Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II
| Autor: | Charlotte Chan |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Degree (graph theory) Multiplicative group General Mathematics 010102 general mathematics General Physics and Astronomy Unipotent 01 natural sciences Transfer (group theory) Scheme (mathematics) 0103 physical sciences Bijection Division algebra 010307 mathematical physics 0101 mathematics Mathematics::Representation Theory Realization (systems) Mathematics |
| Zdroj: | Selecta Mathematica. 24:3175-3216 |
| ISSN: | 1420-9020 1022-1824 |
| Popis: | In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the $$D^\times $$ -representations $$H_\bullet (X)$$ by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of $$D^\times $$ . There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of $$D^{\times }$$ given by $$\theta \mapsto H_\bullet (X)[\theta ]$$ . For a broad class of characters $$\theta ,$$ we show that the representation $$H_\bullet (X)[\theta ]$$ is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from $$\theta $$ using $$\chi $$ -data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full text from SpringerLink