3
$\begingroup$

Is there an example of an algebraic group $G$ with maximal torus $T$ and Weyl group $W$ of type $B_n$, (a specific $n$ is fine), and a character $\lambda\in X=\hom(T,\mathbb{G}_m)$ such that $\operatorname{stab}_W(\lambda)\cong\mathbb{Z}/(2)\times\mathbb{Z}/(2)$?

The stabilizer being with respect to the action of the Weyl group on $X$ given by $^w\chi(t)=\chi(t^w)$, for $\chi\in X$ and $t\in T$. Thank you.

  • 0
    Pick a dominant $\lambda$ such that it is orthogonal to two orthogonal simple roots, and non-orthogonal to the other simple roots. Remember that simple roots that aren't adjacent in the Dynkin diagram are orthogonal. The general result is that the stabilizer of a dominant weight is generated by the simple reflections stabilizing it. – 2017-02-23
  • 1
    @JyrkiLahtonen Thanks, so I assume you need $n\geq 3$ to ensure two orthogonal simple roots? To be concrete, would for example the group $SO(7)$, then have such a weight as you describe? The Weyl group would have type $B_3$, and if $\omega_1,\omega_2,\omega_3$ are fundamental weights, one can take $\lambda$ to be any nonnegative integer multiple of $\omega_2$? – 2017-02-23
  • 0
    That sounds correct to me, Ankita Desari. – 2017-02-23

1 Answers 1

4

Let $\{\alpha_1, \ldots, \alpha_l\}$ be a base for the root system with corresponding fundamental dominant weights $\omega_1, \ldots, \omega_l$.

Write a dominant weight $\lambda$ as a sum of fundamental dominant weights, say $\lambda = \sum_{i = 1}^l a_i \omega_i$ where $a_i \in \mathbb{Z}_{\geq 0}$. Then we know (see for example Humphreys' book on Lie algebras) that $\operatorname{Stab}_W(\lambda)$ is generated by the simple reflections $\sigma_{\alpha_i}$ with $a_i = 0$. Now it should be easy for you to find many examples of $\lambda$ with $\operatorname{Stab}_W(\lambda) \cong \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$.

  • 0
    Thanks! Are these dominant weights the same things as characters on a maximal torus? I'm used to weights in the context of Lie algebras, and characters in the context of algebraic groups. – 2017-02-23
  • 0
    @AnkitaDesari: Here weights are characters of the maximal torus. A dominant weight is a weight $\lambda$ such that $\langle \lambda, \alpha^\vee \rangle \geq 0$ for all simple roots $\alpha$. – 2017-02-26