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Are all cuspidals induced?

Quite late to the party, however, to add to Paul's answer, for Question 3, the Bernstein conjecture (as far as I am aware) relies on the idea that we can decompose the category of smooth ...
Zander's user avatar
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Does every nilpotent lie in the span of simple root vectors?

No, not every nilpotent element lies in the span of simple root vectors. In general, the Lie algebra of a reductive group (G) decomposes into a direct sum of its semisimple and nilpotent parts under ...
user525479's user avatar
4 votes
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Does every nilpotent lie in the span of simple root vectors?

Let $\mathfrak g$ have Cartan subalgebra $\mathfrak h$, let $X_\alpha\in\mathfrak g_\alpha$ be a non-zero vector. Then you are looking at nilpotent elements of the form $\sum_{\alpha\in \Theta}X_\...
Kenta Suzuki's user avatar
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3 votes

Does every nilpotent lie in the span of simple root vectors?

Actually I think this has to fail for $G_2$. Using fundamental coweights, we can construct a semisimple element that scales the simple root vectors $u_1, u_2$ by arbitrary nonzero constants $c_1, c_2$....
Alexander's user avatar
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9 votes
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Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Yes for reductive $G$, and this follows quickly from the dynamical characterization of parabolics: A subgroup $H$ of a reductive group $G$ is parabolic if and only if ...
Will Sawin's user avatar
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2 votes
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Are isomorphic maximal tori stably conjugate?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Gal{Gal}$Let $(B, S)$ be a Borel–torus pair in $G \mathrel{:=} \mathsf G_2$, let $\alpha_1$ be ...
LSpice's user avatar
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