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Normality and non-normality of group compactifications in simple projective spaces. (English. French summary) Zbl 1300.14054
Let $$G$$ 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.

##### MSC:
 14M27 Compactifications; symmetric and spherical varieties 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
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