Periodic Reporting for period 2 - GReatLaP (p-adic Groups, Representations, and the Langlands Program)
Periodo di rendicontazione: 2024-04-01 al 2025-09-30
Number theory is the study of the integers and objects built out of them. Groups are abstract mathematical objects which encode symmetries, e.g. the symmetries of a cube or crystal, and representation theory is the study of groups using concrete constructions such as matrices. Although number theory and representation theory thus arise as very different areas of mathematics, the Langlands correspondence predicts a deep, fascinating connection between them. The ideas surrounding the Langlands correspondence are the driving force for many groundbreaking advances in mathematics.
The symmetry objects on the representation theory side of the (local) Langlands correspondence are called p-adic groups. The classification of representations of p-adic groups is the fundamental problem on this side of the correspondence. Despite a lot of recent progress in this area, we still do not understand the full picture, in particular when p is small. This project is making significant progress in finding the not yet encountered representations and revealing the structure of all representations together, including their interactions. This is solving important conjectures about the shape of these representations and opens the door to a plethora of applications and new developments. It will also enable new progress on various aspects of different incarnations of the Langlands correspondence.
The symmetry objects on the representation theory side of the (local) Langlands correspondence are called p-adic groups. The classification of representations of p-adic groups is the fundamental problem on this side of the correspondence. Despite a lot of recent progress in this area, we still do not understand the full picture, in particular when p is small. This project is making significant progress in finding the not yet encountered representations and revealing the structure of all representations together, including their interactions. This is solving important conjectures about the shape of these representations and opens the door to a plethora of applications and new developments. It will also enable new progress on various aspects of different incarnations of the Langlands correspondence.
The PI and her collaborators have produced three substantial and important preprints available on arxiv.org and the PI’s homepage:
“Structure of Hecke algebras arising from types”
“Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms”
“Construction of tame supercuspidal representations in arbitrary residue characteristic”
These preprints unravel new structures of representations of p-adic groups.
More precisely, the first two describe the category of representations of p-adic groups in terms of very explicit, well understood data (finite dimensional modules over algebras with explicit generators and relations), assuming the prime p is not too small. Moreover, they thereby introduce a new tool for the area and for applications to the Langlands correspondence, which consists of reducing many questions about infinite dimensional representations of p-adic groups to finite dimensional representations of well understood finite groups.
The third paper fills the gap of a missing construction of representations if the prime p is 2. The previous techniques in the area only worked for odd primes p and it had been an open question for the past 25 years on how to deal with the case p=2.
The PI has also several projects in progress that shine additional new light on the representation theory of p-adic groups and their relation with the Langlands program.
“Structure of Hecke algebras arising from types”
“Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms”
“Construction of tame supercuspidal representations in arbitrary residue characteristic”
These preprints unravel new structures of representations of p-adic groups.
More precisely, the first two describe the category of representations of p-adic groups in terms of very explicit, well understood data (finite dimensional modules over algebras with explicit generators and relations), assuming the prime p is not too small. Moreover, they thereby introduce a new tool for the area and for applications to the Langlands correspondence, which consists of reducing many questions about infinite dimensional representations of p-adic groups to finite dimensional representations of well understood finite groups.
The third paper fills the gap of a missing construction of representations if the prime p is 2. The previous techniques in the area only worked for odd primes p and it had been an open question for the past 25 years on how to deal with the case p=2.
The PI has also several projects in progress that shine additional new light on the representation theory of p-adic groups and their relation with the Langlands program.
The progress contained in the above preprints goes significantly beyond the prior state of the art of the research area. An even better understanding of the representation theory of p-adic groups and its connection with the Langlands program is expected by the end of the project.