# Comparison of length functions on Weyl groups

Let $$G$$ be a connected reductive group over an algebraically closed field $$k$$ (with nice enough characteristic), and let $$\sigma:G\to G$$ be a finite order automorphism of $$G$$. The connected component $$G_0$$ of the fixed-point subgroup $$G^\sigma$$ is also a reductive group. By a theorem of Steinberg, there exists a Borel subgroup $$B\subset G$$ and maximal torus $$T\subset B$$ such that both $$B$$ and $$T$$ are $$\sigma$$-stable: ie: $$\sigma(B)=B$$ and $$\sigma(T)=T$$. Furthermore, $$B_0=G_0\cap B$$ is a Borel of $$G_0$$ and $$T$$ may be chosen so that $$T_0=G_0\cap T$$ is a maximal torus of $$G_0$$.

Now in this set up, $$\sigma$$ acts on the roots $$\Phi=\Phi(G,T)$$, preserving the positive and simple roots $$\Delta\subset \Phi^+\subset \Phi$$ determined by $$B$$. It also acts on the Weyl group $$W=W(G,T)$$, and there is a natural subgroup $$W_0\subset W$$ such that $$W_0=W(G_0,T_0)$$. This is contained in the fixed-point set $$W^\sigma$$, but is generally smaller. Finally, let $$l$$ (resp. $$l_0$$) be the length function of $$W$$ (resp. $$W_0$$).

My question is:

For $$w\in W_0\subset W$$ what is known about the comparison between the lengths $$l_0(w)$$ and $$l(w)$$?

Obviously, $$l_0(w)\leq l(w)$$, but I am curious about when stronger estimates may hold. I would be interested in any reference where such questions are considered.

A case I am particularly interested in is when $$\sigma^2=1$$, and there is a maximal torus $$A\subset G$$ such that $$\sigma(a)=a^{-1}$$ for all $$a\in A$$. Such involutions are called split, and in this case it seems like we can choose $$(B,T)$$ so that $$2l_0(w)\leq l(w),$$

which would have some nice consequences. I've verified this for type $$A$$, $$G_2$$, and a couple other small rank cases, but haven't found a general argument. Is this estimate known, or clearly false?

• Are these subgroups of the Weyl group generated by reflections? If so, those are well studied. – Matt Samuel Nov 7 '18 at 23:25
• Yes of course, they are Weyl groups associated to a reductive subgroup, so are generated by reflections. Is there a good reference you might point me to? – WSL Nov 8 '18 at 0:46
• What I mean is: are the reflections in the subgroup also reflections in the larger group. This is not automatic. – Matt Samuel Nov 8 '18 at 1:01
• Matthew Dyer has written papers on reflection subgroups. I think there's some stuff about it in my defunct blog that I haven't looked at in a year. No nontrivial length function bound that I know of. – Matt Samuel Nov 9 '18 at 23:32