Relative representation theory and distributions on reductive groups over local fields

Periodic Reporting for period 4 - RelRepDist (Relative representation theory and distributions on reductive groups over local fields)

Berichtszeitraum: 2019-09-01 bis 2020-12-31

The project lies in the crossroads of the major parts of mathematics - analysis, algebra, geometry, and number theory. More precisely, we study non-commutative harmonic analysis on quotients of reductive algebraic groups. This means that we consider a certain well-behaved class of infinite non-compact but finite-dimensional groups, and study their actions on the space of functions on their quotients. More precisely, we study the decomposition of the space of functions to certain building blocks called irreducible representations. This area is part of group representation theory. This theory can be described as the study of symmetries of linear spaces. In our case, the linear spaces are infinite-dimensional and sometimes have a Frechet structure. Another name for my field is relative representation theory, or symmetry breaking.

The importance of the field comes from the importance of the linearization technique. Non-linear objects are usually difficult to study and even more difficult to approximate numerically. Thus one usually approximates non-linear objects by linear ones. Many interesting objects in science possess natural symmetries, e.g. geometric objects, molecules or the laws of physics. Then, the derived linear objects continue to possess these symmetries. The knowledge of the action of these linear symmetries sheds some light on the original object. For these reasons, group representation theory has found numerous applications in other areas of mathematics, as well as in chemistry, physics and computer science. Our project is mainly curiosity-driven, but along the course of the project applications to string theory, number theory, and integral geometry were found.

To formulate the main objectives let F be a local field, e.g. the field of real numbers or the field of p-adic numbers. Let G be an algebraic reductive F-group, e.g. the group of invertible n-by-n matrices over F. Objectives:
1. Develop tools for the study of G-equivariant generalized functions on spaces with G-action, that will work uniformly for all F.
2. Study the action of G on spaces of functions on quotients of G by its unipotent subgroups.

"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."

We have progress beyond the state of the art in three directions:

1. The first application of the Bernstein center to the study of equivariant generalized functions arising from representations.
2. The first application of the root exchange technique in the archimedean case.
3. A systematization of the root exchange technique, that is an algorithm of its application for all reductive groups over all characteristic zero local and global fields and a proof that the algorithm terminates.
4. Fourier expansion of minimal and next-to-minimal automorphic forms on split simply-laced groups.
5. A general necessary condition for distinction of representations of real reductive groups with respect to spherical subgroups, in terms of the nilpotent orbit attached to the representation (a microlocal geometric invariant).

group of matrices acting on its quotient by its maximal compact subgroup

Periodic Reporting for period 4 - RelRepDist (Relative representation theory and distributions on reductive groups over local fields)

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