Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

Abstract

We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks.

Free group representations: duplicity on the boundary

“…Many papers construct specific families of representations and prove them irreducible. See for example [Yos51], [PS86], [FTP82], [FTS94], [PS96], [KS96], [Pas01], [KS04], and [BG16]. Some of these papers also prove inequivalence of representations, either within or between families.…”

“…Proving this requires different methods than those presented here. See [KS01], [KSS16] and [BG16]. On the other hand, representations which have a single imperfect realization can, if an appropriate Finite Trace Condition holds, be attacked with the same methods as in the case of duplicity.…”

Some spherical functions on hyperbolic groups

“…We also need the "boundary version" of Theorem 2.8 (see [15,Lemma 3.1]): Theorem 2.10. For any R > 0 large enough, there exists a sequence of measures β n,R : Γ → R + such that :…”

“…The Property RD used in [15] to prove Schur's orthogonality relations is replaced by Inequality (6.1), Theorem 6.2 and 6.3. Let R > 0 large enough.…”

“…The representation π 0 is unitary and might be thought as an analog of the endpoint of principal series for SL(2, R). It appears in different contexts as a boundary representation and it is also called quasi-regular representation (or Koopman representation)see [18], [3], [32], [10], [12], [13], [15], [14], [29], [41] and [42] for boundary representations and see [24] and [25] for other quasi-regular representations. More recently in [19], boundary representations of hyperbolic groups have been used to obtain results in the general setting of locally compact groups.…”