Random matrix theory, the exceptional Lie groups, and L-functions

TL;DR

This paper explores the connection between random matrix theory and number-theoretic L-functions by extending known relationships from classical groups to exceptional Lie groups, specifically G_2, and analyzing their characteristic polynomial distributions.

Contribution

It extends the link between random matrix theory and L-functions to exceptional Lie groups, providing explicit calculations for G_2 and relating these to finite field L-functions.

Findings

Characteristic polynomial distributions for G_2 are computed using Macdonald identities.

A family of L-functions over finite fields is shown to have similar value distributions to G_2 characteristic polynomials.

The methods extend to all exceptional Lie groups, broadening the scope of the theory.

Abstract

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groups