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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 …

If $\alpha$ is a long root (e.g. if $\mathfrak g$ is simply-laced), then $e_\alpha$ is in the $G$-orbit of the high weight vectors. Projectively, its orbit looks like $G/P_\Theta$ where $P_\theta$ is …

Yes, if $H$ is closed (i.e. not some irrational-flow subgroup inside the closed subgroup $Z(G)$).

$G_2 \supset SO(4)$

Consider the case that $H$ is a maximal torus of $G'$, and your $G = G' \times H$. (Well, you said $G,H$ semisimple, but I'm going to pretend you meant reductive, because really you should have.) Then …

Much of the literature on the representation theory of the group $GL_n$ is really about the representation theory of the monoid $M_n$. (One might argue that it's really about the representation theory …

Start by deformation-retracting $G$ to its maximal compact, so that $G/T$ will be a flag manifold with a Bruhat decomposition, obtainable by Morse theory as in Agol's answer. If what you want anyway …

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 …

As Mariano's comment indicates, you need more conditions. The usual one is that $G$ is a torus. Using $G$ compact, you can average a metric to get a $G$-invariant metric. Then the exponential map giv …

What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line? I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits …

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 …

The answer is already no for $G=U(2)$ acting on $Sym^6(\mathbb C^2)$. Given a subgroup $\Gamma \leq U(n) \cong SU(n) \times U(1) / Z_n$, we can enlarge it by projecting to $SU(n)/Z_n$ and $U(1)/Z_n$, …

This is another version of Reimundo Heluani's answer, avoiding perverse sheaves. In general, the $H$-orbits on $H/P \times H/Q$ are indexed by $W_P \backslash W_H/W_Q$. If $H = {}^L G(\mathcal K)$, s …

I don't know quite what would count as a nice formula. Here are a couple: Consider the polytope of $(n-3)$-tuples $(d_1,\ldots,d_{n-3})$ of nonnegative numbers, such that $(a_1,a_2,d_1),(d_{n-3},a_{ …

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 …