Ramified automorphic induction and zero-dimensional affine Deligne-Lusztig varieties

TL;DR

This paper constructs extended affine Deligne-Lusztig varieties for reductive groups over local fields, generalizing previous work, and demonstrates their role in automorphic induction, especially for GL_2 with tamely ramified tori.

Contribution

It introduces a new family of varieties generalizing affine Deligne-Lusztig varieties to ramified tori and connects them to automorphic induction, with explicit results for GL_2.

Findings

Construction of extended affine Deligne-Lusztig varieties for ramified tori

Realization of automorphic induction for GL_2

Zero-dimensional, reduced varieties for tamely ramified tori

Abstract

To any connected reductive group G over a non-archimedean local field F (of characteristic p > 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For G = GL_2, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence are zero-dimensional and reduced, i.e., just disjoint unions of points.