New answers tagged reductive-groups
1
vote
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
For a reductive group the category of representations is semisimple so the algebra of endomorphisms of the fiber functor is just a product of matrix algebras, one for each irreducible representation.
...
- 137k
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 ...
- 9
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_\...
- 1,897
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 $c_1, c_2$....
- 943
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 ...
- 137k
Top 50 recent answers are included