In this paper, we define Sobolev spaces on a locally compact unimodular group in link with the spherical Fourier transform of type $\delta$. Properties these are obtained. Analogues embedding theorems proved.

We show how the knowledge of the Fourier coefficients of the Cherednik kernel leads to combinatorial formulas for generalized exponents. We recover known formulas for generalized exponents of irreducible representations parameterized by dominant roots, and obtain new formulas for the generalized exponents for irreducible representations parameterized by the dominant elements of the root lattice...