A1 Subgroups of Exceptional Algebraic Groups

Abstract - Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p$. Under some mild restrictions on $p$, we classify all conjugacy classes of closed connected subgroups $X$ of type $A_1$; for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra ${\\mathcal L}(G)$ of $G$. Moreover, we show that ${\\mathcal L}(C_G(X))=C_{{\\mathcal L}(G)}(X)$ for each subgroup $X$.These results build upon recent work of Liebeck and Seitz, who have provided similar detailed information for closed connected subgroups of rank at least $2$. In addition, for any such subgroup $X$ we identify the unipotent class ${\\mathcal C}$ meeting it. Liebeck and Seitz proved that the labelled diagram of $X$, obtained by considering the weights in the action of a maximal torus of $X$ on ${\\mathcal L}(G)$, determines the ($\\mathrm{Aut}\\,G$)-conjugacy class of $X$. We show that in almost all cases the labelled diagram of the class ${\\mathcal C}$ may easily be obtained from that of $X$; furthermore, if ${\\mathcal C}$ is a conjugacy class of elements of order $p$, we establish the existence of a subgroup $X$ meeting $${\\mathcal C}$ and having the same labelled diagram as ${\\mathcal C}$.