Normality and smoothness of simple linear group compactifications
Identifieur interne : 000161 ( Main/Curation ); précédent : 000160; suivant : 000162Normality and smoothness of simple linear group compactifications
Auteurs : Jacopo Gandini [Italie] ; Alessandro Ruzzi [France]Source :
- Mathematische Zeitschrift [ 0025-5874 ] ; 2013-10-01.
Abstract
Abstract: Given a semisimple algebraic group $$G$$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $$G\times G$$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $$\mathbb{P }(\mathrm{End}(V))$$ , where $$V$$ is a finite dimensional rational $$G$$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $$V$$ . In particular, we show that $${\mathrm{Sp}}(2r)$$ (with $$r \geqslant 1$$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.
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DOI: 10.1007/s00209-012-1136-3
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<front><div type="abstract" xml:lang="en">Abstract: Given a semisimple algebraic group $$G$$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $$G\times G$$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $$\mathbb{P }(\mathrm{End}(V))$$ , where $$V$$ is a finite dimensional rational $$G$$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $$V$$ . In particular, we show that $${\mathrm{Sp}}(2r)$$ (with $$r \geqslant 1$$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.</div>
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