Slow exponential growth representations of Sp(n, 1) at the edge of Cowling's strip

Abstract

We obtain a slow exponential growth estimate for the spherical principal series representation ρs of Lie group Sp(n, 1) at the edge (Re(s) = 1) of Cowling's strip (|Re(s)| < 1) on the Sobolev space H α (G/P) when α is the critical value Q/2 = 2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy ρs (s ∈ [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum-Connes conjecture with coefficients for Sp(n, 1).

Some spherical functions on hyperbolic groups

“…Using duality arguments, we provide several spectral estimates based on [10, Theorem 1.2] for the dual system representations studied in this paper. The results can be viewed as a generalization of the result of [40] to the setting of Gromov hyperbolic groups. Notice that the representations π t are typical examples of slow growth representations.…”