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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 …

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 …

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 …

It's slightly nicer to look at $M_n // U_-$ instead of $GL(n) // U_-$, since then we're looking at a subring of invariants inside a polynomial ring. Namely, the subring generated by all determinants t …

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 …

In your question, you may as well take $M=M_j=$ a ($G$-invariant) tubular neighborhood of a single orbit $G/S$. Near the point $S/S$, the space looks like $\mathfrak g/\mathfrak s \times N$, where $N$ …

This is not a complete answer, but grew large for a comment. Duality acts on the simple weights $\{\omega_i\}$, by $-w_0$ (where $w_0$ is the long element of the Weyl group), so we should really grou …

If $G$ is simply connected (so, $SL_n$ in your less general question), then every line bundle $\mathcal L$ is uniquely of the form $G \times^B \mathbb C_\lambda$ where $\mathbb C_\lambda$ is the $T$-i …