambda}$ we need to determine the set of extremal rays of the associated cone ${\mathcal C}({\widetilde}{X}_{\lambda})$.

If ${\alpha}\in {\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$, then ${\alpha}^\vee$ generates an extremal ray of ${\mathcal C}({\widetilde}{X}_{\lambda})$.

If a simple coroot ${\alpha}^\vee \in {\mathcal C}({\widetilde}{X}_{\lambda})$ does not generate an extremal ray, then we can write $${\alpha}^\vee = \sum_{{\beta}\in {\Delta}{\smallsetminus}\{{\alpha}\}} a_{{\beta}}{\beta}^{\vee} - \sum_{{\beta}\in {\Delta}}
    b_{{\beta}}\omega_{{\beta}}^{\vee},$$ with $a_{\beta}, b_{\beta}{\geqslant}0$ for every ${\beta}$: this yields a contradiction since then it would be $\langle {\alpha},{\alpha}^\vee \rangle {\leqslant}0$.

Recall that a convex cone is said to be *simplicial* if it is generated by linearly independent vectors; the following proposition is a particular case of a characterization of $\mathbb{Q}$-factoriality that M. Brion gave in [@Br2] in the general case of a spherical variety. We recall it in the case of our interest.

\[see [[@Br2 Proposition 4.2]]{}\] \[lem Q fattor\] The variety ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial if and only if ${\mathcal C}({\widetilde}{X}_{\lambda})$ is simplicial.

Therefore, since ${\mathcal C}({\widetilde}{X}_{\lambda})$ has maximal dimension, ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial if and only if the number of extremal rays of the associated cone equals the rank of $G$. To describe such rays we need to introduce some more notation; the description will be slightly more complicated if $\Phi$ is of type ${\mathsf D}$ or ${\mathsf E}$.

Denote ${\Delta}^e$ the set of extremal roots of ${\Delta}$ and set ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}) = \bigcup_{i=1}^n I_i$ the decomposition in connected components. Denote $$I^e = \bigcup_{\substack{I_i \neq I_\mathsf{de} \\ I_i \cap {\Delta}^e \neq {\varnothing}}} I_i,$$ where $I_\mathsf{de}$ is defined as follows. If ${\Delta}$ is of type $\sf{D}$ or $\sf{E}$, denote ${\gamma}_\mathsf{de}$ the unique simple root which is adjacent to other three simple roots and, if it exists, denote $I_\mathsf{de} \subset {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ the unique connected component such that ${\gamma}_\mathsf{de} \in I_\mathsf{de}$ and $|I_\mathsf{de} \cap {\Delta}^e| = 1$, otherwise define $I_\mathsf{de}$ to be the empty set. Denote $I^\ast_\mathsf{de} \subset I_\mathsf{de}$ the minimal connected subset such that ${\gamma}_\mathsf{de} \in I^\ast_\mathsf{de}$ and $I^\ast_\mathsf{de} \cap {\Delta}^e \neq {\varnothing}$, or define it to be the empty set otherwise. Finally define $$J({\lambda}) = \big({\Delta}{\smallsetminus}({\overline}{I^e} \cup I^\ast_\mathsf{de})\big) \cup \big({\Delta}^e {\smallsetminus}\operatorname{Supp}({\lambda})\big).$$

\[raggi estremali2\] The extremal rays of ${\mathcal C}({\widetilde}{X}_{\lambda})$ are generated by the simple coroots $\alpha^{\vee}$ with $\alpha\in {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ and by the opposite of fundamental coweights $-\omega_{\alpha}^{\vee}$ with $\alpha\in J({\lambda})$.

Recall that ${\mathcal C}({\widetilde}{X}_{\lambda})$ is generated by the simple coroots $\alpha^{\vee}$ with $\alpha\in {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ together with the fundamental coweights $-\omega_{\alpha}^{\vee}$ with $\alpha\in {\Delta}$ and that every coroot $\alpha^{\vee}$ with $\alpha\in {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ generates an extremal ray of ${\mathcal C}({\widetilde}{X}_{\lambda})$.

A coweight $-\omega_{{\alpha}}^{\vee}$ does not generate an extremal ray if and only if it can be written as follows $$-\omega_{\alpha}^{\vee}=\sum_{{\beta}\in K}
    a_{{\beta}}{\beta}^{\vee} - \sum_{{\beta}\in H}
    b_{{\beta}} \omega_{{\beta}}^{\vee}$$ with $a_{\beta}>0 $ for every ${\beta}\in K$ and with $b_{\beta}>0$ for every ${\beta}\in H$, for suitable non-empty subsets $K\subset {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ and $H \subset {\Delta}{\smallsetminus}\{{\alpha}\}$. Since the right member of the equality is negative against every simple root in $\partial{K}$, we get $\partial{K} = \{{\alpha}\}$.

Notice that $K$ is connected. Indeed if $K' \subset K$ is a connected component then $\partial{K'} = \{{\alpha}\}$ and $\sum_{{\beta}\in K'} a_{{\beta}} \langle {\alpha}, {\beta}^{\vee} \rangle < 0$: therefore if $K$ contains two connected components it must be $$\sum_{{\beta}\in K} a_{{\beta}} \langle {\alpha}, {\beta}^{\vee} \rangle {\leqslant}-2.$$ On the other hand $\langle {\alpha}, \omega_{{\beta}}^{\vee} \rangle = 0$ for every ${\beta}\in H$, therefore if $K$ is not connected it follows $$-1 = - \langle {\alpha}, \omega_{\alpha}^{\vee} \rangle 
    = \sum_{{\beta}\in K} a_{{\beta}} \langle {\alpha}, {\beta}^{\vee} \rangle {\leqslant}-2.$$

Since $\partial{K}$ is one single root, $K$ contains an extreme of ${\Delta}$, thus we get $K\subset I^{e} \cup I_\mathsf{de}$. Suppose that ${\gamma}_\mathsf{de} \in K \subset I_\mathsf{de}$: then we get a contradiction since it would be $|\partial{K}|=2$. Therefore we get $K\subset I^{e} \cup (I^\ast_\mathsf{de} {\smallsetminus}\{{\gamma}_\mathsf{de}\})$ and ${\alpha}\in {\overline}{I^{e}} \cup I^\ast_\mathsf{de}$. Such a subset $K$ cannot exist if ${\alpha}\in {\Delta}^{e}{\smallsetminus}\operatorname{Supp}({\lambda})$, otherwise it would be $K = {\Delta}{\smallsetminus}\{{\alpha}\}$ which intersects $\operatorname{Supp}({\lambda})$. We get then that every $-{\omega}_{\alpha}^\vee$ with ${\alpha}\in J({\lambda})$ generates an extremal ray of ${\mathcal C}({\widetilde}{X}_{\lambda})$.

Suppose conversely that ${\alpha}\not \in J({\lambda})$. Then we can construct a connected subset $K\subset I^{e} \cup (I^\ast_\mathsf{de} {\smallsetminus}\{{\gamma}_\mathsf{de}\})$ such that $\partial{K} = \{{\alpha}\}$. If ${\gamma}\in K\cap {\Delta}^e$, consider the fundamental coweight $({\omega}_{\gamma}^K)^\vee$ associated to ${\gamma}$ in the irreducible root subsystem associated to $K$: then we get $$({\omega}_{\gamma}^K)^\vee = \sum_{{\beta}\in K}a_{\beta}{\beta}^\vee = {\omega}_{\gamma}^\vee - m {\omega}^\vee_{\alpha},$$ where $a_{\beta}>0$ are rational coefficients and where $m>0$ is an integer. Therefore $-{\omega}_{\alpha}^\vee$ does not generate an extremal ray of ${\mathcal C}({\widetilde}{X}_{\lambda})$.

\[Q-fattorialita\] The variety ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial if and only if the following conditions are fulfilled:

-   $\operatorname{Supp}({\lambda})$ is connected;

-   If $\operatorname{Supp}({\lambda})$ contains a unique element, then this element is an extreme of ${\Delta}$;

-   If ${\Delta}$ is of type ${\mathsf D}$ or ${\mathsf E}$, then $\operatorname{Supp}({\lambda})$ contains ${\gamma}_\mathsf{de}$ and at least two simple roots adjacent to ${\gamma}_\mathsf{de}$.

By Proposition \[lem Q fattor\] together with Lemma \[raggi estremali2\] ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial if and only if $|\operatorname{Supp}({\lambda})| = |J({\lambda})|$.

Suppose that ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial. Consider the dominant weight ${\lambda}' = \sum_{{\alpha}\not \in I^e \cup I^\ast_\mathsf{de}} {\omega}_{\alpha}$: then $J({\lambda}') = J({\lambda})$ and $$|{\Delta}| = |J({\lambda})| + |{\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})| {\geqslant}|J({\lambda}')| + |{\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}')| {\geqslant}|{\Delta}|,$$ which implies $\operatorname{Supp}({\lambda}) = \operatorname{Supp}({\lambda}')$. This shows ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}) = I^e \cup I^\ast_\mathsf{de}$, and we get the following decomposition of $J({\lambda})$: $$J({\lambda}) \cap \operatorname{Supp}({\lambda}) = {\Delta}{\smallsetminus}({\overline}{I^e}\cup I^\ast_\mathsf{de}), \qquad    \quad
    J({\lambda}) {\smallsetminus}\operatorname{Supp}({\lambda}) = {\Delta}^e {\smallsetminus}\operatorname{Supp}({\lambda}).$$

If $I_\mathsf{de} \neq {\varnothing}$, set $I_\mathsf{de} \cap {\Delta}^e = \{{\alpha}_\mathsf{de}\}$. Define a surjective map $F: J({\lambda}) {\smallsetminus}\{{\alpha}_\mathsf{de}\} {\longrightarrow}\operatorname{Supp}({\lambda})$ as follows: $F$ is the identity on $J({\lambda}) \cap \operatorname{Supp}({\lambda})$, while if ${\alpha}\in J({\lambda}) {\smallsetminus}\operatorname{Supp}({\lambda})$ consider the connected component $K\subset {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ containing ${\alpha}$ and define $F({\alpha})$ by the relation $\partial{K} = \{F({\alpha})\}$: since ${\alpha}\neq {\alpha}_\mathsf{de}$, it must be $|\partial{K}|=1$. Therefore $F$ is well defined and it is surjective since $\operatorname{Supp}({\lambda}) {\smallsetminus}J({\lambda}) = \partial{I^e}$. Therefore ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}) = I^e$ and we get i). Being surjective, $F$ has to be injective as well; this easily implies both ii) and iii).

Suppose conversely that $\operatorname{Supp}({\lambda})$ is connected, or equivalently that ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}) = I^e$: then ii) and iii) imply $|{\Delta}^e {\smallsetminus}\operatorname{Supp}({\lambda})| = |\partial{I^e}|$. This shows that ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial, since then $|J({\lambda})| + |{\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})| = |{\Delta}|$.

\[cor:raggi estremali\] If ${\widetilde}{X}_{\lambda}$ is $\mathbb{Q}$-factorial, the extremal rays of ${\mathcal C}({\widetilde}{X}_{\lambda})$ are generated by:

-   the coroots $\alpha^{\vee}$ with $\alpha\in {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$,

-   the coweights $-\omega_{\alpha}^{\vee}$ with $\alpha\in
                    \operatorname{Supp}({\lambda})^\circ \cup \big({\Delta}^e {\smallsetminus}\operatorname{Supp}({\lambda})\big)$.

Smoothness {#sezione smoothness}
----------

Suppose that ${\Sigma}=\{{\lambda},{\lambda}_1,\ldots,{\lambda}_s\}$ is a simple set of dominant weights, where ${\lambda}$ is the maximal one. In this section we will prove the following generalization of Theorem B.

\[smooth Xsigma\] The variety $X_{\Sigma}$ is smooth if and only if $X_{\lambda}$ is normal, $\mathbb{Q}$-factorial and every connected component of ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ has type ${\mathsf A}$.

$X_{\Sigma}$ is smooth if and only if $X_{\lambda}$ is smooth.

To prove Theorem \[smooth Xsigma\], we will make use of a characterization of smoothness for arbitrary group compactifications given by D. Timashev in [@Ti]. For convenience, we will use a generalization of it which can be found in [@Ru] in the more general context of symmetric spaces. We recall it in the case of a simple group compactification.

\[smooth-timashev\] The variety ${\widetilde}{X}_{{\lambda}}$ is smooth if and only if the following conditions are fulfilled:

-   All connected components of ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ are of type ${\mathsf A}$ and there are no more than $|\operatorname{Supp}(\lambda)|$ of them.

-   The cone ${\mathcal C}({\widetilde}{X}_{\lambda})$ is simplicial and it is generated by a basis of the coweight lattice $\Lambda^{\vee}$.

-   One can enumerate the simple roots in order of their positions at Dynkin diagrams of connected components $I_k = \{\alpha_{1}^{k},\ldots,\alpha_{n_{k}}^{k}\}$ of ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ , $k=1,\ldots,n$, and partition the basis of the free semigroup ${\mathcal C}({\widetilde}{X}_{\lambda})^{\vee}\cap \mathbb{Z}{\Delta}$ into subsets $\{\pi_{1}^{k},\ldots,\pi_{n_{k}+1}^{k}\}$, $k=1,\ldots,p$, $p{\geqslant}n$, in such a way that $\langle\pi_{j}^{k}, (\alpha_{i}^{h})^\vee \rangle = \delta_{i,j}\delta_{h,k}$ and $\pi_{j}^{k} - \frac{j}{n_{k}+1} \pi_{n_{k}+1}^{k}$ is the $j$-th fundamental weight of the root system generated by $\{\alpha_{1}^{k},...,\alpha_{n_{k}}^{k}\}$ for all $j,k$.

First, we prove that the conditions are necessary; since we only have to prove that $X_{\lambda}$ is normal, we may assume that ${\Delta}$ is non-simply laced. By Theorem \[smooth-timashev\] i), $\operatorname{Supp}({\lambda})$ contains at least one of the two simple roots $\alpha_S$, ${\alpha}_L$; suppose that $\operatorname{Supp}({\lambda})$ contains $\alpha_L$ but not ${\alpha}_S$. Denote $K =\{{\alpha}_1,\ldots,{\alpha}_l\} \subset {\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ the connected component which contains ${\alpha}_S$ and number its simple roots starting from ${\alpha}_S$: therefore ${\alpha}_1 = {\alpha}_S$ and ${\alpha}_l \in {\Delta}^e$, moreover ${\overline}{K}$ is either of type ${\sf C}_{l+1}$ or of type $\sf{G}_2$. Set ${\omega}^\vee = (l+1)({\omega}^K_l)^\vee$, where $({\omega}^K_l)^\vee$ is the fundamental coweight associated to ${\alpha}_l$ in the root subsystem $\Phi_K$ associated to $K$; then $${\omega}^\vee = \sum_{i=1}^{l} i {\alpha}^\vee_{i} =
(l+1) {\omega}^\vee_{{\alpha}_l} -m {\omega}_{{\alpha}_L}^\vee.$$ where $m = 2$ if ${\overline}{K}$ is of type ${\sf C}_{l+1}$ (with $l{\geqslant}1$) and $m=3$ if ${\overline}{K}$ is of type ${\sf G}_2$.

If ${\overline}{K}$ is not of type ${\sf B}_2$, then ${\Delta}$ is either of type ${\sf C}_r$ (with $r>2$) or of type ${\sf F}_4$ or of type ${\sf G}_2$ and every simple coroot ${\beta}^\vee \in {\Delta}^\vee$ is a primitive element in $\Lambda^{\vee}$ (i.e. there does not exist $\pi^\vee\in {\Lambda}^\vee$ which satisfies $t\pi^\vee = {\beta}^\vee$ with $t>1$): therefore by Lemma \[raggi estremali2\] together with Theorem \[smooth-timashev\] ii) $\{{\alpha}^\vee_1, \ldots, {\alpha}^\vee_l, {\omega}^\vee_{{\alpha}_l} \}$ is part of a basis of ${\Lambda}^\vee$ and we get a contradiction since then the equality above would imply ${\omega}_{{\alpha}_L}^\vee\not \in {\Lambda}^\vee$. Otherwise ${\overline}{K}$ is of type ${\sf B}_2$, thus ${\Delta}$ is of type ${\sf B}_r$ and $\frac{1}{2}{\alpha}_S^\vee \in {\Lambda}^\vee$: then we get a contradiction since by Theorem \[smooth-timashev\] iii) there exists $\pi \in {\mathcal C}({\widetilde}{X}_{\lambda})^{\vee}\cap \mathbb{Z}{\Delta}$ such that $\langle \pi, {\alpha}_S^\vee \rangle = 1$.

Let’s prove now that conditions of Theorem \[smooth-timashev\] are verified if $X_{\lambda}$ is normal, $\mathbb{Q}$-factorial and ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ has type ${\mathsf A}$. Set $N = {\mathcal C}({\widetilde}{X}_{\lambda}) \cap {\Lambda}^\vee$ the monoid generated by the primitive elements of the extremal rays of ${\mathcal C}({\widetilde}{X}_{\lambda})$.

To prove condition i), it is enough to notice as in Proposition \[Q-fattorialita\] that, since $\operatorname{Supp}({\lambda})$ is connected, we have ${\Delta}{\smallsetminus}\operatorname{Supp}({\lambda}) = I^e$ and the number of its connected components equals $|{\Delta}^e {\smallsetminus}\operatorname{Supp}({\lambda})| {\leqslant}|J({\lambda})| = |\operatorname{Supp}({\lambda})|$.

To prove condition ii), let’s show that, if ${\beta}\in {\Delta}{\smallsetminus}J({\lambda}) = {\overline}{I^e}{\smallsetminus}{\Delta}^e$, then $-\omega_{{\beta}}^{\vee} \in N$. Denote $I =\{{\alpha}_1,\ldots,{\alpha}_l\} \subset {\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ the connected component which contains ${\beta}$ in its closure and number its simple roots starting from the extreme of $I$ which is not an extreme of ${\Delta}$; therefore ${\alpha}_l \in {\Delta}^e$. Let $j$ be such that ${\beta}= {\alpha}_j$ or set $j = 0$ if ${\beta}\in \operatorname{Supp}({\lambda})$. Set $K=\{{\alpha}_{j+1},\ldots,{\alpha}_{l}\}$ and set ${\omega}^\vee = (l-j+1)({\omega}^K_l)^\vee$, where $({\omega}^K_l)^\vee$ is the fundamental weight associated to ${\alpha}_l$ in the root subsystem $\Phi_K$ associated to $K$; then $${\omega}^\vee = \sum_{i=1}^{l-j} i {\alpha}^\vee_{j+i} =
(l-j+1){\omega}^\vee_{{\alpha}_l} + \langle {\beta},{\alpha}_{j+1}^\vee \rangle {\omega}_{{\beta}}^\vee.$$ Since $X_{\lambda}$ is normal, by Theorem A we get $\langle {\beta},{\alpha}_{j+1}^\vee \rangle = -1$; therefore by Corollary \[cor:raggi estremali\] $-{\omega}_{{\beta}}^\vee \in N$.

Finally let’s show that condition iii) holds. Suppose that $K = \{{\alpha}_1, \ldots, {\alpha}_l \} \subset {\Delta}{\smallsetminus}\operatorname{Supp}({\lambda})$ is a connected component, where the simple roots in $K$ are numbered starting from the extreme of $K$ which is not an extreme of ${\Delta}$, and define $$\pi_i^K =
\left\{ \begin{array}{ll}
        ({\alpha}_i^\vee)^\ast & \textrm{ if $i{\leqslant}l$} \\
        (-{\omega}^\vee_{{\alpha}_l})^\ast & \textrm{ if $i=l+1$}
\end{array} \right.$$ where, if $\{v_1, \ldots, v_r\}$ is a basis of ${\Lambda}^\vee$, $\{v_1^\ast, \ldots, v_r^\ast\}$ denotes the dual basis of ${\Lambda}$. Therefore, if $\omega_{j}^{K}$ is the $j$-th fundamental weight of $\Phi_{K}$, we have $\omega_{j}^{K} = \pi^K_j - \frac{j}{l+1} \pi^K_{l+1}$.

Remarks and generalizations
===========================

In this section we will consider the more general situation of compactifications of symmetric varieties.

Let $G$ be as before and ${\sigma}:G\to G$ an involution of $G$. We denote by $H^\circ$ the subgroup of points fixed by ${\sigma}$ and by $H$ its normalizer. The notation is not completely coherent with those of previous sections: $G$ plays now the role that $G\times G$ played before, while $H^\circ$ has now the role that the diagonal of $G\times G$ had before.

Let $\Omega^+$ be the set of dominant weights ${\lambda}$ such that ${V({{\lambda}})}$ has a non-zero vector fixed by $H^\circ$ and $\Omega$ the sublattice of ${\Lambda}$ generated by $\Omega^+$. The monoid $\Omega^+$ (resp. the lattice $\Omega$) is in a natural way the set of dominant weights (resp. the set of weights) of a (possibly non-reduced) root system ${\widetilde \Phi}$, which is called the [*restricted root system*]{}. For ${\lambda}\in \Omega^+$ we can consider the (unique) point $x_{\lambda}\in {\mathbb P}({V({{\lambda}})})$ fixed by $H$ and define $X_{\lambda}$ as the closure of the $G$-orbit of $x_{\lambda}$ in ${\mathbb P}({V({{\lambda}})})$.

Proposition \[prp:supporto\] generalizes to this more general situation without any further comment.

Normality of $X_{\lambda}$ and the closure of a maximal torus orbit.
--------------------------------------------------------------------

Let $T\subset G$ be a maximal torus such that the dimension of $TH$ is maximal and let $Z_{\lambda}= {\overline}{T\,x_{\lambda}} \subset X_{\lambda}$. In [@Ru], it is proved that when $X_{\lambda}$ is normal then $Z_{\lambda}$ also is normal. The converse of this result does not hold in general. Indeed $Z_{\lambda}$ is always normal in the case of the $G\times G$-compactification of $G_\mathrm{ad}$.

Generalization to symmetric varieties: normality {#ssez:simmetrichenormalita}
------------------------------------------------

The wonderful compactification has been defined in the more general situation of symmetric varieties and the description of the normalization of $X_{\lambda}$ generalizes thanks to the results contained in [@CM] and [@CDM] (which generalize [@Ka] and [@DC]). In particular, Lemma \[lem:general-normality\] holds here in general. However, in the case of symmetric varieties we do not have a clear description of the multiplication of sections as in Lemma \[lem:coefficientimatriciali\]. In particular, we have no analogue of Proposition \[prp:normalita\].

One may wonder whether the normality of $X_{\lambda}$ is equivalent to the analogous combinatorial condition on the weight ${\lambda}$, that is, ${\lambda}$ satisfies condition $(\star)$ w.r.t. the root system ${\widetilde \Phi}$; here is a counterexample.

Let $G$ be of type ${\sf B}_2$ and let ${\sigma}$ be the involution of type B I: thus $G/H \simeq \mathrm{SO}(5)/\mathrm{S}\big(\mathrm{O}(3)\times \mathrm{O}(2)\big)$ and ${\widetilde\Delta}=2{\Delta}$. Consider ${\lambda}=2{\omega}_1\in{\Omega}^+$; then $X_{\lambda}$ is a normal embedding of $G/H$.

Denote by ${\leqslant}_{\sigma}$ the dominance order w.r.t. the root system ${\widetilde \Phi}$ and suppose that $X_{\lambda}$ is normal. Then ${\lambda}$ satisfies

> *for all $\mu\in{\Omega}^+$ such that $\mu{\leqslant}_{\sigma}{\lambda}$ there exists $n\in{\mathbb N}$ such that ${V({\mu+(n-1){\lambda}})}\subset
>   \mathrm S^n({V({{\lambda}})})$.*

If one assumes that the multiplication map is as generic as possible, then also the converse is true.

Generalization to symmetric varieties: smoothness
-------------------------------------------------

In the setting of normal compactifications of symmetric varieties $G/H^\circ$, fix a maximal torus $T$ such that $TH^\circ$ has maximal dimension and a Borel subgroup $B\supset T$ such that $BH^\circ \subset G$ is dense. If $X$ is a simple normal compactification of $G/H$, denote ${\mathcal D}(X)$ the set of $B$-stable and not $G$-stable prime divisors of $X$ which contain the closed orbit. Denote $\rho : {\mathcal D}(X) \to {\Omega}^\vee$ the map defined by the evaluation of functions; by [@Vu Proposition 1] $\rho({\mathcal D}(X))$ is a basis of the restricted coroot system ${\widetilde}{\Phi}^\vee$. Since the map $\rho$ is not always injective, following the criterion of $\mathbb{Q}$-factoriality in [@Br2] in order to generalize Proposition \[Q-fattorialita\] we only need to assume that $\rho$ is injective on ${\mathcal D}(X)$, and the proof is the same. Such proposition is true also for compactifications of $G/H^\circ$, and not only of $G/H$, since $\mathbb Q$-factoriality concerns no integrality questions.

Theorem \[smooth Xsigma\] also can be generalized to this setting with the same proof, but we do not have anymore the equivalence between property $(\star)$ and the normality of $X_{\lambda}$. Thus the theorem has to be reformulated as follows (recall that a simple normal spherical variety is always quasi-projective).

\[smooth general\] A simple normal compactification $X$ of $G/H$ is smooth if and only if it is $\mathbb{Q}$-factorial, ${\Delta}{\smallsetminus}\rho({\mathcal D}(X))$ satisfies $(\star)$ and every connected component of $\rho({\mathcal D}(X))$ has type ${\mathsf A}$.

[KKLV]{}

N. Bourbaki, [*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV, V, VI*]{}, Actualités Scientifiques et Industrielles **1337**, Hermann Paris 1968.

M. Brion, [*Variétés sphériques et théorie de Mori*]{}, Duke Math. J. **72** (1993) no. 2, 369–404.

R. Chirivì, C. De Concini and A. Maffei, [*On normality of cones over symmetric varieties*]{}, Tohoku Math. J. (2) **58** (2006) no. 4, 599–616.

R. Chiriv[ì]{} and A. Maffei, [ *Projective normality of complete symmetric varieties*]{}, Duke Math. J. **122** (2004), 93–123.

C. De Concini, [*Normality and non normality of certain semigroups and orbit closures*]{}, Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci. **132**, Springer Berlin 2004, 15–35.

C. De Concini and C. Procesi, [*Complete symmetric varieties*]{}, Invariant Theory, Lecture Notes in Math. **996**, Springer, Berlin, 1983, 1–44.

S.S. Kannan, [*Projective normality of the wonderful compactification of semisimple adjoint groups*]{}, Math. Z. **239** (2002) 673–682.

F. Knop, [*The Luna-Vust theory of spherical embeddings*]{}, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 225–249, Manoj Prakashan, Madras, 1991.

F. Knop, H. Kraft, D. Luna and T. Vust, [*Local properties of algebraic group actions*]{}, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem. **13**, Birkhäuser, Basel, 1989, 63–75.

A. Ruzzi, [*Smooth projective symmetric varieties with Picard number equal to one*]{}. To appear in Internat. J. Math.

J.R. Stembridge, [*The partial order of dominant weights*]{}, Adv. Math. **136** (1998) no. 2, 340–364.

D.A. Timashev, [*Equivariant compactifications of reductive groups*]{}, Sb. Math. **194** (2003) no. 3-4, 589–616.

Th. Vust, [*Plongements d’espaces symétriques algébriques: une classification*]{}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) **17** (1990), no. 2, 165–195.
