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A finite subgroup will map to a finite subgroup of $PSL_2(\mathbb Z)$, which is a free product $Z_2 * Z_3$. I believe I have been told that finite subgroups of free products of finite groups are conju …

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 $U(1)/Z_n$, …

$E_8$, before string theory. The Monster, before the Moonshine Module, or at least before the Griess algebra. The 3-connected group that maps to a compact simple Lie group $K$, inducing isomorphisms …

Let $C$ be a finite-dimensional field extension of the real numbers, so $(C \setminus 0)/{\mathbb R}_+$ is a compact abelian Lie group, and a sphere of dimension $\dim_{\mathbb R} C - 1$. If $C \neq …

I believe the right reference is Borel-de Siebenthal. A finite-dimensional proof is as follows. The space of conjugacy classes $G/\sim$ can be identified with $T/W = (Lie(T)/\Lambda)/W = Lie(T)/(\Lam …

Let $$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$ be the braid group on three strands, and consider the surjection $$\phi : Br_3 \twoheadrightarrow SL_2( …

More generally, let $\overline{BwP}/P$ be an orbit closure in a partial flag manifold. If $w$ is minimal in the coset $w W_P$, then $\overline{BwB}/B \to \overline{BwP}/P$ is birational, so it suffice …

The mnemonic I use: if the diagram has a natural involution, then $-w_0$ induces it, otherwise $w_0 = -1$. The only place this fails is in $D_n$, where one can switch the antlers, but shouldn't always …

If $G$ is compact, then as explained in comments $G/ad$ is $T/W$. If further $G$ is simply-connected, hence a product of simple factors, then $T/W$ is a corresponding product of simplices. The point …

I often find it more useful to say $SU(3)$ is a $T^2$ bundle over the manifold of flags in ${\mathbb C}^3$ (itself a ${\mathbb CP}^1$-bundle over ${\mathbb CP}^2$). Partly this is because $T^2$'s homo …

The Schubert classes on $G/P$ are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique $T$-fixed point. The $T$-fixed points on $G/P$ ar …