arXiv subjects
Publications and source records attributed to Roman Avdeev.
We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.
The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on the spaces of regular sections of homogeneous linear bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces G/H, where G is a simply connected non-simple semisimple complex algebraic group and H a connected closed subgroup of it. In all the cases we also find the highest weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.
In 1994, based on Roberts' counterexample to Hilbert's fourteenth problem, A'Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and prove that H is epimorphic in SL(V). In particular, the homogeneous space SL(V)/H provides a new example of a homogeneous space with epimorphic stabilizer that admits no projective embeddings with small boundary.
A new approach to classification of solvable spherical subgroups of semisimple algebraic groups is considered. This approach is completely different from the known approach by D. Luna and provides an explicit classification.