Intégrale orbitale pondérée via l'induite de Lusztig-Spaltenstein généralisée
Yan-Der Lu
TL;DR
The work develops a Lie-algebraic construction of weighted orbital integrals for inner forms of GL by leveraging generalized Lusztig-Spaltenstein induction. It introduces a Lie-algebra version of Richardson and Lusztig-Spaltenstein parabolics, defines weight systems via (G,M)-families, and proves that the resulting local integrals are tempered distributions, with rigorous normalization and measure considerations. It further provides a direct Lie-algebraic alternative to Arthur’s descent-based approach, proves the tempered-distribution property, and establishes equivalence with Arthur’s definitions in elliptic/semi-simple contexts, including semi-local extensions. The results illuminate the analytic behavior of orbital integrals, enable potential Fourier-analytic extensions, and offer practical tools for trace-formula computations in the GL setting, broadening applicability beyond p-adic fields to general local fields of characteristic zero.
Abstract
In this article, we present two novel approaches to constructing weighted orbital integrals of an inner form of a general linear group. Our method utilizes generalized Lustig-Spaltenstein induction. Furthermore, we will prove that a weighted orbital integral on the Lie algebra constitutes a tempered distribution. We also demonstrate that our new definitions and Arthur's original definition are consistent.