Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 307–329 | Cite as

Normality and smoothness of simple linear group compactifications

Article

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Dipartimento Di Matematica “Guido Castelnuovo”“Sapienza” Università Di RomaRomaItaly
  2. 2.Laboratoire De MathématiquesUniversité Blaise Pascal, UMR 6620Aubière CedexFrance

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