# Unitary Representations of Real Reductive Groups and Mixed Hodge Modules

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Texas at Austin (TX) · $139,915

## Abstract

Representation theory can be broadly understood as the mathematical study of symmetry. The symmetries that arise in quantum mechanics are called "unitary representations." One of the major unsolved problems in representation theory is to classify unitary representations. This problem has driven a considerable portion of all research in representation theory over the past eighty years; its solution would have far-reaching ramifications in several neighboring fields, including number theory, harmonic analysis, signal processing, and theoretical physics. The PI proposes to develop a new geometric approach to this problem. This approach will lead to substantial new insights into the structure of unitary representations and, hopefully, a solution to the problem of computing the unitary dual. The project also provides research training opportunities for graduate students.

There are two main existing approaches to the study of unitary representations: the orbit method philosophy of Kirillov and Kostant, and the Hodge theory approach of Schmid and Vilonen. The orbit method seeks to parameterize the unitary dual of a Lie group G in terms of (roughly speaking) orbits for G on the dual space of its Lie algebra. The Hodge theory approach seeks to understand unitary representations by localizing over the flag variety and applying tools from Hodge theory. These approaches work along two different axes. Whereas the orbit method provides mainly hints as to where one should look for unitar

## Key facts

- **NSF award ID:** 2501977
- **Awardee organization:** University of Texas at Austin (TX)
- **SAM.gov UEI:** V6AFQPN18437
- **PI:** Lucas Mason-Brown
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $139,915
- **Funds obligated:** $139,915
- **Transaction type:** Standard Grant
- **Period:** 08/15/2025 → 07/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2501977

## Citation

> US National Science Foundation, Award 2501977, Unitary Representations of Real Reductive Groups and Mixed Hodge Modules. Retrieved via AI Analytics 2026-06-06 from https://api.ai-analytics.org/grant/nsf/2501977. Licensed CC0.

---

*[NSF Awards dataset](/datasets/nsf-awards) · CC0 1.0*