# Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?

The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can attach a Hecke algebra $\mathcal{H}(W,S)$ (this is an exercise in Bourbaki's Groupes et algèbres de Lie IV-VI) with generators $T_w$ ($w \in W$) and relations

\begin{align*} T_w T_{w'} & = T_{ww'} \text{ if } \ell(w)+\ell(w') = \ell(ww') \\ T_s^2 & = a_s + b_s T_s \end{align*}

Sometimes, these abstract Hecke algebras turn out to be isomorphic to `convolution type Hecke algebras'

$$\mathcal{H}(G,B) = \text{End}_G(\text{ind}^G_B \Bbb{1})$$

for a group $G$ and a subgroup $B$ (I am being deliberately vague here), and one therefore gets a functor $$\text{Rep}(G) \longrightarrow \text{R-Mod}(\mathcal{H}(W,S)),\quad V \mapsto V^B$$ relating representations of the abstract Hecke algebra and of the group $G$. This is for instance the case if $(W,S)$ comes from a BN-pair (by another Bourbaki exercise).

• I don't know much about Kac-Moody algebras/groups, but perhaps $PGL_2(Z)$ arises from a BN-pair through something like the following: there's a Kac-Moody Lie algebra (Frenkel-Feingold, Math. Ann. 1983) with Weyl group $PGL_2(Z)$ as desired. Then, I dunno... maybe Tits associates a Kac-Moody group (via a few constructions), and someone else (perhaps) describes "twin buildings" and BN-pairs. I'd start sniffing in those directions, if an expert doesn't weigh in. Good luck! Hoping for a better answer soon. Sep 11, 2017 at 5:20
• How would $H_3$ or $H_4$ be related to a BN-pair? Sep 11, 2017 at 10:06
• Thanks @Marty, indeed Tits apparently associates a group scheme to every generalized Cartan matrix (in "Uniqueness and presentation of Kac-Moody groups over fields") that gives a BN-pair (twin BN-pairs in fact) with Coxeter group corresponding to the Cartan matrix, hence there apparently (if I understood his article correctly) does exist a group with a BN-Pair whose Coxeter group is $\text{PGL}_2(\Bbb{Z})$. The question then remains, how to describe this group more explicitly. Someone must have already done this, I assume. Sep 11, 2017 at 11:46

$PGL_2(\mathbf{Z})$ is a Coxeter group with presentation $\langle s_1,s_2,s_3 \mid s_i^2=(s_1s_2)^3=(s_1s_3)^2=1 \rangle$, i.e. with labels $\{2,3,\infty\}$. To each Coxeter group $W$ with labels in $\{2,3,4,6,\infty\}$ there exists a Kac--Moody $G$ having a (twin) BN-pair with Weyl Group $W$, as constructed by J. Tits. In particular, $PGL_2(\mathbf{Z})$ is the Weyl group of a Kac-Moody group of rank 3.

While the definition of a Kac-Moody group might appear quite technical, there are two ways of handwaving to convey a flavour of what a Kac-Moody group is like:

(i) A semisimple complex Lie algebra has a nice presentation which can be written down from its Dynkin diagram. Using this presentation for more general Dynkin diagrams, one gets an infinite-dimensional complex Lie algebra $L$ generated by some generators $\langle e_i, f_i, h_i\rangle$. Then the associated adjoint complex Kac-Moody group is by definition the subgroup of Aut$(L)$ generated by ad$(t\cdot e_i)$, ad $(t\cdot f_i)$ for $t \in \mathbf C$.

(ii) One can write down a presentation for $SL_n(\mathbf C)$ with generators the elementary matrices and certain relations which can be read off from the group's associated Dynkin diagram. Then a Kac-Moody group can similiarly be written down in terms of generators and relations for more general Dynkin diagrams.

Caprace-Rémy's "Groups with a twin root datum" gives a very nice introduction to these matters.

On the other hand, by Tits' famous classification of spherical buildings there do not exist thick buildings of types $H_3$ or $H_4$, hence no $BN$-pairs of this type, see e.g. this paper by R. Weiss.

• Nice answer! Allcock-Carbone have given a finite presentation of Kac-Moody groups corresponding to simply laced hyperbolic Dynkin diagrams. The diagram for the Feingold-Frenkel algebra (with Weyl group $W = \text{PGL}_2(\Bbb{Z})$) isn't simply laced and so the results don't apply, but unpublished work of Murray-Carbone promises to give an explicit recursive enumeration of the prenilpotent pairs that figure in Tits presentation. Sep 12, 2017 at 23:11