Unitary representations of reductive p-adic groups: an algorithm

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Representation theory is the study of symmetries in linear spaces. The symmetries of an object or a physical system can be encoded into various algebraic structures, such as groups, together with their "representations" (actions on linear spaces). The typical questions in the theory are how these actions are built from the most basic constituents and to study these "atoms", i.e., the irreducible representations. This proposal concerns the classification of irreducible (unitary) representations.More precisely, the aim is to devise a finite algorithm for the determination of all irreducible unitary representations of reductive p-adic groups (think of the invertible square matrices with coefficients in the field of p-adic numbers). From a historical perspective, the classification of unitary representations of (noncompact) semisimple groups ("the unitary dual" problem) is one of the most important unsolved classical problem in representation theory. The origins of this question can be traced back to Gelfand's programme of "abstract harmonic analysis'' from the 1930's and to Wigner's work on the representations of the Lorentz group in Physics. In the last 10 years, new ideas have emerged in the work of Adams, van Leeuwen, Trapa, and Vogan who produced an effective algorithm for deciding the unitarisability question for real reductive groups, and in the work of Schmid and Vilonen who gave a geometric interpretation (in terms of Hodge theory for D-modules) of unitarisability. For applications to automorphic forms, it is imperative to have a similarly precise understanding of the unitary dual of reductive p-adic groups. Thus this proposal advances a corresponding programme for unitary representations of p-adic reductive groups in the framework of the "Langlands correspondence", which is a vast set of conjectures central to much of modern Mathematics. While there is a formal part of the algorithm by Vogan et al which can be easily translated to the p-adic setting, the core problems are deep and require different methods and new ideas. In addition to the satisfaction of having an answer to this classical question in representation theory, the algorithm will uncover new connections between the geometric and arithmetic sides of the Langlands programme and therefore it could have a transformative impact on research in representation theory and in automorphic forms.