Proof of the Knop conjecture

In this paper we prove the Knop conjecture asserting that two smooth affine spherical varieties with the same weight monoids are equivariantly isomorphic. We also state and prove a uniqueness property for not necessarily smooth affine spherical varietiesComment: 31 pages, v2,v3 typos fixed, minor mistakes are corrected, references added v4 20 pages, crucial changes made. The proof now uses results of AG/0703543. A new result concerning a uniqueness property of not necessarily smooth affine spherical varieties is added, v6 final version, to appear in Annales de l Institut Fourie

Embeddings of homogeneous spaces into irreducible modules

Let $G$ be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of $G$ to be embeddable into an irreducible $G$-module. In addition, for an affine homogeneous space we find a criterium for a closed embedding to existComment: v2 8 pages, a gap in the proof is corrected, some examples are added v3 new theorem answering whether there is a closed embedding is adde

Computation of the Cartan spaces of affine homogeneous spaces

Let $G$ be a reductive algebraic group and $H$ its reductive subgroup. Fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Cartan space \a_{G,G/H} is, by definition, the subspace of \Lie(T)^* generated by the weights of $B$-semiinvariant rational functions on $G/H$. We compute the spaces \a_{G,G/H}.Comment: v1 20 pages, v2 minor corrections are mad