AI 中文总结
本文针对线性实约化群或元辛群，基于相关循环对特殊幂幺表示进行几何分类，通过变形量子化给出其哈里什 - 钱德拉模的统一构造，证明相关模有不可约循环且酉可化，突出卢茨蒂格猜想应用。
AI 中文摘要
对于任意线性实约化群或元辛群，本文依据相关循环，对与朗兰兹或元辛对偶李代数中拟特殊幂零轨道相关的弱亚瑟/亚当斯 - 巴尔巴什 - 沃根（ABV）包中的特殊幂幺表示进行几何分类。通过对特定拉格朗日子簇上允许向量丛的变形量子化，给出这些表示的哈里什 - 钱德拉模的统一构造，与基里洛夫、科斯坦特和沃根的余伴随轨道方法理念一致。结果表明，所有此类哈里什 - 钱德拉模都有不可约相关循环，且根据混合霍奇模理论结果是酉可化的。推测所有ABV包的酉性可归结为对偶轨道为特殊的情形。我们的方法突出了卢茨蒂格关于特殊片几何的猜想的应用，该猜想已被克拉夫特和普罗塞西证明对经典李代数成立，朱托、利维、索默斯和作者证明对所有情形成立。
英文摘要
For any linear real reductive group or metapletic group, this article gives a geometric classification of special unipotent representations in the weak Arthur/Adams-Barbasch-Vogan (ABV) packets attached to quasi-distinguished nilpotent orbits in the Langlands or metaplectic dual Lie algebra in terms of their associated cycles. We provide a uniform construction of the Harish-Chandra modules of these representations via deformation quantization of admissible vector bundles over certain Lagrangian subvarieties of the affinizations of the universal covers of special nilpotent orbits in question, which aligns with the coadjoint orbit method philosophy of Kirillov, Kostant, and Vogan. As consequences, all such Harish-Chandra modules have irreducible associated cycles, and are unitarizable by results from the theory of mixed Hodge modules. Conjecturally, the unitarity of all ABV packets can be reduced to the case when the dual orbits are distinguished. Our approach highlights the application of Lusztig's conjecture on the geometry of special pieces, which has been proven for classical Lie algebras by Kraft and Procesi, and for all cases by Juteau, Levy, Sommers and the author.
Comments83 pages, 100% organic man-made proofs
