## New answers tagged reductive-groups

0
votes

### 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 ...

0
votes

### 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 ...

4
votes

Accepted

### 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_\...

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 $c_1, c_2$....

9
votes

Accepted

### 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 ...

2
votes

Accepted

### Are isomorphic maximal tori stably conjugate?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Gal{Gal}$Let $(B, S)$ be a Borelâ€“torus pair in $G \mathrel{:=} \mathsf G_2$, let $\alpha_1$ be ...

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