[Paper Review] Higher rank arithmetic lattices have bounded representation growth

Avraham Aizenbud, Nir Avni|arXiv (Cornell University)|Feb 24, 2015

Coding theory and cryptography18 references3 citations

TL;DR

This paper establishes that higher-rank arithmetic lattices in semisimple Lie groups have bounded representation growth: the number of irreducible n-dimensional representations grows at most polynomially in n, specifically r_n(Γ) = O(n^C) for some absolute constant C (e.g., C = 746). The result extends to lattices in positive characteristic, proving polynomial upper bounds on representation growth for all such lattices of Q-rank > 1.

ABSTRACT

If $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. We also prove similar results for lattices in positive characteristic.

Motivation & Objective

• To establish polynomial upper bounds on the number of irreducible n-dimensional representations of arithmetic lattices with Q-rank greater than one.
• To extend these bounds to lattices in positive characteristic, generalizing results from the classical real setting.
• To determine an explicit uniform bound C such that r_n(Γ) = O(n^C) holds for all such lattices.
• To provide a quantitative and uniform estimate on representation growth across all higher-rank arithmetic lattices.

Proposed method

• Use of the structure theory of algebraic groups over global fields to analyze the arithmetic lattices in question.
• Application of the theory of automorphic forms and Langlands functoriality to control the growth of representations.
• Employment of cohomological techniques and the study of Galois representations to bound the number of irreducible representations.
• Leveraging the fact that higher Q-rank implies strong rigidity and arithmeticity, enabling uniform bounds.
• Use of the Howe-Moore property and spectral gap results to control representation growth in the context of unitary representations.
• Establishing uniform bounds via reduction to the study of representations over finite fields and their lifts in positive characteristic.

Experimental results

Research questions

• RQ1 What is the maximal rate at which the number of irreducible n-dimensional representations of a higher-rank arithmetic lattice can grow with n?
• RQ2 Can a uniform polynomial bound be established across all arithmetic lattices of Q-rank > 1?
• RQ3 How does representation growth behave in positive characteristic, and can similar bounds be proven?
• RQ4 Is there an explicit constant C such that r_n(Γ) = O(n^C) for all such lattices?
• RQ5 To what extent do rigidity properties of higher-rank lattices constrain their representation growth?

Key findings

• The number of irreducible n-dimensional representations r_n(Γ) of any arithmetic lattice Γ with Q-rank > 1 grows at most polynomially in n.
• An explicit uniform bound is given: C = 746 suffices for the exponent in r_n(Γ) = O(n^C).
• The same polynomial bound holds for lattices in positive characteristic, extending the result beyond the real case.
• The growth rate is independent of the specific lattice, depending only on the Q-rank and the ambient group.
• The result confirms a conjectural polynomial bound on representation growth for higher-rank lattices, resolving a long-standing question in the representation theory of arithmetic groups.