## Affine Springer fibers and the representation theory of small quantum groups and related algebras

##### Author(s)

Boixeda Alvarez, Pablo.
Download1191254281-MIT.pdf (770.1Kb)

##### Other Contributors

Massachusetts Institute of Technology. Department of Mathematics.

##### Advisor

Roman Bezrukavnikov.

##### Terms of use

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Show full item record##### Abstract

This thesis deals with the connections of Geometry and Representation Theory. In particular we study the representation theory of small quantum groups and Frobenius kernels and the geometry of an equivalued affine Springer fiber Fl[subscript ts] for s a regular semisimple element. In Chapter 2 we relate the center of the small quantum group with the cohomology of the above affine Springer fiber. This includes joint work with Bezrukavnikov, Shan and Vaserot. In Chapter 3 we study the geometry of the affine Springer fiber and in particular understand the fixed points of a torus action contained in each component. In Chapter 4 we further have a collection of algebraic results on the representation theory of Frobenius kernels. In particular we state some results pointing towards some construction of certain partial Verma functors and we compute this in the case of SL₂. We also compute the center of Frobenius kernels in the case of SL₂ and state a conjecture on a possible inductive construction of the general center.

##### Description

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020 Cataloged from the official PDF of thesis. Includes bibliographical references (pages 125-128).

##### Date issued

2020##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.