Counting parabolic principal G-bundles with nilpotent sections over P^1

Singh, Rahul (2022) Counting parabolic principal G-bundles with nilpotent sections over P^1. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit’s calculations is the case of $\mathbb{P}^1$ and two marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials. It is a natural question to generalize Mellit’s calculations to arbitrary reductive groups.

We consider the case of $\mathbb{P}^1$ with two marked points and an arbitrary split connected reductive group $G$ over $\mathbb{F}_q$. Firstly, we give an explicit formula for the number of $\mathbb{F}_q$-rational points of generalized Steinberg varieties of $G$. Secondly, for each principal $G$-bundle over $\mathbb{P}^1$, we give an explicit formula counting the number of triples consisting of parabolic structures at $0$ and $\infty$ and compatible nilpotent sections of the associated adjoint bundle.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Singh, Rahul ras292@pitt.edu ras292
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Fedorov, Roman fedorov@pitt.edu fedorov Committee Member Hales, Thomas hales@pitt.edu hales Committee Member Kaveh, Kiumars kaveh@pitt.edu kaveh Committee Member Schiffmann, Olivier olivier.schiffmann@math.u-psud.fr
Date:	20 July 2022
Date Type:	Publication
Defense Date:	6 April 2022
Approval Date:	20 July 2022
Submission Date:	7 April 2022
Access Restriction:	1 year -- Restrict access to University of Pittsburgh for a period of 1 year.
Number of Pages:	80
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	Keywords: Reductive group, principal $G$-bundle, parabolic structure, generalized Springer variety, generalized Steinberg variety, affine fibration, stratification, coproduct, symmetric function.
Date Deposited:	20 Jul 2022 21:44
Last Modified:	20 Jul 2022 21:44
URI:	http://d-scholarship.pitt.edu/id/eprint/42543

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