Let G be a complex reductive group. A normal G-variety X is called spherical
if a Borel subgroup of G has a dense orbit in X. Of particular interest are
spherical varieties which are smooth and affine since they form local models
for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of
G. In this paper, we classify all smooth affine spherical varieties up to
coverings, central tori, and C*-fibrations.Comment: v1: 23 pages, uses texdraw; v2: 25 pages, introduction updated, Lemma
7.2 fixed, references added, typos correcte
With this chapter we provide a compact yet complete survey of two most remarkable "representation theorems": every arguesian projective geometry is represented by an essentially unique vector space, and every arguesian Hilbert geometry is represented by an essentially unique generalized Hilbert space. C. Piron's original representation theorem for propositional systems is then a corollary: it says that every irreducible, complete, atomistic, orthomodular lattice satisfying the covering law and of rank at least 4 is isomorphic to the lattice of closed subspaces of an essentially unique generalized Hilbert space. Piron's theorem combines abstract projective geometry with lattice theory. In fact, throughout this chapter we present the basic lattice theoretic aspects of abstract projective geometry: we prove the categorical equivalence of projective geometries and projective lattices, and the triple categorical equivalence of Hilbert geometries, Hilbert lattices and propositional systems.
Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.
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