Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Loseu, Ivan

Abstract

This dissertation explores various aspects of Springer fibers and their deep connections to geometric and combinatorial representation theory. It combines the works of the author during his PhD. Throughout these works, we explain sophisticated roles of Springer fibers in modern topics like modular representation theory and symplectic duality. Additionally, we resolve several classical questions concerning the Springer correspondence and the cohomology of Springer fibers. The first project centers around a discretization of a Springer fiber $\mathcal{B}_e$. This discretization is a finite set $Y_e$ that has appeared in various contexts in representation theory. A key theme of our result is that $Y_e$ also discretizes a distinguished fixed-point variety of $\mathcal{B}_e$. For certain families of Springer fibers, we describe $Y_e$ explicitly using full exceptional collections in the derived category of coherent sheaves on the fixed-point variety. This perspective provides a novel categorical model for the discretization. The second project studies the action of a finite group on the irreducible components of $\mathcal{B}_e$. We obtain an explicit classification of stabilizers in this action, proving a conjecture of Lusztig and Sommers. This suggests an unexplored connection between Springer fibers, component group actions, and Kazhdan–Lusztig cells in finite Weyl groups. The third project focuses on the Hikita conjecture for nilpotent orbits, which predicts a graded isomorphism between the cohomology of a Springer fiber and the ring of functions on the scheme-theoretic intersection of a nilpotent orbit closure with a Cartan subalgebra. We provide an almost complete classification of cases where this isomorphism holds by analyzing cohomological surjectivity and flatness conditions.

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