Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
This dissertation covers two mostly independent topics motivated by the theory of character sheaves on reductive algebraic groups.
The first topic is a generalization of Mirković’s theory of character sheaves on reductive Lie algebras. Due to their simple geometric characterization, character sheaves on Lie algebras can be thought of as a simplified model for Lusztig’s theory of character sheaves on algebraic groups. We extend the theory to the case of sheaves with positive characteristic coefficients. Along the way, we give a characterization of these character sheaves in terms of Fourier transforms of orbital sheaves. There is another classical description of character sheaves in terms of admissible sheaves. We prove that this description admits a modular analogue and gives rise to connections between modular character sheaves and the modular generalized Springer correspondence of Achar–Henderson–Juteau–Riche.
The second topic is a study of three different categorifications of the monodromic Hecke algebra. The first categorification is geometric and defined in terms of parity sheaves. The second categorification is algebraic and a generalization of Abe’s incarnation of Soergel bimodules. The last categorification is diagrammatic and a generalization of the Elias– Williamson diagrammatic calculus. We will further compare these categorifications with one another. When well-defined, we will show that all three categorifications are actually equivalent– generalizing results of Abe and Riche–Williamson. Finally, we establish a modular analogue of Lusztig and Yun’s monodromic-endoscopic equivalence which roughly states that these categorifications can be described in terms of categorifications of unipotent Hecke algebras associated to endoscopic groups.
Date
5-14-2026
Recommended Citation
Sandvik, Colton C., "Modular Character Sheaves on Lie Algebras and Monodromic Hecke Categories" (2026). LSU Doctoral Dissertations. 7084.
https://repository.lsu.edu/gradschool_dissertations/7084
Committee Chair
Achar, Pramod N.
LSU Acknowledgement
1
LSU Accessibility Acknowledgment
1