2024/10/24 by Gurbir Dhillon, Dhillon, Gurbir, Joakim Faergeman +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.18360
openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a reductive group G, we introduce a notion of singular support for cocomplete dualizable DG-categories equipped with a strong G-action. This is done by considering the singular support of the sheaves of matrix coefficients arising from the action. We focus particularly on dualizable G-categories whose singular support lies in the nilpotent cone of \mathfrakg^* and refer to these as nilpotent G-categories. For such categories, we give a characterization of the singular support in terms of the vanishing of its generalized Whittaker models. We study parabolic induction and restriction functors of nilpotent G-categories and show that they interact with singular support in a desired way. We prove that if an orbit is maximal in the singular support of a nilpotent G-category C, the Hochschild homology of the generalized Whittaker model of C coincides with the microstalk of the character sheaf of C at that orbit. This should be considered a categorified analogue of a result of Moeglin-Waldspurger that the dimension of the generalized Whittaker model of a smooth admissible representation of a reductive group over a non-Archimedean local field of characteristic zero coincides with the Fourier coefficient in the wave-front set of that orbit. As a consequence, we give another proof of a theorem of Bezrukavnikov-Losev, classifying finite-dimensional modules for W-algebras with fixed regular central character. More precisely, we realize the (rationalized) Grothendieck group of this category as a certain subrepresentation of the Springer representation. Along the way, we show that the Springer action of the Weyl group on the twisted Grothendieck--Springer sheaves is the categorical trace of the wall crossing functors, extending an observation of Zhu for integral central characters.