Cuspidal character sheaves on graded Lie algebras
Wille Liu, Cheng-Chiang Tsai, Kari Vilonen, Ting Xue
TL;DR
The paper develops a uniform framework for cuspidal character sheaves on graded Lie algebras, showing that all such sheaves arise from nearby-cycle data followed by a Fourier--Sato transform. Central to the construction are nil-supercuspidal data $(\mathrm{c},\chi)$, the associated nearby-cycle sheaves $\mathcal{P}_{\chi}$, and the adjunction formula linking parabolic restriction with dagger-transformed objects; combined with hyperbolic and spiral restriction, this yields a Springer-type correspondence classifying cuspidals via endomorphism algebras. The main theorem asserts that every simple subsheaf or quotient of $\mathcal{P}_{\chi}^{\dag}$ is cuspidal, and every cuspidal character sheaf on $\mathfrak{g}_1$ arises in this way; it also proves structural results about the supports (distinguished strata) and provides a pathway to a block decomposition in terms of Lusztig--Yun data. Applications connect these geometric objects to homogeneous affine Springer fibres and degenerate double affine Hecke algebras, giving a correspondence between finite-dimensional DAHA modules and cuspidal character sheaves, and confirming conjectures in the Vinberg-type I setting. The work thus advances a generalized Springer theory for graded Lie algebras and yields classification results that tie representation-theoretic objects to geometric and topological invariants of graded components.
Abstract
We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier--Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the classification of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras.