Modelling classical types: Algebraic group actions via algebras with symmetries

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Let G be a simple classical complex Lie group with Lie algebra g. We fix a standard parabolic subgroup P in G and consider its conjugation action on the nilpotent cone N of nilpotent matrices in g of the same size.This action restricts to certain varieties in N, for example to the variety N(x), where x is an integer, of elements n of nilpotency degree x (that is, n^x=0) and to the variety N_P of nilpotent matrices in the Lie algebra of P. We want to understand these group actions.In type A, when without loss of generality G=GL_n (for G=SL_n, the setup is the same), we can translate the group actions to Representation Theory: For each action, we find a bijection between the orbits and certain isomorphism classes of representations of a quiver with relations. Since these bijections are given by associated fibre bundles, orbit closure relations and codimensions are preserved.We have found a similar translation for symplectic and orthogonal G: in these cases there are bijections to certain symmetric isomorphism classes of symmetric representations of symmetric quivers with relations, that is, to isomorphism classes of representations of algebras with self-duality.Concerning the actions mentioned before, depending on the type of G, we want to reach three main goals:I A finiteness criterion which lists all cases in which the described group action only admits a finite number of orbits.II To understand the finite cases in detail, for example while parametrizing the orbits by combinatorial objects, by describing degenerations, by finding singularities in the orbit closures, and by calculation of resolutions of singularities and intersection cohomology.III In the infinite cases, we intent to describe generic normal forms with which we define and understand semi-invariants which generate the parabolic semi-invariant rings. Explicit quotients and equivariant cohomology will be calculated.For type A, there are many known representation-theoretic results which we can use for the examination of the mentioned goals. To understand the actions for the other classical types, we have to extend the symmetric representation theory and the basic focus of this project, thus, lies in this area. The case of the general linear group provides many clues in which way such expansion might be useful and possible; but we also know that new phenomena are to be expected. For example, we proved that the degeneration order in the even orthogonal case is not induced by type A.