Saturation rank for finite group schemes and varieties of elementary subalgebras

Let $G$ be a connected reductive algebraic $\Bk$-group over an algebraically closed field $\Bk$ of characteristic $p > 0$, and $\mfg:=\mrlie(G)$ be its Lie algebra. Suppose in addition that $G$ is standard, i.e. $p$ is good for $G$, the derived subgroup of $G$ is simply connected and there exists a non-degenerate $G$-invariant symmetric bilinear form for $\mfg$. Following \cite{CFP} and \cite{PS} we acquaint ourselves with the projective varieties $\mbE(r,\mfg)$ and how they relate to the representation theory of $\mfg$. It is the purpose of this thesis to investigate the geometric properties of $\mbE(r,\mfg)$. By refining the definition of saturation rank in the context of finite groups schemes, we reveal parallel descriptions for constant finite group schemes and infinitesimal group schemes. When our consideration concentrates on restricted Lie algebras, i.e. infinitesimal group schemes of height $\leq 1$, we identify the rank with the semisimple rank of $G$ in case of reductive Lie algebras. This characterization is dedicated to establishing the relationship between $\mbE(r,\mfg)$ and the restricted nullcone $V(\mfg)$. As a supplement, we also consider the non-reductive case, i.e. the proper parabolic subgroups. In order to study more generally the rank on the $r$-th Frobenius kernel $G_{(r)}$, we specialize to $G=\SL_{n}$ and $r=2$, revealing an interesting result. We also identify the varieties $\mbE(\rsm,\mfg)$ for $G$ being simple and of classical type when $\rsm$ equals $\rrm-1$, where $\rrm$ is the maximal dimension of elementary subalgebras of $\mfg$. Our goal is to determine the irreducible components of $\mbE(\rsm,\mfg)$, and to pursue links and similarities with the result of Pevtsova and Stark on $\mbE(\rrm,\mfg)$. Having done this, the irreducibility of $\mbE(\rsm,\mfg)$ for related types will be seen.



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