The Deligne-Kazhdan philosophy and the Langlands conjectures in positive characteristic

Radhika Ganapathy, Purdue University

Abstract

Two non-archimedean local fields are m-close if their rings of integers modulo the m-th power of their respective maximal ideals are isomorphic. The Deligne-Kazhdan philosophy can be loosely stated as follows: If F and F' are m-close, then the representation theory of Galois groups and reductive algebraic groups over F and F' are the same, “up to level m". In this thesis, we attempt to understand the relationship between this philosophy and the local Langlands conjectures. In the second chapter of this thesis, we establish that the local Langlands correspondence (LLC) for GLn(F) is compatible with this philosophy. To do this, we crucially use Lemaire's work on transferring generic representations of GLn over close local fields. Lemaire's work relies on a Hecke algebra presentation written down by Howe for GLn(F). The remaining chapters of the thesis focus on generalizing the work of Howe and Lemaire to any split reductive group with simply connected derived subgroup. More precisely, we write down a presentation of the Hecke algebra [special characters omitted](G, Im), where Im is the m-th Iwahori congruence subgroup of G. We then establish a variant the Kazhdan isomorphism for this Hecke algebra. Using this variant of the Kazhdan isomorphism, we generalize Lemaire's work on transferring generic representations over close local fields.

Degree

Ph.D.

Advisors

Yu, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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