A family of singular oscillatory integral operators and failure of weak amenability

Abstract

A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$ dimensional Heisenberg group and $SL(2,\Bbb R)$ acts via the irreducible representation of dimension $2n$ fixing the center of $H_n$. We show that these groups fail to be weakly amenable. Following an idea of Haagerup for the case $n=1$ one can reduce matters to the problem of obtaining nontrivial uniform bounds for a family of singular oscillatory integral operators with product type singularities and polynomial phases. The result on the family $G_n$ and various other previously known results are used to settle the question of weak amenability for a large class of Lie groups, including the algebraic groups; we assume that the Levi-part has finite center.