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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Acknowledgments
We would like to thank A. Maffei for fruitful conversations on the subject. As well, we would like to thank the referee for his careful reading and for useful suggestions and remarks.
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Gandini, J., Ruzzi, A. Normality and smoothness of simple linear group compactifications. Math. Z. 275, 307–329 (2013). https://doi.org/10.1007/s00209-012-1136-3
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DOI: https://doi.org/10.1007/s00209-012-1136-3