Arithmetic of Cuspidal Representations

Abstract

Here, F is a non-Archimedean local field of residual characteristic p. We are concerned with the irreducible, cuspidal representations of the general linear groups \(\mathrm {GL}(n, F)\). A complete classification of these representations has been known for a long time. It is achieved using rather complicated objects, the simple types and simple characters. The methods it requires have been useful more widely, and the general scheme is now known to apply to many more groups, including \(\mathrm {GL}(m, D)\) (where D is a central F-division algebra), orthogonal groups \(\mathrm {SO}(n)\), symplectic groups \(\mathrm {Sp}(2n)\), (both for p not equal to 2) and even a couple of exceptional groups. In some cases, it is known that the common classification conforms to the requirements of Functoriality. The most interesting, and presently the most difficult, instance of Functoriality is the basic connection between the irreducible cuspidal representations of \(\mathrm {GL}(n, F)\) and the irreducible, n-dimensional representations of the Weil group of F. These notes describe the classification of the cuspidal representations, introducing the results and techniques currently necessary for making this connection more explicit, given that it is known to exist.