# On connectedness of intersection of subgroups

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?

• 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. Jan 19, 2019 at 21:09
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.
• Just to be clear, that's the discrete group $\mathbb F_p$, not the group $\mathbf G_a$, even if it happens that $k = \mathbb F_p$. Jan 21, 2019 at 12:24
• (Err, I mean "the constant group scheme $\mathbb F_p$".) Jan 21, 2019 at 22:51