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

Maass forms for $SL(n,\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators ${\mathcal{D}}_{m,n}$ of orders $1\le m\le n$ for $GL(n,\mathbb{R})$. For any $1\le m\le n$ 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.