## Notes of Uri Baders’s Cambridge lecture 11-05-2017

Unitary representations of reflection groups and deformations

Joint with Jan Dymara.

1. Boundary representations

The leading example is the following reflection group. Start with the right-angled pentagon tesselation of hyperbolic plane. Reflections in the side generate right-angled Coxeter group ${\langle s_1,\ldots,s_5|s_i^2,[s_i,s_{i+1}]\rangle}$. It is a lattice in ${PGl(2,{\mathbb R})}$. It is Gromov hyperbolic. It acts on the circle preserving the Lebesgue measure class. The Radon-Nikodym derivatives of group elements are denoted by ${\frac{d\gamma_*\nu}{d\nu}(x)}$.

In general, for a measure class preserving action of a group ${\Gamma}$ on a measure space ${(X,\nu)}$, the quasi-regular representation on ${L^2(X,\nu)}$,

$\displaystyle \begin{array}{rcl} f\mapsto(\frac{d\gamma_*\nu}{d\nu}(x))^{1/2}f(\gamma^{-1}x) \end{array}$

is unitary.

Theorem 1 (Cowling-Steger 1991) For lattices in ${PGl(2,{\mathbb R})}$, the quasi-regular representation on ${L^2(S^1)}$ is irreducible.

The proof uses induction and the representation theory of ${PSl(2,{\mathbb R})}$.

This was generalized by Bekka and Cowling in 2002: restrictions of unitary representations of Lie groups to lattices.

I want to study more general situations, with no ambient group. For a given metric on ${\Gamma}$, one constructs a measure on ${\partial\Gamma}$, the Patterson-Sy-ullivan measure which is quasi-invariant (measure class is preserved). Whence a boundary quasi-regular representation. In fact, a variety of representations, since it depends on the metric. In 2011, with Roman Muchnik, we studied boundary representations of fundamental groups of negatively curved manifolds and proved irreducibility. We show that metrics give isomorphic boundary representations iff they have the same length spectrum.

Recently, Lukas Garncarek showed that boundary representations of hyperbolic groups are irreducible and classified by the metric up to rough similarity.

2. How to prove irreducibility?

Schur: a representation is irreducible iff the centralizer of ${\Gamma}$ in bounded operators ${B(H)}$ consists of constants.

The double centralizer theorem states that the weak operator topology (WOT) closure of the linear span ${\Gamma}$ in ${B(H)}$ is equal to the double centralizer ${Z_{B(H)}Z_{B(H)}(\Gamma)}$. Hence it suffices to prove weak operator density of ${\Gamma}$ in ${B(H)}$.

With Muchnik, we express every rank 1 bounded operator as a WOT limit of linear combinations of ${\Gamma}$.

2.1. The flip trick

Proposition 2 (Bader-Boyer-Garncarek) A representation ${\rho}$ of ${\Gamma}$ on ${H}$ is irreducible iff the flip belongs to the WOT closure of the linear span of ${\rho(\Gamma)\otimes\rho(\Gamma)}$ in ${B(H\hat\otimes H)}$.

Indeed, ${\rho}$ irreducible implies ${\rho\otimes\rho}$ irreducible which implies that ${\overline{\rho(\Gamma)\otimes\rho(\Gamma)}=B(H\hat\otimes H)}$. Conversely, if ${\rho}$ is not irreducible, ${H=H_1\oplus H_2}$, ${H_1\hat\otimes H_2}$ is a non flip invariant subspace of ${H\hat\otimes H}$

Conjecture. For boundary representations

$\displaystyle \begin{array}{rcl} \frac{1}{|\Gamma_n|}\sum_{\gamma\in\Gamma_n}\frac{\rho(\gamma)\otimes\rho(\gamma^{-1})}{\langle\gamma 1,1\rangle^2} \end{array}$

converges to the flip in ${L^2(\partial\Gamma\times\partial\Gamma)}$.

Given ${f\in L^2(\partial\Gamma)}$, we show that rank one operator ${T:f\mapsto \langle f,1\rangle 1}$ is the WOT limit of

$\displaystyle \begin{array}{rcl} \frac{1}{|\Gamma_n|}\sum_{\gamma\in\Gamma_n}\frac{\rho(\gamma)}{\langle\gamma 1,1\rangle}. \end{array}$

More generally, if ${\chi}$ and ${\chi'}$ are functions of small support in the unit tangent bundle, they define functions on the boundary, a similar sum over a suitable subset of ${\Gamma}$ converges to ${f\mapsto \langle f,\chi\rangle\chi'}$, by equidistribution. Such operators are dense.

Garncarek has a smarter method.

3. Back to Coxeter groups

Deformations arise from Iwahori-Hecke algebras.

Let ${W}$ be an infinite Coxeter group. Let ${q}$ be a parameter. Define a new multiplication ${e_s e_w=e_{sw}}$ or ${qe_{sw}+(1-q)e_w}$ (depending wether ${\ell(sw)>\ell(w)}$ or not) on ${{\mathbb C} W}$, denoted by ${H_q}$. This also allows to deform the boundary representation.

$\displaystyle \begin{array}{rcl} (\rho_q(\gamma)(f))(x)=... \end{array}$

defines a representation of ${H_q}$ in ${B(L^2(\partial W))}$.

Theorem 3 (Bader-Dymara) For ${q\geq 1}$, these are all irreducible.

These representations are related to representations of locally compact groups acting on hyperbolic buildings, transitively on pairs (chamber,apartment). Studying biinvariant functions (under maximal compact subgroup) leads to this representation of ${H_q}$, where ${q=}$ thickness ${-1}$.

4. Questions

Lubotzky: does ergodicity on pairs suffice to imply irreducibility? We dream of that this would work (for instance, actions on Poisson boundaries are doubly ergodic), but we cannot prove it yet.

Could this extend to subgroups in Lie groups which are not lattices? Need to change to a measure concentrated on the limit set.

Burger maybe this would work for representations of the complementary series which are closure to the trivial.