A Note on the Asymptotic Expansion of Matrix Coefficients over $p$-adic Fields

TL;DR

This paper explores the asymptotic behavior of matrix coefficients over p-adic fields, connecting Casselman's theorem with explicit finite sum expansions, especially for general linear groups.

Contribution

It explicitly derives the asymptotic expansion of matrix coefficients for general linear groups over p-adic fields, building on Casselman's theorem.

Findings

Explicit expansion formulas for matrix coefficients of GL(n)

Connection between Casselman's theorem and finite sum representations

Enhanced understanding of asymptotic behavior in p-adic representation theory

Abstract

In this note, presented as a ``community service", followed by the PhD research of the author, we draw the relation between Casselman's theorem regarding the asymptotic behavior of matrix coefficients of reductive algebraic groups over $p$-adic fields and its expression as a finite sum of finite functions. In addition, we write the expansion explicitly for general linear groups.