Slow exponential growth representations of Sp(n, 1) at the edge of Cowlings strip

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 Cowlings 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 authors program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).