Unitary representations of reductive p-adic groups: an algorithm
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
Representation theory is the study of symmetries in linear spaces. The symmetries of an object or a physical system can be encoded into various algebraic structures, such as groups, together with their "representations" (actions on linear spaces). The typical questions in the theory are how these actions are built from the most basic constituents and to study these "atoms", i.e., the irreducible representations. This proposal concerns the classification of irreducible (unitary) representations.
More precisely, the aim is to devise a finite algorithm for the determination of all irreducible unitary representations of reductive p-adic groups (think of the invertible square matrices with coefficients in the field of p-adic numbers). From a historical perspective, the classification of unitary representations of (noncompact) semisimple groups ("the unitary dual" problem) is one of the most important unsolved classical problem in representation theory. The origins of this question can be traced back to Gelfand's programme of "abstract harmonic analysis'' from the 1930's and to Wigner's work on the representations of the Lorentz group in Physics. In the last 10 years, new ideas have emerged in the work of Adams, van Leeuwen, Trapa, and Vogan who produced an effective algorithm for deciding the unitarisability question for real reductive groups, and in the work of Schmid and Vilonen who gave a geometric interpretation (in terms of Hodge theory for D-modules) of unitarisability. For applications to automorphic forms, it is imperative to have a similarly precise understanding of the unitary dual of reductive p-adic groups. Thus this proposal advances a corresponding programme for unitary representations of p-adic reductive groups in the framework of the "Langlands correspondence", which is a vast set of conjectures central to much of modern Mathematics. While there is a formal part of the algorithm by Vogan et al which can be easily translated to the p-adic setting, the core problems are deep and require different methods and new ideas. In addition to the satisfaction of having an answer to this classical question in representation theory, the algorithm will uncover new connections between the geometric and arithmetic sides of the Langlands programme and therefore it could have a transformative impact on research in representation theory and in automorphic forms.
More precisely, the aim is to devise a finite algorithm for the determination of all irreducible unitary representations of reductive p-adic groups (think of the invertible square matrices with coefficients in the field of p-adic numbers). From a historical perspective, the classification of unitary representations of (noncompact) semisimple groups ("the unitary dual" problem) is one of the most important unsolved classical problem in representation theory. The origins of this question can be traced back to Gelfand's programme of "abstract harmonic analysis'' from the 1930's and to Wigner's work on the representations of the Lorentz group in Physics. In the last 10 years, new ideas have emerged in the work of Adams, van Leeuwen, Trapa, and Vogan who produced an effective algorithm for deciding the unitarisability question for real reductive groups, and in the work of Schmid and Vilonen who gave a geometric interpretation (in terms of Hodge theory for D-modules) of unitarisability. For applications to automorphic forms, it is imperative to have a similarly precise understanding of the unitary dual of reductive p-adic groups. Thus this proposal advances a corresponding programme for unitary representations of p-adic reductive groups in the framework of the "Langlands correspondence", which is a vast set of conjectures central to much of modern Mathematics. While there is a formal part of the algorithm by Vogan et al which can be easily translated to the p-adic setting, the core problems are deep and require different methods and new ideas. In addition to the satisfaction of having an answer to this classical question in representation theory, the algorithm will uncover new connections between the geometric and arithmetic sides of the Langlands programme and therefore it could have a transformative impact on research in representation theory and in automorphic forms.
Organisations
- University of Oxford (Lead Research Organisation)
- Columbia University (Collaboration)
- Institut de Mathématiques de Jussieu (Collaboration)
- National University of Singapore (Collaboration)
- University of Padova (Collaboration)
- Cornell University (Collaboration)
- Massachusetts Institute of Technology (Collaboration)
People |
ORCID iD |
Dan Ciubotaru (Principal Investigator) |
Publications
Aubert, A.-M.
A nonabelian Fourier transform for tempered unipotent representations
in arXiv:2021
Ciubotaru D
(2022)
The wavefront sets of Iwahori-spherical representations of reductive p-adic groups
in arXiv:2112.14354v4
Ciubotaru D
(2022)
The Wavefront Sets of Unipotent Supercuspidal Representations
in arXiv:2206.08628
Ciubotaru D
(2023)
Local character expansions via positive depth Barbasch-Moy theory
Ciubotaru D
(2023)
Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups II
Ciubotaru D
(2023)
Some Unipotent Arthur Packets for Reductive p-adic Groups
in International Mathematics Research Notices
Ciubotaru D
(2022)
Weyl groups, the Dirac inequality, and isolated unitary unramified representations
in Indagationes Mathematicae
Ciubotaru D
(2022)
On the generalized Ramanujan conjecture over function fields
in arXiv:2204.06053
Ciubotaru D
The nonabelian Fourier transform for elliptic unipotent representations of exceptional p-adic groups
in arXiv:2006.13540
Ciubotaru D.
Some unipotent Arthur packets for reductive p-adic groups I
in arXiv:2021
Description | Some of the oldest unsolved questions in number theory can be rephrased in modern mathematical language in terms of the interplay between representation theory (the study of symmetries) and Galois theory (originated from the study of polynomial equations). We have found new patterns in this programme. More precisely, we have observed a remarkable interaction between two apparently unrelated mathematical (Fourier) transformations which likely has deep mathematical meaning. We have also made progress on some old conjectures of Arthur in the field, by constructing interesting special cases of `Arthur packets'. We found new relations between the abstract harmonic analysis of a p-adic Lie group and the local Langlands programme. Our work on the classification of unitary representations has already had an important impact in number theory in relation with certain versions of the famous generalised Ramanujan conjecture. |
Exploitation Route | The results have serious applications to number theory, in the area of automorphic forms, for example to an instance of the generalised form of Ramanujan's conjecture, which we proved in a joint paper (2022) with Michael Harris. |
Sectors | Education |
Description | DPhil supervision - Xin Zhao |
Geographic Reach | Multiple continents/international |
Policy Influence Type | Influenced training of practitioners or researchers |
Impact | The DPhil student has acquired state of the art skills in mathematical research. |
Description | DPhil supervision Mick Gielen |
Geographic Reach | Multiple continents/international |
Policy Influence Type | Influenced training of practitioners or researchers |
Impact | The DPhil candidate is gaining new state of the art skills in this area of mathematics. |
Description | Marie Curie Fellowships panelist |
Geographic Reach | Europe |
Policy Influence Type | Participation in a guidance/advisory committee |
Impact | The Marie Curie fellowships are some of the most prestigious European postdoctoral grants. They made a substantial difference in the career and development of the successful recipients. |
Description | NSF panel |
Geographic Reach | North America |
Policy Influence Type | Participation in a guidance/advisory committee |
Impact | The panel ranked the submitted research proposals and made recommendations for funding. |
Description | Ph.D. supervision - Ruben La |
Geographic Reach | National |
Policy Influence Type | Influenced training of practitioners or researchers |
Description | Supervision DPhil student - Elena Collaciani |
Geographic Reach | Europe |
Policy Influence Type | Influenced training of practitioners or researchers |
Impact | The PhD student is acquiring state of the art skills in this area of mathematical research. |
Description | L-packets and the nonabelian Fourier transform |
Organisation | Institut de Mathématiques de Jussieu |
Country | France |
Sector | Public |
PI Contribution | With Anne-Marie Aubert (Paris) and Roman Bezrukavnikov (MIT), I have been studying one aspect of the Langlands programme, namely the relation between the classification of irreducible representations of p-adic groups via L-packets and the restriction to maximal compact subgroups. My contribution is the expertise with branching of representations. |
Collaborator Contribution | Aubert is an expert in the local Langlands programme. Bezrukavnikov is an expert in geometric representation theory. |
Impact | One joint paper with Aubert and Romano, submitted for publication (preprint listed in the publications section). On-going work with Bezrukavnikov should be reported in a preprint this year. |
Start Year | 2021 |
Description | L-packets and the nonabelian Fourier transform |
Organisation | Massachusetts Institute of Technology |
Country | United States |
Sector | Academic/University |
PI Contribution | With Anne-Marie Aubert (Paris) and Roman Bezrukavnikov (MIT), I have been studying one aspect of the Langlands programme, namely the relation between the classification of irreducible representations of p-adic groups via L-packets and the restriction to maximal compact subgroups. My contribution is the expertise with branching of representations. |
Collaborator Contribution | Aubert is an expert in the local Langlands programme. Bezrukavnikov is an expert in geometric representation theory. |
Impact | One joint paper with Aubert and Romano, submitted for publication (preprint listed in the publications section). On-going work with Bezrukavnikov should be reported in a preprint this year. |
Start Year | 2021 |
Description | Local Langlands correspondence and local character expansion |
Organisation | Massachusetts Institute of Technology |
Country | United States |
Sector | Academic/University |
PI Contribution | I have started a collaboration with Prof Ju-lee Kim at MIT. The goal is to study the relation between the local character expansion and the local Langlands programme. |
Collaborator Contribution | Prof Kim is an expert in abstract harmonic analysis. |
Impact | This collaboration has only recently began. We have settled on a conjectural description of the relation between the local character expansion and the Langlands parametrization and we are in the process of verifying in the case of the general linear group. The results will make the subject of a first joint paper. |
Start Year | 2023 |
Description | Representations of finite group of Lie type |
Organisation | University of Padova |
Country | Italy |
Sector | Academic/University |
PI Contribution | I have begun a collaboration with Prof Giovanna Carnovale in Padova to study the representation theory of finite groups of Lie type following the work of Lusztig. Part of this, I am co-supervising a PhD student in Padova, Elena Collaciani, who will visit Oxford this year to accelerate the progress on this problem. |
Collaborator Contribution | Prof Carnovale is an expert in geometric representation theory and tools from this area are needed for the project. |
Impact | The collaboration only began recently. When the results are available, they will make the subject of a joint paper. One. outcome of the collaboration is the training of a new PhD student in this research area. |
Start Year | 2023 |
Description | Spherical unitary dual |
Organisation | Cornell University |
Country | United States |
Sector | Academic/University |
PI Contribution | I contribute my expertise in the classification of unitary representations of Hecke algebras. |
Collaborator Contribution | The partner, Dan Barbasch (Cornell), is an expert in the unitary representation theory of Lie groups. |
Impact | We have produce several Mathematica programs to calculate signatures of hermitian forms for representations. The goal is to obtain a classification of the spherical unitary dual of complex exceptional groups. |
Start Year | 2021 |
Description | Unitary representations and numbers the Ramanujan conjecture |
Organisation | Columbia University |
Country | United States |
Sector | Academic/University |
PI Contribution | My expertise is in the classification of unitary representations of p-adic groups, this is what is applied to the project. |
Collaborator Contribution | The partner, Michael Harris (Columbia), is an expert in automorphic forms and the Langlands programme. |
Impact | We have a preprint that it's undergoing the final revisions before we post it on the arxiv. |
Start Year | 2021 |
Description | Wavefront sets of representations of p-adic groups |
Organisation | National University of Singapore |
Country | Singapore |
Sector | Academic/University |
PI Contribution | This is a project in collaboration with Dr Lucas Mason-Brown (Oxford) and Dr Emile Okada (Singapore). The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the unipotent representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an explicit description of the wavefront set of an irreducible spherical representation with real Satake parameter. We apply these results to the computations of unipotent Arthur packets, motivated by Arthur's conjectures. |
Collaborator Contribution | Emile Okada, a former DPhil student of mine, is an expert in the classification of nilpotent orbits and the Bruhat-Tits building, and their applications to abstract harmonic analysis. |
Impact | We have produced three preprints, posted on arXiv, with one more to come soon. We have reported on this work at several seminars and international conferences. |
Start Year | 2021 |
Description | Algebra and Representation Theory Seminar Oxford |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Professional Practitioners |
Results and Impact | I am the organiser of the Algebra and Representation Theory seminar in Oxford, where we have invited talks (mostly outside speakers) weekly during term time. The seminar brings together the local experts in this research area with the speaker and about a dozen DPhil students and sometimes a few undergraduate students. |
Year(s) Of Engagement Activity | 2021,2022,2023 |
URL | https://www.maths.ox.ac.uk/events/past/623 |
Description | BIRS workshop Canada |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | I attended online and gave an invited talk at the workshop "Langlands Program: Number Theory and Representation Theory", BIRS, Oaxaca, Mexico, November 2022. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.birs.ca/events/2022/5-day-workshops/22w5178 |
Description | Conference Amsterdam |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | I attended and gave an invited talk at the international conference "From E6 to E60", a conference in honour of Eric Opdam, Amsterdam, September 2022. |
Year(s) Of Engagement Activity | 2022 |
URL | https://staff.fnwi.uva.nl/r.r.j.bocklandt/E60/ |
Description | Conference talk - Singapore |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | I participated in the workshop "Representation Theory, Combinatorics and Geometry", IMS Singapore, January 2023, where I gave an invited talk. |
Year(s) Of Engagement Activity | 2023 |
URL | https://ims.nus.edu.sg/events/representation-theory-combinatorics-and-geometry/ |
Description | Groups and Equations - Taster session |
Form Of Engagement Activity | Participation in an open day or visit at my research institution |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Schools |
Results and Impact | 80 A-level students attended this talk, aimed at giving them an idea about the mathematics studied at university and the application process at University of Oxford. |
Year(s) Of Engagement Activity | 2023 |
URL | https://www.some.ox.ac.uk/study-here/access-outreach/open-days/ |
Description | Open days Somerville |
Form Of Engagement Activity | Participation in an open day or visit at my research institution |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Schools |
Results and Impact | I take part every year, twice a year, to the open days organised by Somerville College for prospective students and their parents. I give a short presentation about the tutorials in Mathematics in college and answer questions from the audience regarding admissions, access, and the Mathematics degree. |
Year(s) Of Engagement Activity | 2021,2022,2023 |
Description | Pure Maths Colloquium - Sheffield |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Professional Practitioners |
Results and Impact | I gave an invited colloquium talk in the Maths Department at Sheffield University, November 2022. |
Year(s) Of Engagement Activity | 2022 |
Description | Satellite conference - vICM 2022 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | I gave an invited online talk at Satellite conference of vICM 2022, Geometric Representation Theory, online, July 2022. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.hairer.org/ICMSCG/ |
Description | Taster sessions: Groups and Equations |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | I gave an outreach talk (February 2022) organised by my college, Somerville, for A-level students in Mathematics interested in studying Maths at university. The talk was about groups and their use in Galois theory for deciding solvability of equations. |
Year(s) Of Engagement Activity | 2022 |