Spectrality of Product Domains

A set $\Omega$ in $R^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential
functions. Which sets $\Omega$ are spectral? This question is known as "Fuglede's spectral set problem".
In the talk we will be focusing on the case of product domains, namely, when $\Omega = AxB$.
In this case, it is conjectured that $\Omega$ is spectral if and only if the factors A and B are both spectral.
We will discuss some new results, joint with Nir Lev, supporting this conjecture, and their applications
to the study of spectrality of convex polytopes.