Sur les plongements de Demazure. (On the Demazure embeddings) (Q1858275)

Let \({\mathfrak g}\) be a semisimple Lie algebra over \(\mathbb{C},\) and let \( {\mathfrak h}\) be a Lie subalgebra of \({\mathfrak g}\) of dimension \(m\). Let \(G\) be the neutral component of Aut \({\mathfrak g}\) and \(H\) the subgroup of \(G\) which stabilizes \({\mathfrak h.}\) Denote by \(\overline{G. {\mathfrak h}}\) the closure of the orbit of \(G.{\mathfrak h}\) in the Grassmannian Gr\(_{m}({\mathfrak g})\) -- this is the Demazure embedding of \(G/H.\) Here, it is shown that if \({\mathfrak g}\) is of type \(A\) (that is, \({\mathfrak g}\) is isomorphic to a product of Lie algebras of the form sl\(_{n}\)), then for all Lie subalgebras \({\mathfrak h}\) of \({\mathfrak g}\) which are spherical and equal to their normalizer we have that the Demazure embedding \(\overline{G. {\mathfrak h}}\) is ``wonderful'' (``magnifique''), i.e. \( \overline{G.{\mathfrak h}}\) is projective and smooth; \(G\) has a dense orbit in \( \overline{G.{\mathfrak h}}\) whose complement is the union of irreducible divisors having a nonempty transversal intersection; and for all \(x,x^{\prime }\in X\)\ if \(\{i:x\in D_{i}\}=\{i:x^{\prime }=D_{i}\}\) then \( G.x=G.x^{\prime }.\) The paper consists of a review of spherical and wonderful varieties and Demazure morphisms, followed by a treatment of localization, particularly the localization of wonderful varieties, followed by a proof of the main theorem given above. The author points out that nothing seems to be known in the case where \({\mathfrak h}\) is not spherical (but still equal to its normalizer), and it is asked if the adherence of the orbit is always smooth.