Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions
Variants of Kirillov theory express characters of appropriate representations of appropriate groups in terms of Fourier transforms of explicitly given orbital integrals on Lie algebras. In the case of Murnaghan-Kirillov theory, one considers irreducible, admissible representations of reductive \(p\)-adic groups, and the expressions are only valid on a neighborhood of \(0\). For some representations, a single orbital integral does not suffice, and one must instead consider linear combinations. One would like to know these linear combinations explicitly for all representations. If certain tameness hypotheses are satisfied, then the problem essentially reduces to the case of ``depth-zero'' representations, which are associated to cuspidal representations of finite reductive groups. The paper under review expresses the characters of such representations in terms of distributions that are attached to generalized Green functions of finite groups. This further reduces the problem to that of expressing these new distributions in terms of orbital integrals. The methods are inspired by earlier work of \textit{J.-L. Waldspurger} [Intégrales Orbitales Nilpotentes et Endoscopie pour les Groupes Classiques non Ramifiés. Astérisque. 269. Paris: Société Mathématique de France (2001; Zbl 0965.22012)].
scientific article; zbMATH DE number 5982617
scientific article