# Pieri Integral Formula and Asymptotics of Jack Unitary Characters

## TL;DR

This paper introduces Jack (unitary) characters, establishes integral formulas including a Pieri integral formula, and analyzes their asymptotic behavior as signatures grow large, extending known results in representation theory.

## Contribution

It develops new integral representations and a Pieri integral formula for Jack characters, facilitating their asymptotic analysis and connecting to spherical functions of Gelfand pairs.

## Key findings

- Established integral formulas for Jack characters
- Proved the Pieri integral formula as a dual to Pieri rule
- Demonstrated uniform limits of Jack characters in large signature limit

## Abstract

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, we study the asymptotics of Jack characters as the corresponding signatures grow to infinity in the sense of Vershik-Kerov. We prove the existence of a small $\delta > 0$ such that the Jack characters of $m$ variables have a uniform limit on the $\delta$-neighborhood of the $m$-dimensional torus. Our result specializes to a theorem of Okounkov and Olshanski.