Characters of p-adic reductive groups in the local Langlands correspondence

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This project falls within the EPSRC Algebra research area. It fits within the influential and massive Langlands program, which attempts to connect and unify large parts of mathematics, in particular geometry, number theory, analysis and representation theory. More specifically the project connects to the local Langlands correspondence, which relates admissible representations of (p-adic) reductive groups to the so-called Langlands parameters. We are interested in the relation between the characters of such representations and their associated Langlands parameters.To study this relation we will consider the local character expansion due to Harish-Chandra and Howe. This expresses the distribution character of a (smooth admissible) representation of a reductive p-adic group as a linear combination of Fourier transforms of nilpotent orbital integrals. Much is known about the domain of validity of the character expansion (by the works of Waldspurger and DeBacker), but the coefficients of the linear combination have largely remained mysterious. To get a handle on these expansions we shall attempt to relate some of their coefficients (i.e., the leading coefficients) to a certain growth rate (the "canonical dimension"), namely the growth rate of the dimensions of the finite-dimensional subspaces of the representation obtained by considering the vectors fixed under a descending chain of certain open compact subgroups. This growth rate received attention in the case of real reductive groups and in the context of the mod p Langlands correspondence, however this avenue of research remains unexplored in the case of complex representations of reductive p-adic groups. Using the real reductive case and the mod p case as footholds, we should be able to develop a satisfactory theory of the growth rate of the dimensions of the fixed point subspaces also in the case of complex representations of reductive p-adic groups. We expect this theory to shed a light on the local character expansion and we can then use this insight to examine the relation between the character of a representation and its associated Langlands parameters, thus helping to elucidate some part of the local Langlands correspondence. The local Langlands correspondence for p-adic groups is still incomplete, but an important part of it, the classification of the so-called unipotent, is now known in full generality by the work of Kazhdan, Lusztig, Reeder, Waldspurger, Opdam, Solleveld and others. This gives us a good and representative source of examples to test the connection between the canonical dimension and the Langlands parameters.In the same vein, another important invariant attached to a character of the representation is the wavefront set, which is another measure of growth. We would like to relate the canonical dimension to the wavefront set, such relations being known for real groups, but not for p-adic groups. In particular, it will be interesting to investigate the relation with the recent work of Ciubotaru, Mason-Brown, and Okada on the wavefront set of unipotent representations.These results are likely to have an impact in the representation theory of reductive groups over local fields and in number theory, in the general framework of the Langlands programme.