Journal of Lie Theory

Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal ${\check\Delta}_G = \{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[{n\over2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property of J. Ludwig and L. Turowska [Growth and smooth spectral synthesis in the Fourier algebras of Lie groups, Studia Math. 176 (2006) 139--158]. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed by B. E. Forrest, E. Samei and N. Spronk [Convolutions on compact groups and Fourier algebras of coset spaces, Studia Math. to appear; arXiv:0705.4277]. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.