3
$\begingroup$

I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected algebraic subgroups of a linear algebraic subgroup defined over a perfect field?

Indeed, the general question is motivated by the following special case of it: Let $G$ be a semisimple group defined over $k$, which is the algebraic closure of a finite field. Let $U\subset G$ be a connected unipotent subgroup and $H\subset G$ be a parabolic (or any connected) subgroup, then is $U\cap H$ is still connected?

Any comments are very welcome! Thanks in advance!

$\endgroup$
2
  • $\begingroup$ In general, I think you're out of luck. (However, perhaps one could handle the case where $U$ is the unipotent radical of a parabolic, and hence filtered by $\mathfrak{gl}_1$'s, and that such groups have no smooth, finite subgroups?) It may be of interest to know Proposition 14.22(a) of Borel, which states that the intersection of two parabolics is connected; that seems close to your situation. $\endgroup$
    – LSpice
    Jan 19, 2019 at 21:09
  • $\begingroup$ Also intersection with Levi subgroups of reductive groups often behaves well, as one shows using the fact they are centralisers of tori. $\endgroup$
    – LSpice
    Jan 19, 2019 at 21:10

1 Answer 1

4
$\begingroup$

Let $p=\mathrm{char}\,k$ and $G=GL(3,k)$. Then $$ \{\begin{pmatrix}1&0&0\\t-t^p&1&t\\0&0&1\end{pmatrix} \mid t\in k\} $$ is a group isomorphic to $\mathbf{G}_a$. Its intersection with the Borel subgroup of upper triangular matrices is $\mathbb{F}_p$.

Usually, the connectedness of $U\cap P$ is shown by exhibiting a torus which normalizes both $U$ and $P$ and whose action on $U$ is contracting.

$\endgroup$
2
  • $\begingroup$ Just to be clear, that's the discrete group $\mathbb F_p$, not the group $\mathbf G_a$, even if it happens that $k = \mathbb F_p$. $\endgroup$
    – LSpice
    Jan 21, 2019 at 12:24
  • $\begingroup$ (Err, I mean "the constant group scheme $\mathbb F_p$".) $\endgroup$
    – LSpice
    Jan 21, 2019 at 22:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.