Optimal $SL(2)$-homomorphisms
By George J. McNinch. 2003
54 citations recorded by the source corpus.
Let $G$ be a semisimple group over an algebraically closed field of \emph{very good} characteristic for $G$. In the context of geometric invariant theory, G. Kempf and -- independently -- G. Rousseau have associated optimal cocharacters of $G$ to an unstable vector in a linear $G$-representation. If the nilpotent element $X \in \Lie(G)$ lies in the image of the differential of a homomorphism $\SL_2 \to G$, we say that homomorphism is optimal for $X$, or simply optimal, provided that its restriction to a suitable torus of $\SL_2$ is optimal for $X$ in the sense of geometric invariant theory. We show here that any two $\SL_2$-homomorphisms which are optimal for $X$ are conjugate under the connected centralizer of $X$. This implies, for example, that there is a unique conjugacy class of \emph{principal homomorphisms} for $G$. We show that the image of an optimal $\SL_2$-homomorphism is a \emph{completely reducible} subgroup of $G$; this is a notion defined recently by J-P. Serre. Finally, if $G$ is defined over the (arbitrary) subfield $K$ of $k$, and if $X \in \Lie(G)(K)$ is a $K$-rational nilpotent element with $X^{[p]}=0$, we show that there is an optimal homomorphism for $X$ which is defined over $K$.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f330feb853b65e100a7bbbc7395db1e6e5ad33c7
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups (2011, background): “In his appendix to [24], Serre shows that any two Springer maps give the same bijection between G-orbits on N and on U .”
- On the Smoothness of Centralizers in Reductive Groups (2010, background): “me of the form H= H1 ×T with a torus T and a semisimple group H1 for which the characteristic of kis very good (this notion is due to McNinch, see [McN08]; they are also called “strongly standard” in [McN05]). •Gis D-standard, i.e. it is separably isogenous to a group CH(D)◦, where His as in the cas”
- Existence of moduli spaces for algebraic stacks (2018, unknown): “we can modify YR such that its generic fiber YK is semisimple. To see this let us consider the Jordan decomposition YK = Y s K + Y n K. By our assumptions on the characteristic of K the main result of [McN05] shows that there exists a parabolic subgroup P+ λ ⊂G defined by a cocharacter λ: Gm →GK such ”