Explicit Formulas for the Casimir Eigenvalues of $SL(n,\mathbb{Z})$-Maass Forms

TL;DR

This paper derives explicit formulas for Casimir eigenvalues of Maass forms on $SL(n, ext{Z})$, generalizing known results for the Laplacian using a novel graph-theoretic approach.

Contribution

It provides explicit eigenvalue formulas for all Casimir operators of Maass forms on $SL(n, ext{Z})$, extending previous Laplacian results with a new graph-based proof.

Findings

Derived formulas for Casimir eigenvalues in terms of Langlands parameters.

Unified approach for all Casimir operators of order m.

Connected differential operators to graph partitions.

Abstract

Maass forms for $SL(n,\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\mathcal{D}_{m,n}$ of orders $1 \leq m \leq n$ for $GL(n,\mathbb{R})$. For any $1 \leq m \leq n$ and Maass form $\phi$ for $SL(n,\mathbb{Z})$, we provide a formula for the eigenvalue of $\mathcal{D}_{m,n}$ associated with $\phi$ in terms of the Langlands parameters of $\phi$. In the case $m=2$, we recover the formula for the Laplace eigenvalue of a Maass form due to Terras, the Casimir differential operator of order $2$ being the Laplacian. Our proof takes a graph-theoretic approach, relating the action of every elementary differential operator of order $m$ for $GL(n,\mathbb{R})$ to the partitions of a directed, edge-ordered graph with $m$ edges and at most $m$ vertices.