Limit multiplicities for ${\rm SL}_2(\mathcal{O}_F)$ in ${\rm SL}_2(\mathbb{R}^{r_1}\oplus\mathbb{C}^{r_2})$

TL;DR

This paper establishes that for a fixed archimedean signature, the family of lattices formed by ${\rm SL}_2(\mathcal{O}_F)$ over number fields exhibits the limit multiplicity property in the corresponding Lie group, advancing understanding of their spectral distribution.

Contribution

It proves the limit multiplicity property for families of lattices ${\rm SL}_2(\mathcal{O}_F)$ with fixed archimedean signature, a significant step in automorphic forms and representation theory.

Findings

Lattices ${\rm SL}_2(\mathcal{O}_F)$ have the limit multiplicity property.

The result holds uniformly over number fields with fixed archimedean signature.

Advances the understanding of spectral distribution of these lattices.

Abstract

We prove that the family of lattices ${\rm SL}_2(\mathcal{O}_F)$, $F$ running over number fields with fixed archimedean signature $(r_1, r_2)$, in ${\rm SL}_2(\mathbb{R}^{r_1}\oplus\mathbb{C}^{r_2})$ has the limit multiplicity property.