Boundary unitary representations-irreducibility and rigidity

Boundary unitary representations—right-angled hyperbolic buildings

“…Our goal now is to prove irreducibility of the representation ρ 0 of the Hecke algebra H = H (G, K ) on the space V K . The argument follows very closely an irreducibility argument in [1]. For easier comparison, we adjust our notation to match [1].…”

“…Our proof is a modification of the proof given in [1]. The main tool which allows us to use [1] almost verbatim is the following pointwise inequality between functions on ∂W :…”

“…We begin with an explanation of (w| ∩ h), and then show the pointwise inequality T q w 1 ≤ C w 1 We can now repeat the argument with the points 1 and w interchanged. The point hw plays the role of the point h, and the distance from w to ∩h is |w − hw|/2.…”

“…The argument follows very closely an irreducibility argument in [1]. For easier comparison, we adjust our notation to match [1]. Thus, the boundary ∂W will be denoted B , and the measure l will be called ν.…”

“…It turns out that the associated Hecke algebra representation could be seen as a deformation of the unitary representation of W alluded to above. We prove the irreducibility of this deformed representation using the tools developed in [1] (see also the recent generalization [15]). Our exact theorem is THEOREM 1.…”