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Demazure embeddings are smooth. (English) Zbl 1183.14068

Let \(G\) be a connected semisimple algebraic group of adjoint type over an algebraically closed field of characteristic zero, and \(\mathrm{g}\) be its Lie algebra. Recall that a subalgebra \(\mathrm{h}\subseteq\mathrm{g}\) is said to be spherical if there is a Borel subalgebra \(\mathrm{b}\subseteq\mathrm{g}\) such that \(\mathrm{h}+\mathrm{b}=\mathrm{g}\). Suppose that the spherical subalgebra \(\mathrm{h}\) coincides with its normalizer in \(\mathrm{g}\). The closure \(\overline{G\mathrm{h}}\) of the orbit \(G\mathrm{h}\) in the corresponding Grassmann variety is called the Demazure embedding of \(G\mathrm{h}\).
M. Brion [J. Algebra 134, No. 1, 115–143 (1990; Zbl 0729.14038)] conjectured that the Demazure embedding is smooth. Positive results in this direction are contained in the works of M. Demazure, C. De Concini, C. Procesi, D. Luna, P. Bravi and G. Pezzini. The main theorem of the article states that the Demazure embedding is indeed smooth. In the proof, the results of D. Luna [J. Algebra 258, No. 1, 205–215 (2002; Zbl 1014.17009)] and G. Pezzini [Math. Z. 255, No. 4, 793–812 (2007; Zbl 1122.14036)] are used.

MSC:

14M27 Compactifications; symmetric and spherical varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
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