Deformation of orbits in minimal sheets

Keywords: Algebraic group theory, algebraic geometry, classical invariant theory
The main object of study of this work are orbits in so-called minimal sheets in irreducible representations of semisimple groups. Let $G$ be a semisimple group. The notion of sheets goes back to Dixmier: Given a $G$-module $V$, the union of all orbits in $V$ of a fixed dimension is a locally closed subset. Its irreducible components are called sheets of $V$. We call a sheet minimal if it contains an orbit in $V$ of minimal strictly positive dimension among all orbits in $V$.
In Chapter I, some notation is fixed and some basic results are proved.
In Chapter II, we describe minimal sheets in simple $G$-modules, and study $G$-stable deformations of orbits in minimal sheets by means of an invariant Hilbert scheme. Invariant Hilbert Schemes have been introduced by Alexeev and Brion in 2005. These are quasi-projective schemes representing functors of families of $G$-schemes with prescribed Hilbert function. The discussion in Chapter II is closely related to the work of Jansou in the following way: Choose once and for all a highest weight vector $v_\lambda \in V(\lambda)$ for each dominant weight $\lambda \in \Lambda^+$, and let $X_\lambda = \overline{G v_\lambda} \subset V(\lambda)$ be the closure of the orbit $G v_\lambda$ of $v_\lambda$ in $V(\lambda)$. In his thesis Jansou investigates $G$-stable deformations of $X_\lambda$ in $V(\lambda)$. If $h_\lambda$ denotes the Hilbert function of $X_\lambda$, then Jansou proves that the invariant Hilbert scheme $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine space of dimension 0 or 1, depending on $G$ and $\lambda$. Furthermore, he gives a complete list of all pairs $(G,\lambda)$ such that $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine line. In the sequel, we call these weights Jansou-weights.
The orbit $Gv_\lambda$ is of minimal strictly positive dimension among all $G$-orbits in $V(\lambda)$. There exist other orbit of the same dimension as $Gv_\lambda$ in $V(\lambda)$ if and only if $\lambda$ is an integral multiple of a Jansou-weight. Here, we start with a general orbit $X$ of minimal strictly positive dimension in a fixed simple $G$-module $V(\lambda)$, and we study $G$-stable deformations of $X$. In particular, we conjecture that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in the closure of the sheet of $X$ is an affine space of dimension either 0 or 1. This will stand in contrast to the fact that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in $V(\lambda)$ can look much more complicated. This is the content of Chapter III, in which we will focus on the group $\SL_2$, and compute some corresponding invariant Hilbert schemes. In particular, we study deformations of orbits of the form $SL_2 \cdot x^{d/2}y^{d/2}$ in the space $k[x,y]_d = V(d)$ of binary forms of degree $d$. It turns out that easiest accessible case is when $d$ is a multiple of 4, and even in this case the corresponding invariant Hilbert scheme can become very complicated. This reflects the principle that even in `simple' cases for invariant Hilbert schemes all possible sort of `bad' things (different irreducible components, non-reduced points, singularities) occur. (This `bad' behavior is also encountered in the case of the classical Grothendieck Hilbert scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial.) In Chapter III Classical Invariant Theory is often used, and some computations are computer-based.
Finally, in Chapter IV we turn our attention to not necessarily simple modules. In the multiplicity-free case important work has been done by Bravi and Cupit-Foutou. We translate some of their results to the case of not necessarily multiplicity-free modules. This corrects a result by Alexeev and Brion. Chapter IV is independent from the preceding chapters.