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I also don't know what "projective octave level" means but judging by the names of the groups and their dimensions, here is my guess: elliptic motion group - real compact group $F_4$ hyperbolic motion …

Compact Riemannian symmetric spaces admitting a Lie group of diffeomorphisms $G$ properly containing the isometry group are essentialy (up to covers) the symmetric $R$-spaces, which are of the form $G …

Exhausting account on homogeneous spaces of the form $G/P$ with $G$ semisimple and $P$ parabolic is given in this book. The relationship between $P$ and $L$ is also explained there in detail so presum …

I am reading notes by David Vogan on Unitary representations and Complex analysis (pdf / dvi). The setting is as follows (page 23): Let $X$ be a $(\mathfrak{g},K)$-module and let $X(\mu)$ denote its …

Your question can be rephrased as "When is the induction the same as coinduction?" This has appeared on MathOverflow before and fancy answer to your question can be found here: When are induction and …

In Spinors and Calibrations by F. Reese Harvey, you can find proof (p. 283) of $S^7 \simeq Spin(7)/G_2.$ It takes the same approach as Bryant's notes mentioned in the comments but it is much more deta …

Regarding your second question, the cases where $C_{min} = C_{max}$ for cases where $G$ is a real form a complex semisimple Lie group have been classified by Misyureva. Basically, you get that $V$ is …

For which simple Lie groups there exists a left-invariant complex structure?

Yes, there should be an explicit expression. Let me sketch how to get it. Start with the Iwasawa decomposition to write your matrix $M$ as a product of three matrices $M = KAN$ where $N$ is nilpotent, …

This is not an answer. Just a few well known facts. Each $P$-invariant subset is also invariant with respect to the Levi part $L$ of $P$ and hence it decomposes into irreducibles for $L$. The repres …

According to theorem 2.15, all Cartan subalgebras of a complex semi-simple Lie algebras are conjugated. I.e. there exists $\alpha \in \mathrm{Inn}(\mathfrak{g})$ such that $\mathfrak{h}_1 = \alpha(\ma …

More generally, if $A$ is a bounded $G$-invariant operator acting on a unitary representation $(\rho,\mathcal{H})$ of $G$ and $A^*$ is it's adjoint (i.e. $(Ax,y) = (x,A^*y)$ for all $x,y\in \mathcal{H …

There seems to be a construction of $\mathfrak{e}_8$ using some sort of geometrical objects in Configurations of lines and models of Lie algebras by Laurent Manivel.

Such representations are actually quite rare and are superset of the minuscule representations.

The Clifford / Weyl algebra duality stems from the symmetric / antisymmetric duality. For me the most natural way to generalize this to $GL(V)$ is kind of vacuous, i.e. the full tensor algebra $\bigot …