Normality and non-normality of group compactifications in simple projective spaces.

*(English. French summary)*Zbl 1300.14054Let \(G\) be a simply connected semisimple algebraic group over an algebraically closed field of characteristic zero and \(V(\lambda)\) be a simple \(G\)-module of highest weight \(\lambda\). Then \(\text{End}V(\lambda)\) is a simple \((G\times G)\)-module and \(\mathbb{P}(\mathrm{End}V(\lambda))\) is a \((G\times G)\)-variety. The orbit closure \(X_{\lambda}=\overline{(G\times G)[{\roman{id}}_{V(\lambda)}]}\subset\mathbb{P}(\text{End}V(\lambda))\) is an equivariant compactification of the image of \(G\) in \(\mathrm{PGL}(V(\lambda))\) considered as a symmetric space. The nature of singularities of the varieties \(X_{\lambda}\) is studied in the paper under review.

The following theorems are the main results of the paper: (A) \(X_{\lambda}\) is normal if and only if, whenever the support \(\text{Supp}(\lambda)\) of the highest weight \(\lambda\) contains a long simple root in a non-simply laced component of the Dynkin diagram of \(G\), it contains the short simple root which is adjacent to a long one in the component; (B) \(X_{\lambda}\) is smooth if and only if it is normal and the following three conditions hold:

(1) the intersection of \(\text{Supp}(\lambda)\) with each component of the Dynkin diagram is connected and is an extreme vertex of the component whenever it is a one-element set;

(2) \(\text{Supp}(\lambda)\) contains every junction vertex together with at least two neighboring vertices;

(3) the complement of \(\text{Supp}(\lambda)\) is a subdiagram having all components of type A.

These results specify general criteria for normality and smoothness of projective compactifications of reductive groups obtained by D. A. Timashev [Sb. Math. 194, No. 4, 589–616 (2003); translation from Mat. Sb. 194, No. 4, 119–146 (2003; Zbl 1074.14043)]. A criterion for projective normality of \(X_{\lambda}\) was obtained by S. S. Kannan [Math. Z. 239, No. 4, 673–682 (2002; Zbl 0997.14012)]: \(X_{\lambda}\) is projectively normal if and only if \(\lambda\) is minuscule. It should be noted that the criteria obtained in the paper are of pure combinatorial nature and easy to check. In fact, Theorems A and B are extended in the paper to a wider class of simple projective group compactifications \(X_{\Sigma}=\overline{(G\times G)[{\roman{id}}_V]}\subset\mathbb{P}(\text{End}V)\), where \(V=\bigoplus_{\lambda\in\Sigma}V(\lambda)\) is any (multiplicity-free) \(G\)-module and the set of highest weights \(\Sigma\) contains a unique maximal element \(\lambda_{\max}\) with respect to the dominance order. (“Simple” means here “having a unique closed orbit”.) In particular, it follows that \(X_{\Sigma}\) is smooth if and only if \(X_{\lambda_{\max}}\) is smooth.

Methods of the proofs include using wonderful compactifications of semisimple groups, results of Kannan on projective normality, Timashev’s criterion for smoothness, etc. Further generalizations to completions of arbitrary symmetric spaces \(G/H\) in \(\mathbb{P}(V(\lambda))\) are also possible, but require different methods.

The following theorems are the main results of the paper: (A) \(X_{\lambda}\) is normal if and only if, whenever the support \(\text{Supp}(\lambda)\) of the highest weight \(\lambda\) contains a long simple root in a non-simply laced component of the Dynkin diagram of \(G\), it contains the short simple root which is adjacent to a long one in the component; (B) \(X_{\lambda}\) is smooth if and only if it is normal and the following three conditions hold:

(1) the intersection of \(\text{Supp}(\lambda)\) with each component of the Dynkin diagram is connected and is an extreme vertex of the component whenever it is a one-element set;

(2) \(\text{Supp}(\lambda)\) contains every junction vertex together with at least two neighboring vertices;

(3) the complement of \(\text{Supp}(\lambda)\) is a subdiagram having all components of type A.

These results specify general criteria for normality and smoothness of projective compactifications of reductive groups obtained by D. A. Timashev [Sb. Math. 194, No. 4, 589–616 (2003); translation from Mat. Sb. 194, No. 4, 119–146 (2003; Zbl 1074.14043)]. A criterion for projective normality of \(X_{\lambda}\) was obtained by S. S. Kannan [Math. Z. 239, No. 4, 673–682 (2002; Zbl 0997.14012)]: \(X_{\lambda}\) is projectively normal if and only if \(\lambda\) is minuscule. It should be noted that the criteria obtained in the paper are of pure combinatorial nature and easy to check. In fact, Theorems A and B are extended in the paper to a wider class of simple projective group compactifications \(X_{\Sigma}=\overline{(G\times G)[{\roman{id}}_V]}\subset\mathbb{P}(\text{End}V)\), where \(V=\bigoplus_{\lambda\in\Sigma}V(\lambda)\) is any (multiplicity-free) \(G\)-module and the set of highest weights \(\Sigma\) contains a unique maximal element \(\lambda_{\max}\) with respect to the dominance order. (“Simple” means here “having a unique closed orbit”.) In particular, it follows that \(X_{\Sigma}\) is smooth if and only if \(X_{\lambda_{\max}}\) is smooth.

Methods of the proofs include using wonderful compactifications of semisimple groups, results of Kannan on projective normality, Timashev’s criterion for smoothness, etc. Further generalizations to completions of arbitrary symmetric spaces \(G/H\) in \(\mathbb{P}(V(\lambda))\) are also possible, but require different methods.

Reviewer: Dmitri A. Timashev (Moskva)

##### MSC:

14M27 | Compactifications; symmetric and spherical varieties |

14L30 | Group actions on varieties or schemes (quotients) |

14M17 | Homogeneous spaces and generalizations |

##### Keywords:

semisimple algebraic groups; projective representations; group compactifications; wonderful varieties; symmetric spaces##### References:

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