[Paper Review] Fourier Multiplier Norms of Spherical Functions on the Generalized Lorentz Groups

Troels Steenstrup|ArXiv.org|Nov 25, 2009

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper provides a closed-form expression for the completely bounded Fourier multiplier norm of spherical functions on the generalized Lorentz groups $SO_0(1,n)$ for $n \geq 2$, using representation-theoretic and harmonic analysis techniques. The key result shows that these norms are unbounded as a function of the spectral parameter, implying no uniform bound exists, and this extends to $SU(1,n)$, $Sp(1,n)$, and $F_{4(-20)}$, proving the existence of completely bounded Fourier multipliers that are not coefficients of uniformly bounded representations.

ABSTRACT

Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the remaining connected, real rank one, simple Lie groups with finite center, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space.

Motivation & Objective

• To compute the completely bounded Fourier multiplier norm of spherical functions on the generalized Lorentz groups $SO_0(1,n)$ for $n \geq 2$.
• To determine whether there exists a uniform bound on these norms across all spherical functions.
• To extend the unboundedness result to other semisimple Lie groups: $SU(1,n)$, $Sp(1,n)$, and $F_{4(-20)}$.
• To establish the existence of completely bounded Fourier multipliers that are not coefficients of uniformly bounded unitary representations on Hilbert spaces.

Proposed method

• Use the characterization of completely bounded Fourier multipliers via the existence of bounded maps $P, Q: G \to \mathscr{H}$ such that $\varphi(y^{-1}x) = \langle P(x), Q(y) \rangle$.
• Apply the Gelfand pair structure of $(SO_0(1,n), SO(n))$ to analyze $K$-bi-invariant spherical functions.
• Employ integral representations involving the Haar measure on $K = SO(n)$ to express the multiplier norm as an average over $K$.
• Use the theory of special functions, particularly the Gamma function, to derive a closed-form expression for the norm.
• Leverage symmetry and invariance properties of the spherical functions to reduce the norm computation to a measure-theoretic average over group cosets.
• Extend the result to other groups via structural similarities in their spherical function theory and representation-theoretic properties.

Experimental results

Research questions

• RQ1 What is the exact value of the completely bounded Fourier multiplier norm for spherical functions on $SO_0(1,n)$ for $n \geq 2$?
• RQ2 Is there a uniform upper bound on the completely bounded Fourier multiplier norms of all spherical functions on $SO_0(1,n)$?
• RQ3 Can the unboundedness of the multiplier norms on $SO_0(1,n)$ be extended to other semisimple Lie groups such as $SU(1,n)$, $Sp(1,n)$, and $F_{4(-20)}$?
• RQ4 Does the existence of unbounded completely bounded Fourier multipliers imply the non-existence of uniformly bounded representations whose coefficients realize these multipliers?

Key findings

• The completely bounded Fourier multiplier norm of the spherical function $\varphi_s$ on $SO_0(1,n)$ is given by a closed-form expression involving the Gamma function: \[ \|\varphi_s\|_{M_0A(G)} = \frac{\Gamma\left(\frac{m}{2}+\mathrm{Re}(s)\right)\Gamma\left(\frac{m}{2}-\mathrm{Re}(s)\right)\Gamma\left(\frac{m}{2}+i\mathrm{Im}(s)\right)\Gamma\left(\frac{m}{2}-i\mathrm{Im}(s)\right)}{\Gamma\left(\frac{m}{2}\right)^2 \left|\Gamma\left(\frac{m}{2}+s\right)\Gamma\left(\frac{m}{2}-s\right)\right|} \] for $|\mathrm{Re}(s)| < \frac{m}{2}$, where $m = n-1$.
• For $s = \pm \frac{m}{2}$, the norm is exactly 1.
• The norm is unbounded as $\mathrm{Re}(s)$ approaches $\pm \frac{m}{2}$, implying no uniform bound exists across all spherical functions on $SO_0(1,n)$.
• The unboundedness result extends to $SU(1,n)$, $Sp(1,n)$ (for $n \geq 2$), and the exceptional group $F_{4(-20)}$, showing the same phenomenon holds in these settings.
• As a consequence, there exist completely bounded Fourier multipliers on these groups that are not coefficients of any uniformly bounded representation on a Hilbert space.
• The construction relies on extending a multiplier from a discrete subgroup $\Gamma$ to a larger group $\Gamma'$, preserving complete boundedness while ensuring it cannot arise from a uniformly bounded representation.