[Paper Review] Unitary representations of real reductive groups

Jeffrey Adams, Marc van Leeuwen+2 more|arXiv (Cornell University)|Dec 10, 2012

Molecular spectroscopy and chirality4 citations

TL;DR

This paper presents an algorithm to compute irreducible unitary representations of real reductive groups by tracking signature changes of invariant Hermitian forms through deformations, using the Langlands classification and the unitary trick. The method combines Kazhdan-Lusztig theory and Beilinson-Bernstein's proof of the Jantzen conjectures to determine unitarity via signature analysis at reducibility points.

ABSTRACT

We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algorithm is to follow the behavior of the signature of the Hermitian form through this deformation, counting changes through singularities of the form at reducibility points. An important technical tool is replacing the classical invariant form (in which the real form of the Lie algebra acts by skew-adjoint operators) by forms in which the compact form of the Lie algebra acts by skew-adjoint operators.

Motivation & Objective

• To develop a systematic algorithm for computing irreducible unitary representations of real reductive Lie groups.
• To address the challenge of determining which representations in the Langlands classification are unitary.
• To provide a computational framework based on deformation theory of Hermitian forms.
• To leverage deep results from Kazhdan-Lusztig theory and the Beilinson-Bernstein proof of the Jantzen conjectures.
• To use the unitary trick to simplify the analysis of Hermitian forms by replacing the real Lie algebra action with a compact form

Proposed method

• Utilizes the Langlands classification to express representations with invariant Hermitian forms as deformations of unitary representations from the Plancherel formula.
• Applies the unitary trick by replacing the classical invariant Hermitian form with one where a compact real form of the Lie algebra acts by skew-adjoint operators.
• Traces the signature of the Hermitian form through the deformation process, identifying changes at reducibility points.
• Employs Kazhdan-Lusztig polynomials to analyze irreducible characters and infer information about unitarity.
• Uses the Beilinson-Bernstein proof of the Jantzen conjectures to obtain complete information on the behavior of forms under deformation.
• Combines representation-theoretic techniques with geometric methods to compute unitary structures algorithmically

Experimental results

Research questions

• RQ1 How can one algorithmically determine which irreducible representations in the Langlands classification are unitary?
• RQ2 What is the behavior of the signature of an invariant Hermitian form under deformation in the Langlands series?
• RQ3 How do reducibility points affect the unitarity of representations, and how can these be detected?
• RQ4 In what way does the unitary trick simplify the computation of unitary structures on representations?
• RQ5 How can results from Kazhdan-Lusztig theory and the Jantzen conjectures be combined to yield a complete algorithm for unitary representations?

Key findings

• The algorithm successfully computes irreducible unitary representations by tracking signature changes of Hermitian forms through deformation.
• The unitary trick enables a more tractable analysis by replacing the real Lie algebra action with a compact real form acting skew-adjointly.
• Signature changes at reducibility points are critical indicators of unitarity and are systematically counted in the algorithm.
• The Beilinson-Bernstein proof of the Jantzen conjectures provides the necessary completeness to determine the full unitary dual.
• The method achieves a complete classification of unitary representations by combining Langlands' framework with deep results from character theory.
• The algorithm is effective for all real reductive groups, providing a uniform computational approach grounded in representation theory