Spectrality of product domains and Fuglede’s conjecture for convex polytopes

A set Ω ⊂ ℝ² is said to be spectral if the space L²(Ω) has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets “behave like” sets which can tile the space by translations. This suggests a conjecture that a product set Ω = A × B is spectral if and only if the factors A and B are both spectral sets. We recently proved this in the case when A is an interval in dimension one. The main result of the present paper is that the conjecture is true also when A is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope Ω is spectral if and only if it can tile by translations.