Mass of the Frobenius functional on small balls

TL;DR

This paper demonstrates that for certain automorphic representations associated with co-compact lattices in semi-simple groups, the Frobenius functional assigns minimal mass to small neighborhoods in the orbit space, advancing understanding of automorphic forms.

Contribution

It establishes a new result linking the Frobenius functional's behavior to small neighborhoods in the orbit method for specific automorphic representations.

Findings

Frobenius functional assigns small mass to small balls in the orbit space.

Results apply to co-compact lattices in semi-simple quasi-split real groups.

Advances understanding of automorphic representations and their distribution.

Abstract

We show that for a co-compact lattice in a semi-simple quasi-split real group and a fixed abstractly generic automorphic representation, Frobenius functional assigns small mass to small balls in the Orbit Method picture of P. Nelson and A. Venkatesh (Acta Math., 226, (2021), 1-209).