Timeline for Maximal connected subgroup of orthogonal group

5 events

when toggle format	what		by	license	comment
Aug 30, 2020 at 8:26	comment	added	Andrew Musso		Thanks a lot @ClaudioGorodski
Aug 29, 2020 at 12:55	comment	added	Claudio Gorodski		If $G=SO(n)$ then $H^0$ (identity component) is $SO(k)\times SO(n-k)$ where $k=\dim L$, say in the real case. Then the adjoint representation on $\mathfrak m=\mathfrak h^\perp$ (wrt invariant inner product) is the tensor product of the standard representations of $SO(k)$ and $SO(n-k)$.
Aug 28, 2020 at 9:04	comment	added	Andrew Musso		Hello, thank you very much for your comment. Would you have a hint about how to show that the action of $H$ on $\mathfrak{m}$ is irreducible in my case, i.e, $H = H_L , G = SO_Q$ and $\mathfrak{m}$ is the reductive complement of the lie algebra of $H_L$?
Aug 27, 2020 at 12:32	comment	added	Claudio Gorodski		I think you can adapt this idea to your case. Consider connected Lie groups $G$ and $H$ where $H$ is a closed subgroup of $G$. Consider the Lie algebras $\mathfrak g$ and $\mathfrak h$ and assume $\mathfrak h$ is a reductive subalgebra of $\mathfrak g$. Let $\mathfrak m$ be a reductive complement. Suppose the adjoint action of $H$ on $\mathfrak m$ has not fixed vectors (e.g. irreducible, as in your case). Then $\mathfrak h$ is a maximal subalgebra of $\mathfrak g$ and $H$ is a maximal connected subgroup of $G$.
Aug 25, 2020 at 15:00	history	asked	Andrew Musso	CC BY-SA 4.0