Star-product formulas on finite groups

Abstract

Given a finite group G, associative and noncommutative products * will be constructed on the function space L²(G), where these star-products are parametrized by the unitary dual Ĝ of G. If R_f:L²(G)→ L²(G) is the convolution operator R_f(f₁)=f*f₁, then the projections of R_f on the irreducible spaces of the elements of Ĝ is equivalent to R_f*, R_f*(f₁)=f*f₁. It is thus natural to consider Harmonic Analysis/Quantum Mechanics on L²(G) using the star-product. In this work, we establish star-product formulas on finite groups, both in the position and momentum representation.