Ordinary $GL_2(F)$-representations in characteristic two via affine Deligne-Lusztig constructions

TL;DR

This paper constructs and compares supercuspidal representations of _2 over characteristic two fields using affine Deligne-Lusztig varieties, revealing new geometric methods for understanding these representations.

Contribution

It introduces a geometric construction of supercuspidal _2 representations via affine Deligne-Lusztig varieties and compares this with existing type-theoretic approaches.

Findings

Constructed many supercuspidal representations in cohomology.

Affine Deligne-Lusztig varieties are zero-dimensional in this setting.

Established a correspondence between geometric and type-theoretic constructions.

Abstract

The group $\GL_2$ over a local field with (residue) characteristic $2$ possesses much more smooth supercuspidal $\ell$-adic representations, than over a local field of residue characteristic $> 2$. One way to construct these representations is via the theory of types of Bushnell-Kutzko. We construct many of them in the cohomology of certain extended affine Deligne-Lusztig varieties attached to $\GL_2$ and wildly ramified maximal tori in it. Then we compare our construction with the type-theoretic one. The corresponding extended affine Deligne-Lusztig varieties were introduced in a preceding article. Also in the present case they turn out to be zero-dimensional.