I attended today a talk of Dorian Goldfeld. In the talk, he mentioned that for a Maass cusp form $\phi$ of $\mathrm{SL}(n, \mathbb{Z})$, with Langlands parameter $\alpha = \left( \alpha_1, \dots, \alpha_n \right)$ (where $\alpha_1 + \dots + \alpha_n = 0 $), the Laplace eigenvalue $\lambda_\Delta$ is given by the formula
$$\lambda_\Delta = \frac{n^3 - n}{24} - \frac{\alpha_1^2 + \alpha_2^2 + \dots + \alpha_n^2}{2}.$$
See arXiv 2007.13268, page 10.
This formula reminded me of a PhD defense talk I went to two years ago about vertex algebras. In that talk, the Virasoro algebra was discussed. In the definition of the Virasoro algebra we have the relations $$ \begin{align} \left[L_n, L_m \right] &= \left(n - m\right)L_{m + n} + \frac{n^3 - n}{12}\cdot \delta_{n,-m}\cdot C, \\ \left[ L_n, C \right] &= 0. \end{align} $$
I was wondering if there is any relation between the two, as the expression $ \frac{n^3 - n}{12}$ appears in both (in the Laplace eigenvalue formula we divide it by $2$, but we also divide the sum of the squares of the Langlands parameter by $2$).
