Pizzetti Formulae for Stiefel Manifolds and Applications

Abstract

Pizzetti’s formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular, we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson–Zuber integral for the coset SO (4)/[SO (2)×SO(2)]. This integral naturally appears in the calculation of the two-point correlation function in the transition of the sta..