Cocenters of p -adic Groups, III: Elliptic and Rigid Cocenters
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Cocenters of p‑adic Groups, III: Elliptic and Rigid Cocenters Dan Ciubotaru1 · Xuhua He2 Received: 2 April 2020 / Accepted: 18 July 2020 © Peking University 2020
Abstract In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive p-adic group is contained in the rigid cocenter. As applications, we prove the trace Paley–Wiener theorem and the abstract Selberg principle for mod-l representations. Keywords p-adic group · Cocenter · Trace Paley–Wiener theorem · Abstract Selberg principle Mathematics Subject Classification 22E50 · 20C08
1 Introduction 1.1 Let F be a nonarchimedean local field of residual characteristic p. Let 𝔾 be a connected reductive group over F and G = 𝔾(F) be the group of F-points. The study of smooth admissible representations of G is a major topic in representation theory. In particular, the representation theory over complex numbers is a part of the local Langlands program, and it has been a central area of research in modern representation theory. The G-representations over an algebraically closed field R of characteristic l ≠ p (in short, the mod-l representations) is also a natural object to study, and it has attracted considerable interest recently due to the applications to number theory, DC was partially supported by EPSRC EP/N033922/1. XH was partially supported by NSF DMS1463852 and DMS-1128155 (from IAS) * Xuhua He [email protected] Dan Ciubotaru [email protected] 1
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China
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e.g., congruences of the modular forms and the mod-l Langlands program. Several important progresses have been achieved in this direction, e.g., Vignéras [24, 26] and Vignéras-Waldspurger [27]. However, less is known compared to the complex representations. One of the major difficulties is that the proofs of many key results for complex representations rely heavily on harmonic analysis methods, which are not always available for mod-l representations. The main purpose of this paper (as well as of the previous papers [13, 14]) is to develop a new approach towards the (complex and mod-l) representation theory of G. The approach is based on the relation between the cocenter H̄ R = HR ∕[HR , HR ] of the Hecke algebra HR = H(G)R over the field R and the Grothendieck group ℜR (G) of representations over R via the trace map
Tr R ∶ H̄ R → ℜR (G)∗ . A detailed analysis on the cocenter side should lead to a deep understanding on the representation side. It is also worth mentioning that the study of the structure of the cocenter is influenced by, and relies on, certain recent developments in arithmetic geometry, in particular, the work on the theory of 𝜎-isocrystals and affine Deligne–Lusztig varieties. 1.2 We explain the main results of this paper. Let M be a standard Levi subgroup. On the side of represent
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