Let *F*_{λ} be a generalized flag variety of a simple Lie group *G* embedded into the projectivization of an irreducible *G*-module *V*_{λ}. We define a flat degeneration *F*_{λ}^{a}, which is a *G*_{a}^{M} variety. Moreover, there exists a larger group *G*^{a} acting on *F*_{λ}^{a}, which is a degeneration of the group *G*. The group *G*^{a} contains *G*_{a}^{M} as a normal subgroup. If *G* is of type *A*, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of *F*_{λ}^{a} is generated by the set of degenerate Plüker relations. We prove that the coordinate ring of *F*_{λ}^{a} is isomorphic to a direct sum of dual PBW-graded *g*-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.

I construct a correspondence between the Schubert cycles on the variety of complete flags in ℂn and some faces of the Gelfand–Zetlin polytope associated with the irreducible representation of SLn(ℂ) with a strictly dominant highest weight. The construction is motivated by the geometric presentation of Schubert cells using Demazure modules due to Bernstein–Gelfand–Gelfand [3]. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand–Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.

The present constitutes the lecture notes from a mini course at the Summer School ”Structures in Lie Representation Theory” from Bremen in August 2009.

The aim of these lectures is to describe algebraic varieties on which an algebraic group acts and the orbit structure is simple. The methods that will be used come from algebraic geometry, and representation theory of Lie algebras and algebraic groups.

We begin by presenting fundamental results on homogeneous varieties under (possibly non-linear) algebraic groups. Then we turn to the class of log homogeneous varieties, recently introduced in [7] and studied further in [8]; here the orbits are the strata defined by a divisor with normal crossings. In particular, we discuss the close relationship between log homogeneous varieties and spherical varieties, and we survey classical examples of spherical homogeneous spaces and their equivariant completions.