Geometrization of Admissible Distributions and the Local Langlands Conjecture

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The long-term objective of my research program is to prove the Langlands Conjecture for non-Archimedean local fields as a consequence of a duality between two categories of perverse sheaves. The local Langlands Conjecture promises a precise relation between Galois representations of local fields and admissible representations of reductive groups over local fields and, as such, is a central problem with important implications in number theory. Many instances of the local Langlands Conjecture are now known, most recently due to spectacular new results by Arthur, but the general case remains open and it seems that the subject may benefit from new perspectives. *** While capitalizing on recent progress, my approach to the Langlands Conjecture for non-Archimedean local fields ultimately relies on a novel geometric and categorical perspective on admissible distributions that is adapted from George Lusztig's work on character sheaves. This adaptation relies on an arsenal of techniques from arithmetic geometry, including smooth integral models for certain algebraic varieties over non-Archimedean local fields, their Greenberg transforms and new refinements of Serre-Hazewinkel class field theory.*** This research program began in 2011 when I was working at the Mathematisches Forschungsinstitut Oberwolfach with collaborators Pramod Achar, Masoud Kamgarpour and Hadi Salmasian. There we understood how to geometrize and categorify complete pure Langlands parameters for all quasisplit groups p-adic G using equivariant perverse sheaves on an ind-variety built from the Langlands group of G. Overwhelmed by the elegance of this picture, we started to think about a `dual' category of equivariant anti-orbital perverse sheaves on a pro-variety built from G itself, and dared to dream that the Langlands Correspondence might be understood using something like the Fourier-Mukai transform between these categories. This idea, which we call the Strasbourg Dream, led David Roe and me to the notion of quasicharacter sheaves for tori over non-Archimedean local fields and the proof that both quasicharacters and Langlands parameters for tori are encoded in quasicharacter sheaves. Even this case is very rich and surprising, encompassing, as it must, both geometric class field theory and pure rational forms of tori.*** The next step in this research program is to find the correct notion of quasicharacter sheaves for quasisplit groups by adapting Lusztig's construction of character sheaves to a category developed recently by Takashi Suzuki. Quasicharacter sheaves will be simple perverse sheaves equipped with an action of the Weil group of the local field (compare with nearby cycles). The key idea is that both Langlands parameters and admissible distributions will be encoded in quasicharacter sheaves, and to use this encoding to establish the Langlands Conjecture for quasisplit groups over non-Archimedean local fields.**