Cuspidal character sheaves on graded Lie algebras II
Wille Liu, Kari Vilonen, Ting Xue
TL;DR
This work delivers a complete classification of cyclic gradings $ rak g=igoplus_{imod m} rak g_i$ that support cuspidal character sheaves for graded simple Lie algebras, and it specifies the corresponding supports. Building on the nearby-cycle framework from LTVX, the authors reduce the problem to nil-supercuspidal data and bi-orbital cuspidal pairs, then treat classical types via a quiver description and multi-segment labelling, while handling exceptional types through Kac coordinates and affine-root data. The paper provides explicit gradings that admit cuspidal character sheaves, describes the supporting strata, and establishes uniqueness of support where applicable; it also yields a complete classification of bi-orbital cuspidal sheaves across classical and exceptional types, including central-character constraints and local-system counts. The results have significant implications for the geometry of affine Springer fibres, representations of double affine Hecke algebras, and the broader program of affine character sheaves. Overall, the work advances explicit geometric classification and paves the way for concrete connections to representation theory and related geometric structures.
Abstract
In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the explicit classification of cuspidal character sheaves for graded Lie algebras.