## Periodic Reporting for period 2 - RelRepDist (Relative representation theory and distributions on reductive groups over local fields)

**Reporting period:**2016-09-01

**to**2018-02-28

## Summary of the context and overall objectives of the project

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 some applications to string 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 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 some applications to string 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.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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, my students S. Carmeli and I. Glazer studied the harmonic analyses on several classical symmetric spaces and obtained bound on multiplicities of the irreducible representations that appear in the spaces of functions on the symmetric spaces.

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. Since the latter have Euler decomposition, this gives some explicit expressions for next-to-minimal automorphic forms, that in turn have physical meaning.

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, my students S. Carmeli and I. Glazer studied the harmonic analyses on several classical symmetric spaces and obtained bound on multiplicities of the irreducible representations that appear in the spaces of functions on the symmetric spaces.

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. Since the latter have Euler decomposition, this gives some explicit expressions for next-to-minimal automorphic forms, that in turn have physical meaning.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

By the end of the project we expect to extend the obtained results, and

1. Deduce Hausdorfness of the Lie algebra homology for a wider variety of cases.

2. Study in depth the real and p-adic next-to-minimal representations, and establish properties similar to the known properties of the minimal representations. We are motivated by some results we already obtained for exceptional groups, and by the physical meaning of these representations.

3. Study the multiplicity one property of classical complex symmetric pairs

4. Establish the existence of degenerate Whittaker models for real reductive groups

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.

By the end of the project we expect to extend the obtained results, and

1. Deduce Hausdorfness of the Lie algebra homology for a wider variety of cases.

2. Study in depth the real and p-adic next-to-minimal representations, and establish properties similar to the known properties of the minimal representations. We are motivated by some results we already obtained for exceptional groups, and by the physical meaning of these representations.

3. Study the multiplicity one property of classical complex symmetric pairs

4. Establish the existence of degenerate Whittaker models for real reductive groups