Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 283-302.

Slow exponential growth representations of $\mathrm{Sp}(n, 1)$ at the edge of Cowling's strip

Authors: Pierre Julg (1), Shintaro Nishikawa (2)
Author institution:(1) Institut Denis Poisson, Universite d'Orleans, Collegium Sciences et Techniques, Batiment de mathematiques, Rue de Chartres B.P. 6759, F-45067 Orleans Cedex 2 - France
(2) School of Mathematical Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, U.K.

Summary: We obtain a slow exponential growth estimate for the spherical principal series representation $\rho_s$ of the Lie group $\mathrm{Sp}(n, 1)$ at the edge $(\mathrm{Re}(s)=1)$ of Cowling's strip $(|\mathrm{Re}(s)|<1)$ on the Sobolev space $\mathcal{H}^\alpha(G/P)$ 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 $\mathrm{Sp}(n,1)$ .

DOI: http://dx.doi.org/10.7900/jot.2022oct12.2437
Keywords: Sp $(n, 1)$ , uniformly bounded representations, slow exponential growth representations

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