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

TL;DR

This paper establishes slow exponential growth estimates for certain representations of Sp(n, 1) at the boundary of Cowling's strip, aiding progress on the Baum–Connes conjecture for this group.

Contribution

It provides new growth estimates for the spherical principal series representations of Sp(n, 1) at the edge of Cowling's strip, crucial for Baum–Connes conjecture research.

Findings

Slow exponential growth estimate at Re(s)=1 for spherical principal series

Growth estimate for homotopy of the principal series

Supports Baum–Connes conjecture approach for Sp(n, 1)

Abstract

We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [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).