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This is a construction in some sense dual to the proposal of Darij Grinberg. Lets consider only $G=GL(n,\mathbb{C})$ for simplicity and take $V$ and $W$ to be subrepresentations of $\bigotimes^k \math …

According to Lemma 9.3, a module $M$ is in $\mathcal{O}^{\mathfrak{p}}$ if and only if it is locally $\mathfrak{n}_I^-$-finite. Since the Verma module $M(\lambda)$ contains infinite-dimensional $\math …

By the PBW theorem we can write $U(\mathfrak{p}) = U(\mathfrak{l})U(\mathfrak{u}).$ By our assumption $U(\mathfrak{u})m$ is finite dimensional for any $m\in M$ and by the second point of the definit …

Bertram Kostant, Lie Algebra Cohomology and the Generalized Borel-Weil Theorem, Ann. of Math., 74, (1961), No. 2, 329-387

Yes. We can choose basis of $\mathrm{Hom}_\mathfrak{g}(V_{\mu}, S^2V_{\lambda})$ and extend it by a basis of $\mathrm{Hom}_\mathfrak{g}(V_{\mu}, \Lambda^2V_{\lambda}).$ Since we have $f = f^S + f^A$ …

There is a "Schur-Weyl theory" for representation of $O(n)$ and $Sp(n)$. The group algebra of the symmetric group is replaced by the Brauer algebra. Basically, you first decompose your tensor product …

There is an article by Jing-Song Huang and Chen-Bo Zhu called Weyl's construction and tensor power decomposition for G2 that builds on an invariant theory for $G_2$ (see the linked review for details) …

arXiv:1006.3407

For every irreducible finite-dimensional representation there's a basis in which the matrices take particularly simple form (this should be in most of the introductory representation theory books). Yo …

Not an answer, just too long for a comment. All representations in sight can be decomposed as direct sums of irreducible representations. By the Schur lemma, mapping between irreducibles can be either …

Maybe you should look at the paper Lie Algebra Cohomology and Generalized Schubert Cells by Bertram Kostant. He computes the decomposition via some sort of algebraic laplacian operator.

This can be found in many places. On one hand there are general constructions of representations like as quotients of (or generalized) Verma modules or as sections of homogeneous line (or vector) bu …

There is infinitely many linkage classes each containing some $\Phi^+_I$-dominant elements. But since any module from $\mathcal{O}$ is finitely generated it will decompose only into finitely many modu …

See proof of Theorem 6.11 of Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$ by James E. Humphreys. This theorem proves what you want in the case $\mathfrak{p}$ is a Borel …

I don't think there is other reason. I think in general group representation theory you consider regular elements as those which have trivial stabilizer. Then you should perhaps speak about "regular w …