Local parameters of supercuspidal representations (Q6610998)

In this paper the authors refine some aspects of Genestier-Lafforgue (and Fargues-Scholze) parameterization of irreducible representations of a connected reductive group \(G\) over a non-Archimedean local field \(F\) of positive characteristic. The Genestier-Lafforgue result is expressed in terms of a so-called semisimple parameter (the domain is the Weil group) which differs from the classical notion of the Langlands correspondence/conjecture which deals with the Weil-Deligne group. The refinement is for the tempered and for the supercuspidal representations in the following way. \N\NFrom the abstract: ``Our first result shows that the Genestier-Lafforgue parameter of a tempered \(\pi\) can be uniquely refined to a tempered L-parameter \(\mathcal{L}(\pi)\), thus giving the unique local Langlands correspondence which is compatible with the Genestier- Lafforgue construction. Our second result establishes ramification properties of \(\mathcal{L}^{ss}(\pi)\) for unramified \(G\) and supercuspidal \(\pi\) constructed by induction from an open compact (modulo center) subgroup. If \(\mathcal{L}^{ss}(\pi)\) is pure in an appropriate sense, we show that \(\mathcal{L}^{ss}(\pi)\) is ramified (unless \(G\) is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show \(\mathcal{L}^{ss}(\pi)\) is wildly ramified.'' \N\NThe proofs are via global argument, using the results on Poincare series which were established in the previous work of Gan and Lomeli. Also, further results on the globalization of discrete series are given in Appendix by Raphael Beuzart-Plessis. The authors also offer a wealth of further information and discussions and pose some conjectures; e.g. Conjecture 11.7 which deals with the relation between the Genestier-Lafforgue parametrization in the positive characteristic case with the Fargues-Scholze parametrization for groups over p-adic fields via Kazhdan's, Deligne's and Ganapathy's results on representations of groups over \(n\)-close fields.