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I. V. Losev MSU, IUM Moscow, Russia ivanlosev@yandex.ru

Uniqueness prop erties for spherical varieties
The base field K is assumed to be algebraically closed and of characteristic zero. Let G be a connected reductive group. Fix a Borel subgroup B G. An irreducible G-variety X is said to be spherical if X is normal and B has an open orbit on X . The last condition is equivalent to K(X )B = K. Note that a spherical G-variety contains an open G-orbit. The goal of this note is to review the author results on uniqueness properties for certain classes of spherical varieties: homogeneous spaces, smooth affine varieties and general affine varieties. The properties are stated in terms of certain combinatorial invariants. Let us describe combinatorial invariants in interest. Fix a maximal torus T B . (B ) The set XG,X := {µ X(T )|K(X )µ = {0}} is called the weight lattice of X . This is a sublattice in X(T ). By the Cartan space of X we mean aG,X := XG,X Z Q. This is a subspace in t(Q) . Next we define the valuation cone of X . Let v be a Q-valued discrete G-invariant valuation of K(X ). One defines the element v a G,X by the formula v , µ = v (fµ ), µ XG,X , fµ K(X )(B ) \ {0}. µ It is known that the map v v is injective. Its image is a finitely generated convex cone in a . We denote this cone by VG,X and call it the valuation cone of X . G,X Let DG,X denote the set of all prime B -stable divisors of X . This is a finite set. To (B ) \ {0}. D DG,X we assign D a G,X by D , µ = ordD (fµ ), µ XG,X , fµ K(X )µ Further, for D DG,X set GD := {g G|g D = D}. Clearly, GD is a parabolic subgroup of G containing B . Choose (g). Below we regard DG,X as an abstract set equipped with two maps D D , D GD . Theorem 1 ([2]). Let X1 , X2 be spherical homogeneous spaces of G. If XG,X1 = X VG,X2 , DG,X1 = DG,X2 , then X1 , X2 are equivariantly isomorphic.
G,X2

, VG,X1 =

Now we consider uniqueness properties for affine spherical varieties. A basic combinatorial invariant of an affine spherical G-variety X is its weight monoid X+ := {|f K[X ]}. It G,X is clear that X+ = { XG,X | D , 0, D DG,X }. G,X The next theorem incorporates uniqueness properties for both smooth and arbitrary affine spherical varieties. It is proved using Theorem 1. Theorem 2 ([1]). Let X1 , X2 be affine spherical G-varieties such that X Suppose at least one of the fol lowing conditions holds: 1. Both X1 , X2 are smooth. 2. VG,X1 = V
G,X
2

+ G,X

1

= X+ 2 . G,X

.

Then X1 , X2 are G-equivariantly isomorphic. References [1] I.V. Losev. Proof of the Knop conjecture. Preprint (2006), arXiv:math.AG/0612561v4, 20 pages. [2] I.V. Losev. Uniqueness property for spherical homogeneous spaces. Preprint (2007), arXiv:math.AG/0703543, 21 pages. 1