Let $G$ be a connected reductive group over a field $k$, $A_0$ a maximal $k$-split torus, $\Phi = \Phi(A_0,G)$, and $\Delta$ a base of $\Phi$. Let $P_0$ be a minimal $k$-parabolic subgroup containing $A_0$ with Levi subgroup $M_0 = Z_G(A_0)$, and let $N_0 = \mathscr R_u(P_0)$ with $\Phi(A_0,N_0) = \Phi^+$.

Let $\theta \subseteq \Delta$, let $w$ be in the relative Weyl group $N_G(A_0)/M_0$, and assume that $w (\theta) \subseteq \Delta$. For $S \subseteq \Delta$, let $P_S = M_S N_S$ be the corresponding standard parabolic subgroup, and let $[S] = \Phi(A_0,M_S)^+$.

I have read that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is normal in $N_{w \theta}$, and I'm trying to understand why.

My approach was to first try the special case when $G$ is split over $k$. In this case, I can argue that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is directly spanned in any order by the root subgroups $U_{\alpha} : \alpha \in E$, where

$$E = \{ \alpha \in \Phi^+ - [w\theta] : w^{-1} \alpha > 0 \} $$

Note that $\Phi(A_0,N_{w\theta}) = \Phi^+ - [w\theta] $.

In order to show that $N_{w\theta}$ normalizes $N_{w \theta} \cap wN_{\theta}w^{-1}$, I was trying to apply Proposition 14.5(3) in Borel, Linear Algebraic Groups. It comes down to showing this (replacing $w \theta$ by $S$, and $w^{-1}$ by $w$:

Let $S \subseteq \Delta$, assume $w(S) \subseteq \Delta$, and let $\alpha, \beta \in \Phi^+ - [S]$. Assume that $\alpha + \beta$ is a root. If $w(\beta) > 0$, and $w(\alpha) < 0$, then $w(\alpha + \beta) > 0$.

I did not have success with this approach. Is there a more sensible way to show that $N_{w \theta} \cap wN_{\theta}w^{-1}$ is normal in $N_{w \theta}$?

  • 4
    $\begingroup$ Since your profile says that you are a 3rd-year grad student with interests in rep. theory, alg. geometry, and number theory, and you have been posing so many questions about reductive groups over fields on MO (seem to be working through Borel's book), are there no professors or other grad students in your department in those areas with whom the overall topic can be discussed? $\endgroup$ – nfdc23 Apr 22 '17 at 4:19

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