# Existence of commuting Chevalley involution

Let $$\mathfrak{g}$$ be a simple finite-dimensional complex Lie algebra, and let $$\theta$$ be a complex linear involution on $$\mathfrak{g}$$. Let $$\mathfrak{a}$$ be a Cartan subspace, and choose a $$\theta$$-stable Cartan subalgebra $$\mathfrak{h}$$ containing $$\mathfrak{a}$$. Finally, make a choice of positive restricted roots in $$\mathfrak{a}^*$$, and extend this to a choice of positive roots in $$\mathfrak{h}^*$$, giving rise to a positive system for the roots of $$\mathfrak{g}$$.

Let $$\omega$$ denote the Chevalley involution of $$\mathfrak{g}$$ with respect to $$\mathfrak{h}$$ and the choice of positive system just described. To be explicit, $$\omega$$ is the unique automorphism of $$\mathfrak{g}$$ such that $$\omega(h)=-h$$ for $$h\in\mathfrak{h}$$, and $$\omega(e_{\pm\alpha})=e_{\mp\alpha}$$ for all simple roots $$\alpha$$.

Is it true that $$\omega$$ commutes with $$\theta$$?

• I think you have made a choice of pinning (choice of non-$0$ vector $e_\alpha$ in each root space for a simple root $\alpha$) that you are not mentioning; and then of course what you really mean is that $[e_\alpha, \omega(e_\alpha)]$ is the coroot $h_\alpha$, and $\omega^2(e_\alpha) = e_\alpha$. (A different choice of pinning gives a different, though conjugate, Cartan involution, and this can affect commuting with $\theta$.) This already forces $h_\alpha$ to be negated by $\omega$; so, since $\mathfrak g$ is simple, $\omega$ acts by negation on $\mathfrak h$—it's not a separate condition. Jan 2 '21 at 16:09
• Yes, you are right, I was being sloppy. My statement wasn't meant to depend on the choice of $\mathfrak{sl}_2$-triples, although as we now see it's false anyway :) Jan 4 '21 at 9:29

I've realized that the answer is no. I should have checked some examples! The case of $$(\mathfrak{gl}(4),\mathfrak{gl}(3)\times\mathfrak{gl}(1))$$ gives a counterexample. Indeed, this is induced by the involution $$\theta$$ given by conjugation by $$\begin{bmatrix}0 & 0 & 0 & i\\0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\-i &0 &0&0\end{bmatrix}.$$ Then $$\theta$$ preserves the Cartan subalgebra of diagonal matrices, which contains a Cartan subspace for this pair. One choice of positive system compatible with the Iwasawa decomposition in this case is the usual one, giving the Borel of upper triangular matrices. Thus a Chevalley involution $$\omega$$ is given by $$X\mapsto-X^t$$.
Now one can compute that $$\omega\theta(e_{12})=-ie_{24}$$ while $$\theta\omega(e_{12})=ie_{24}$$.
• Isn't the centraliser of your involution $\operatorname{GL}_2 \times \operatorname{GL}_1^2$, embedded as $(g, s, t) \mapsto \frac1 2\begin{pmatrix} s + t & & & s - t \\ & 2g \\ t - s & & & s + t \end{pmatrix}$, not $\operatorname{GL}_3 \times \operatorname{GL}_1$? Maybe I misunderstand the notation. I think if you choose your Cartan subalgebra to be the diagonal inside $\operatorname{GL}_2$ times these two $\operatorname{GL}_1^2$'s, then things go better. Jan 2 '21 at 15:50
• Namely, your involution is conjugate to $\operatorname{diag}(\zeta_8, 1, 1, -\zeta_8)$, where $\zeta_8$ is a primitive 8th root of unity, and it is clear that that automorphism behaves well with respect to the standard diagonal torus. Jan 2 '21 at 16:04