Newforms in Cuspidal Representations

TL;DR

This paper extends the theory of newform vectors in cuspidal representations of p-adic groups to include ll-modular cases, providing explicit formulas and demonstrating conductor compatibility with congruences.

Contribution

It generalizes the theory of newforms to ll-modular representations and establishes explicit formulas for certain cuspidal representations.

Findings

Conductor is compatible with congruences modulo ll for supercuspidal ll-modular representations.

Explicit formulas are derived for depth zero and minimax cuspidal representations.

The theory is extended from complex to ll-modular settings, including ramified cases.

Abstract

We consider newform vectors in cuspidal representations of $p$-adic general linear groups. We extend the theory from the complex setting to include~$\ell$-modular representations with~$\ell\neq p$, and prove that the conductor is compatible with congruences modulo~$\ell$ for (ramified) supercuspidal~$\ell$-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.