Cuspidal $\ell$-modular representations of $p$-adic classical groups

TL;DR

This paper constructs all cuspidal representations of p-adic classical groups over fields of characteristic not p, advancing classification and understanding of their induced representations.

Contribution

It provides a complete construction of cuspidal representations via types and advances the classification by identifying when types induce equivalent representations.

Findings

Constructed all cuspidal representations from types.

Identified when cuspidal types induce equivalent representations.

Proved induced representations from general types are quasi-projective.

Abstract

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.