Update: Videos for this lecture series and related lectures can be found here.
The course is one of several LMS-funded lecture series given by early career researchers and aimed at graduate students. The lectures will be given on Zoom and recorded. Netan Dogra's website has more information about related lectures and how to attend.
Course Description
In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (eg in the work of Romano--Thorne) to the local Langlands correspondence (eg in the work of Reeder--Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper "Vinberg's representations and arithmetic invariant theory" and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.
Lecture schedule
Friday, August 28, 2020, 1:30-2:30pm
Tuesday, September 1, 2020, 1:30-2:30pm
Friday, September 4, 2020, 1:30-2:30pm
Lecture notes
Lecture 1 Notes
Lecture 2 Notes
Lecture 3 Notes
Exercises
Lecture 1 Exercises
Lecture 2 Exercises
Lecture 3 Exercises
References
Background on Lie algebras
Course notes from my Cambridge Part III Lie algebras course (typeset by Eve Pound).
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer-Verlag New York-Berlin, 1978.
R. W. Carter, Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, 96, Cambridge University Press, Cambridge, 2005.
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991.
Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989.
Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, American Journal of Mathematics, Vol. 81, No. 4 (Oct., 1959), pp. 973-1032.
Polynomial invariants on Lie algebras
Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
Vinberg theory
E. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709. MR0430168 (55 #3175)
Dmitri I. Panyushev, On invariant theory of θ-groups. J. Algebra, 283(2):655–670, 2005.
Mark Reeder, Paul Levy, Jiu-Kang Yu, and Benedict H. Gross, Gradings of positive rank on simple Lie algebras, Transform. Groups 17 (2012), no. 4, 1123–1190. MR3000483
Slodowy slices
Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980.
Applications to arithmetic statistics
Jack A. Thorne, Vinberg’s representations and arithmetic invariant theory, Algebra Number Theory 7 (2013), no. 9, 2331–2368. MR3152016
Benedict H. Gross, On Bhargava’s representation and Vinberg’s invariant theory, Frontiers of mathematical sciences, 2011, pp. 317–321.
Jack A. Thorne. E6 and the arithmetic of a family of non-hyperelliptic curves of genus 3. Forum Math. Pi, 3:e1, 41, 2015.
Jack A. Thorne. Arithmetic invariant theory and 2-descent for plane quartic curves, Algebra Number Theory 10 (2016), no. 7, 1373–1413. With an appendix by Tasho Kaletha.
Beth Romano and Jack Thorne, E8 and the and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve. arXiv:1804.07702.
Jef Laga, The average size of the 2-Selmer group of a family ofnon-hyperelliptic curves of genus 3, arXiv: 2008.13158.
Applications to p-adic groups
Mark Reeder and Jiu-Kang Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27 (2014), no. 2, 437–477. MR3164986