Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, $\mathrm{D}$ is a division quaternion algebra over $\mathrm{k}$, and $\mathrm{K}$ is a quadratic extension of $\mathrm{k}$?
$\begingroup$
$\endgroup$
3
-
$\begingroup$ I wish I understood these classifications better, but such a subalgebra would "complexify" (i.e., upon passing to a suitable extension of $\mathrm k$) to $A_1 + G_2$, and Theorem 3.1 of Seitz's "Maximal subgroups of exceptional algebraic groups" seems to suggest that there's no such maximal subgroup. There's a lot that's swept under the rug there (like the difference between maximality over $\mathrm k$ and over an extension), but, if I'm reading correctly, it seems to suggest that the answer is 'no.' $\endgroup$LSpice– LSpice2023-10-09 23:16:03 +00:00Commented Oct 9, 2023 at 23:16
-
$\begingroup$ The complexification is $A_1G_2G_2$ (which can be maximal in $E_8$), not $A_1G_2$. $\endgroup$Daniel Sebald– Daniel Sebald2023-10-10 00:30:12 +00:00Commented Oct 10, 2023 at 0:30
-
$\begingroup$ Re, ah, yes, sorry, I missed the passage to $\mathrm K$. $\endgroup$LSpice– LSpice2023-10-10 00:46:59 +00:00Commented Oct 10, 2023 at 0:46
Add a comment
|