Albert Lopez Bruch

A nonabelian Fourier transform for p-adic groups

Under the Local Langlands Correspondence, unipotent representations of a p-adic group admit a natural parametrization in terms of geometric data attached to the dual group. On the other hand, these representations can be described explicitly using the theory of unipotent representations of finite groups of Lie type, developed by Deligne and Lusztig.
A central ingredient in this theory is Lusztig’s nonabelian Fourier transform, a key element in the construction of irreducible characters of finite reductive groups. In this talk, I will explain this algebraic construction and the connection between representations of p-adic groups to those of finite groups of Lie type. Finally, I will discuss recent efforts to construct an analogue of the nonabelian Fourier transform in the p-adic setting, and summarize the progress achieved so far.