## Project Leader

- Clifton Cunningham, University of Calgary

### Graduate Mentor

- Kristaps Balodis, University of Calgary

### Undergraduate Team Members

- E Thompson, University of Calgary

## Problem Statement

This project develops a fundamental tool for representation theory of p-adic groups: computing multiplicity matrices for admissible representations.

Representations of Lie groups play a central role in areas of research ranging from high-energy physics to number theory. In 2008, the Atlas for Lie Groups and Representations was fêted in Science, Nature, Scientific American and the New York Times for computing the multiplicity matrix for the exceptional Lie group $E_8$. At the heart of the $E_8$ result is a tool that responds to this query: find the change of basis matrix relating irreducible representations and standard representations in the Grothendieck group of smooth representations. In fact, the Atlas tool answers this query for Lie groups of all types. This project develops a tool that answers the same query for p-adic groups. Lie groups and p-adic groups are parallel structures that appear together in the Langlands program. In fact, one can effectively argue that studying Lie groups without p-adic groups is like listening to the sound of one hand clapping. This project is the first step in building the hand that matches the Atlas: we call it the p-Atlas. The idea of building an Atlas for p-adic Groups and Representations, or p-Atlas as we call it, was born in the early days of the Voganish Project. This idea is now viable because of developments in the Voganish Project in Summer 2022, resulting in algorithms that compute multiplicity matrices for three infinite families of representations of p-adic $GL(n)$. With this project, we will expand that capability significantly. The Voganish Project is a vibrant, diverse, international, vertically integrated research group with faculty in Canada, USA, France, Korea and China. The PIMS XVMLab format is a perfect fit for this community, to recruit and expand and to develop new computational tools for the Langlands program.

### Final Report

This VXML project was completed by the participants listed above as described in the following documents:

### Details

- Expected team size: 3
- Student Experience Level: Intermediate, students who have had an introduction to proofs

##### Prerequisites

- Advanced Linear Algebra
- Multivariate Calculus
- Abstract Algebra

##### Skills

- Python/Sage
- GitHub (preferred)
- LaTeX (optional)

### Other Information

The objective of this project is to develop and implement algorithms to compute multiplicity matrices for multiple infinite families of representations of p-adic GL(n) building on geometric techniques we have developed and implemented for other families. All students working on this PIMS VXML project will be members of the Voganish Project (http://automorphic.ca) and therefore have access and exposure to an international research community already including undergraduates, graduate students, postdocs and professors.