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Arithmetic wavefront sets and generic L-packets
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DOI：10.1016/j.aim.2025.110668.png)
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Abstract
En 中文
Let G be a classical group defifined over a local field Fof characteristic zero. Let pi be an irreducible admissible representation pi of G(F), which is of Casselman-Wallach type if Fis archimedean. If pi has a generic local L-parameter, we define the arithmetic wavefront set WFari(pi) of pi, which is a subset of F-rational nilpotent orbits in the Lie algebra g(F) under adjoint action of G(F), by means of the arithmetic structures of the enhanced L-parameter (phi, chi) of pi. Those arithmetic structures are discovered by using our method of consecutive descents of enhanced L-parameters, based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. We study the basic structure of WFari(pi) and prove that it is an invariant of pi(Theorem 5.10). Furthermore, those basic structures WFari(pi)(max) are expected to yield the precise F-rational structure of WFari(pi)(max), which has been realized, when Fis archimedean, in Theorems 1.3and 1.4( Theorems 7.10, 7.14, and 9.2). Based on the local Langlands reciprocity, the Wavefront Set Conjecture (Conjecture 1.2and Conjecture 5.14) asserts that the wavefront sets on the L-parameter side should be closed related to those on the representation side, namely, WFwm(pi)(max) = WFari(pi)(max) = WFtr(pi)(max) when pi has a generic local L-parameter, where the algebraic wavefront set WFwm(pi) is defined by C. Moeglin and J.-L. Waldspurger in [88], using the generalized Whittaker models and the analytic wavefront set WFtr(pi) is defined by R. Howe in [53,57] via the wavefront set of the distribution character Theta(pi) of pi, and also by the asymptotic expansion of the distribution character Theta(pi) of pi near the identity ([9,48,56]). Conjecture 1.2is verified for families of interesting cases and provides a means to determine WFwm(pi)(max) and WFtr(pi)(max) in terms of the computationally more accessible WFari(pi)(max). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Keywords:
Classical groups over local fields
Admissible representations and Casselman-Wallach representations
Rationality of local Langlands correspondence
Local Gan-Gross-Prasad conjecture
Nilpotent orbits
Wavefront sets
Journal
A
IF:
1.5
Papers:
329
Citations:
0