"The study of generalized functions on real manifolds is somewhat more difficult than the case of p-adic manifolds. The works of my students B. Elazar and A. Shaviv on Schwartz functions help to reduce this gap. On the other hand, there is one set of tools which is available for real manifolds and not available for p-adic ones. These tools are algebras of differential operators. If the manifold is a reductive group a special role is played by the algebra of bi-invariant differential operators. Bernstein suggested a certain algebra of integral operators that plays a similar role in the p-adic case. In [3] we applied the Bernstein center to the study of equivariant generalized functions on a p-adic reductive group, generalizing several classical results of Harish-Chandra.
Another difficulty that arises in the real case is posed by the topology on the representations. This complicates the use of a basic tool called the Lie algebra homology. This is a collection of vector spaces attached to each representation. A-priory, these vector spaces might be ill-behaved, i.e. not Hausdorff. We managed to show that in several important cases they are in fact Hausdorff.
In addition, with my colleague A. Aizenbud we generalized to the real case a new tool for proving vanishing of invariant distributions.
In addition, my students S. Carmeli and I. Glazer, and my post-docs H. Lu and R. Rubio studied the harmonic analyses on several classical symmetric spaces and obtained bounds on multiplicities of the irreducible representations that appear in the spaces of functions on the symmetric spaces. In particular, they showed that most complex symmetric pairs are Gelfand pairs. We are now working on the remaining ones.
This summarizes our advance towards the first goal.
As for the second goal, we studied the relations between the spaces of functions of the quotients of a fixed reductive group by different unipotent subgroups. These relations are new both in the real and the p-adic case, as well as in the global case of automorphic representations.
This enables to find new models for representations, and to confirm several conjectures for the general linear groups. The global case has found some applications in physics, more precisely the super-string theory. Namely, the global case allows to express certain Fourier coefficients of next-to-minimal automorphic forms through their degenerate Whittaker coefficients. We applied this to deduce that the top Fourier coefficients of minimal and next-to-minimal automorphic forms decompose into Euler product. This gives some explicit expressions for next-to-minimal automorphic forms, that in turn have physical meaning.
Both directions of the project found applications in the generalized doubling method - a new prominent method that constructs integral representations for the tensor product L-functions of classical groups.
The method is developed by Y. Cai, S. Friedberg, D. Ginzburg, and E.Kaplan. Y. Cai was my post-doc, and applied both directions of the project in his collaboration on the generalized doubling method. Later I worked on the local aspects of the method in collaboration with E. Kaplan, again applying and further developing the tools of this project.
We have made 4 steps towards the dissemination of the results:
1. Conference, workshop, and seminar talks (both in-person and virtual) by the group members.
2. Special trimester “Representation Theory of Reductive Groups Over Local Fields and
Applications to Automorphic forms”, j.w. Prof. Joseph Bernstein, Erez Lapid and Avraham
Aizenbud, Weizmann Institute of Science, Israel. During the two months of the trimester we had ~100 visitors, including leading experts in the field from all over the world, and graduate students from many countries.
We had many mini-courses and research seminars during the trimester, as well as multiple informal interactions and collaborations.
3. A publication in the ""EU research"" journal (Jul 21, 2017) briefly described the field and the project to the general public.
4. An online weekly seminar that I run since May 2020 unites representation theorists from all over the world in these difficult times. We have around 30 participants in each Zoom meeting, and ~150 views after the meeting through my youtube channel."
