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The pair (SO(n), SO(n-1))) is a symmetric pair, hence in this case M is a maximal subgroup of SO(n). Thus the connected component of your subgroup must the whole group.

I am sorry to tell I believe the answer is No. Consider $\mathfrak g = \mathfrak{su}(2)$ and its complexification $\mathfrak g_{\mathbb C} = \mathfrak{sl}(2,\mathbb C)$, and $\mathfrak h= \mathfrak{s …

In Vogan's green book, on page 400 it is shown that given an irreducible $(\mathfrak g, K)-$module, then, each lowest $K-$type has multiplicity one. Perhaps this gives a partial answer to your questi …

Check the book Freudenthal Linear Lie groups, in this book you find a complete treatment of Aut(G) for G a real semisimple (connected) Lie group. best regards

also check Onishchik: Lectures on Real Semisimple Lie Algebras and Their Representations.

on page 386 (paragraph 66.7) you find the table Out(G)/Int(G) on page 387 you find D_{l,j} j>1 your Lie algebras so(p,q) when your p or q is even. on page 391 you find so(p,q) when both p,q are odd T …

Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal enve …

If you consider $V^\infty$ with its Fréchet topology. and irreducibility means no closed invariant subspaces except for the trivial ones. Then $V$ is irreducible iff $V^\infty $ is irreducible. In the …

I do not know if you need K-admissibility. You can check if it fails in the following example: write the usual Iwasawa for $\operatorname{SL}_2(R)=KAN$, $MAN$ usual parabolic. Fix $ T$ a continuous li …

Fix a regular element $\lambda$ in $Lie(T)\subset Lie(G)$, then the coadjoint orbit $Ad(G) \lambda$ is isomorphic to $G/T.$ Best,

Let $\mathfrak g$ be a complex simple Lie algebra. We fix Cartan subalgebra $\mathfrak h$ and a system of positive roots $\Psi$ for the root system of the pair $(\mathfrak g, \mathfrak h).$ For each …

Check an old paper of Joseph Wolf in Journal of Mechanics ... early sixties , there you find a detailed answer to your question. I do not have access to Mathscinet for a better answer. Best regards J …

If i am not wrong the following is true, G compact connected Lie group, H the centralizer of a torus in G, then G/H is a projective manifold, does this answer your question??

It is not multiplicity free..., check the green book of Antony Knapp, or else the old book of Zelobenko, compact.....

check Eastwood-Wolf, branchig of ...., Arxive 0812.0822 math[RT] in this paper you find who to compute branching laws useing LiE.