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I don't think the $\pm$ issue is too deep, and I'm punting on it in favor of answering the other question. You can get a hold of dual Verma modules by considering distributions on $G/B$ supported on a …

The construction you describe appears in Tamvakis' The connection between representation theory and Schubert calculus (Enseign. Math. 50 (2004), 267-2860). Basically, instead of working with represent …

If $\lambda,\mu,\nu$ are considered as vectors in $\mathbb Z^n$, then the set of triples forms a convex cone, given by a finite (for each $n$) list of inequalities. (The corresponding statement does n …

There are conjectural ones in the Berenstein-Zelevinsky paper referenced in that one. They have another paper with a general theorem, Tensor product multiplicities, canonical bases and totally positiv …

$\mathrm{Hom}_G(V^*,V) \cong \mathrm{Hom}(V^*,V)^G \cong (V\otimes V)^G \cong (\mathrm{Sym}^2 V\oplus \mathrm{Alt}^2 V)^G$, so the first is nonzero ($V$ is self-dual) iff $V$ possesses a symmetric or …

Every irrep of $SU(n)$ extends to irreps of $U(n)$, and conversely, the restriction of any irrep of $U(n)$ to $SU(n)$ remains irreducible. If your dominant weight of $SU(n)$ is $(a_1,\ldots,a_{n-1})$ …

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V …

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, …

Let $\theta:G\to G$ be an involution of a complex connected reductive Lie group, preserving a maximal torus $T$ (which, for me, lies inside a $\theta$-invariant Borel $B$). Let $K = G^\theta$ be the f …

Ingredients: The composite of Poisson maps is Poisson The action map $G\times G\to G$ is Poisson Your choice of $T,U$ should be Poisson submanifolds of $G$. You didn't say which Poisson structure yo …

Your question is about the vectors in $V_1\otimes V_2$ that provide $1$-dimensional $B_\Delta$-subrepresentations of weight $\mu$, where $B_\Delta$ is the diagonal in $B\times B \leq G\times G$. Let's …

It sounds like you want a datum to associate to a compact Lie group and chosen torus, that's functorially the same as a choice of Borel containing that torus if one were to complexify, without using t …

In general, $V_{k\rho} \cong \bigotimes\limits_{\beta\in \Delta_+} (\mathbb C_{k\beta/2} \oplus \mathbb C_{(k-2)\beta/2} \oplus \ldots \oplus \mathbb C_{-k\beta/2})$ as $T$-representations, provable v …

Use the Haar measure on $G$ (compact!) to average a metric, obtaining a $G\times G$-invariant metric, and thus an identification $\mathfrak g \cong \mathfrak g^*$. Also, the geodesic spray $\mathfrak …

I don't really see how to get there from just compact groups, so in that sense this is not an answer. My take on the question is something like: how might one have guessed the existence of canonical b …