Distinction problems by Iwahori-Hecke algebras

This project falls within the ESPRC Algebra research area. The general subject area of this project is the representation theory of p-adic reductive groups, in the general framework of the local Langlands programme. Motivated by applications to automorphic forms, the aim of the project is to study certain distinction questions for smooth representations of p-adic groups or finite groups of Lie type. Consider a reductive p-adic or a finite group of Lie type G and a closed subgroup H of G and two smooth (complex) irreducible representations \pi of G and \sigma of H. The most basic case is when \sigma is the trivial representation. The question we are interested in is when Hom_H(\pi,\sigma) is non-zero and what its dimension is in the case that it is non-zero. In the most interesting cases, this dimension is equal to 0 or 1. This type of problem has a long history. A very classical example is when H is a maximal compact open subgroup of G, for example G=GL(n,Q_p) and H=GL(n,Z_p). The representations distinguished by this H are called (H-)spherical and the fact that the multiplicity space is at most one-dimensional follows from the fact that the spherical Hecke algebra H(G,H) of compactly supported functions of G which are H-biinvariant is abelian. Another famous example is the case of Whittaker (or generic) representations, in which case, H is the unipotent radical of a Borel subgroup (when G is quasisplit) and \sigma is a nondegenerate character of H [CS80].
One particularly interesting case to consider is when \pi is a smooth representation of G with Iwahori fixed vectors. It turns out that the category of such representations is equivalent to the category of modules over Iwahori-Hecke algebras H(G,I) of compactly supported I-biinvariant functions. As a consequence, we can study such representations from the point of view of Iwahori-Hecke algebras. The main challenge that comes with this approach is transferring the questions we are studying about the smooth representations to the equivalent question in the Iwahori-Hecke algebras setting. The novelty in this research methodology is the emphasis of this algebraic approach from the point of view of Iwahori-Hecke algebras.
There are multiple objectives that we could aim to reach in this project. For instance, one could try to generalise the results of [CS16] who consider representations with Whittaker models. In [CS16], G is a Chevalley group over a p-adic field, H is a unipotent radical of a Borel subgroup of G and \sigma is a non-degenerate character of H. The equivalent problem in the setting of the Iwahori Hecke algebra is to determine the simple modules whose restriction to the finite Hecke algebra contains the Steinberg module. Another objective could be to consider the example with G = GL(2n,\Q_p), H = SP(2n,\Q_p) and \sigma the trivial character, a case studied before via different methods in [OS07]. As a related toy example, one would have to consider the irreducible representations of the symmetric group whose restriction to the hyperoctahedral subgroup contains the trivial representation. More interestingly, insight should be obtained by considering the similar restriction problem for finite groups of Lie type over F_q.
The most ambitious goal would be to find an Iwahori-Hecke algebra common framework which applies to a vast class of distinction problems, which include as particular cases the examples mentioned above.