My one month stay in ICL in spring 2022 was a great opportunity to benefit of an excellent working environment and start collaborations (some of them long term) with colleagues having expertise complementary to mine.
I started a project with Prof. David Helm on a re-interpretation of some parts of Dipendra Prasad's conjecture on distinction of complex representations for local Galois pairs in terms of Bernstein center.
We could set up a coherent framework and state the conjecture, which indeed follows from that of Prasad. With this formulation the hope is to extend Prasad's conjecture to the setting of representations modulo ell or representations in families, and we started working out small rank examples.
I also took the opportunity to visit Prof. Shaun Stevens in East-Anglia to start a project on generalizing the so called Shahidi's local coefficients of generic representations occuring in the Langlands-Shahidi method to the setting of inner forms of GL(n).
It seems there that these coefficients will be conjugacy classes of invertible matrices rather than scalars. We made a conjecture on these conjugacy classes of matrices and their behaviour under the Jacquet-Langlands correspondence.
Finally we started a project with Justin Trias, a postdoctoral fellow at ICL under the supervision of Prof. Helm, on Whittaker models and duality for representations of p-adic reductive groups modulo ell. In particular we proved a multiplicity one result, and many other properties of such representations related with taking contragredient. This led to the a preprint now submitted for publication, and to other problems that we are currently thinking about.
Host: Professor David Helm