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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 …

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 …

check work of David Vogan, and a book of Trappa Vogan ...

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 …

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 …

I believe, the first proof that a K-type occurs at most in finitely many discrete series is in Harish-Chandra

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 …

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 …

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

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

Check the home page of Toshiyuki Kobayashi (link), download the earlier papers of his, over there you fill an answer. Edit: in a 2017 note of Duflo-Galina-Vargas (link behind paywall), you will find …

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.

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

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 …

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