Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II
Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II
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除法代数的 Deligne–Lusztig 构造和局部 Langlands 对应，II
DOI：
10.1016/j.aim.2016.02.021
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
C. Chan
中科院分区：
文献类型：
--
作者：
C. Chan
在1979年，Lusztig提出了一个约化p-adic群的超尖点表示的上同调构造，类似于有限约化群的Deligne-Lusztig理论。本文建立了Lusztig程序的一个新实例。精确地说，设D是局部非阿基米德正特征域K上的四元数代数，X是与D×相关联的p-adic Deligne-Lusztig ind-scheme. K的非分歧二次扩张（的乘法群）的拟特征标与θ <$Hi（X）[θ]给出的D×的表示之间存在着自然的对应.我们表明，这种对应关系是一个双射（域和目标的温和限制后），并匹配的双射局部Langlands和Jacquet-Langlands。
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 D be the quaternion algebra over a local non-Archimedean field K of positive characteristic, and let X be the p-adic Deligne–Lusztig ind-scheme associated to D×. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified quadratic extension of K and representations of D× given by θ↦ H i (X)[θ]. We show that this correspondence is a bijection (after a mild restriction of the domain and target), and matches the bijection given by local Langlands and Jacquet–Langlands.