MATH720 (Spring 2022): Topics in Representation theory.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: MW 11.35-12.50pm, Location: DL 431. The first class is on 8/31, and the last class is on 12/7 (note that the 2nd class is on Friday, Sept 2).

Homeworks:

  • Homework 1, due Sep 26.
  • Homework 2, due Oct 11.
  • Homework 3, due Oct 31.
  • Homework 4, due Nov 17.
  • Homework 5, due Dec 16.

    The class discusses the geometry and representation theory associated to singular symplectic varieties as well as applications to Lie representation theory. Here is a preliminary program:

    0) Motivation: unitary representations of Lie groups and Orbit method. Quantum and classical mechanics. Express introduction to Poisson and Symplectic geometry. Deformation quantization.

    1) Orbits in semisimple Lie algebras. Their classification. sl_2-triples. Fundamental groups and equivariant covers.

    2) Singular symplectic varieties and symplectic resolutions. Motivations and key examples. Q-factorial terminalizations. Lusztig-Spaltenstein induction. Construction of Q-factorial terminalizations for affinizations of nilpotent orbits and their covers via Hamiltonian reduction. Deformations.

    3) Quantizations. Motivations from Representation theory. Classification of quantizations. Quantum Hamiltonian reduction. Quantizations of affinizations of covers of nilpotent orbits.

    4) Harish-Chandra modules and bimodules for semisimple Lie algebras. Harish-Chandra bimodules over quantizations. Classification of Harish-Chandra bimodules with full support. Time permitting, we'll also discuss Harish-Chandra modules.

    Schedule:

  • Aug 31, Lecture 1: Unitary representations; motivation from Quantum Mechanics; Orbit method; Poisson manifolds.
  • Sep 2, Lecture 2: Poisson manifolds, cont'd; Classical Mechanics; Hamiltonian actions and moment maps. [CdS] is a good reference for the material of the lecture, including the basics.
  • Sep 7, Lecture 3: Transitive Hamiltonian actions. Orbit method via Correspondence principle. Deformation quantization.
  • Sep 12, Lecture 4: Filtered and deformation (formal) quantizations, cont'd. Algebraic Orbit method.
  • Sep 14, Lecture 5: Semisimple orbits and Jordan decomposition. sl_2-triples.
  • Sep 19, Lecture 6: The structure of the centralizer. Nilpotent orbits in the classical Lie algebras and their equivariant covers.
  • Sep 21, Lecture 7: Finiteness of the number of nilpotent orbits; algebras of regular functions on nilpotent covers.
  • Sep 26, Lecture 8: Filtered Poisson deformations. Singular symplectic varieties. Spec(C[O]) is singular symplectic.
  • Sep 28, Lecture 9: Singularity Zoo. Intro to Invariant theory.
  • Oct 3, Lecture 10: Nilpotent cone. Categorical quotient for the adjoint G-action and filtered Poisson deformations.
  • Oct 5, Lecture 11: Filtered quantizations of C[N]. General classification result. Namikawa-Cartan space.
  • Oct 10, Lecture 12: Q-factorial terminalizations. Cohomology vanishing.
  • Oct 12, Lecture 13: Hamiltonian reduction. Induced varieties.
  • Oct 17, Lecture 14: Deformations of induced varieties. Induced covers..
  • Oct 24, Lecture 15: Properties of induced covers. Filtered Poisson deformations from induction.
  • Oct 26, Lecture 16: Q-factorial terminalizations from induced varieties, I.
  • Oct 31, Lecture 17: Q-factorial terminalizations from induced varieties, II.
  • Nov 2, Lecture 18: Wrapping-up Poisson deformations and Q-factorial terminalizations; twisted differential operators.
  • Nov 7, Lecture 19: TDO continued. Quantum Hamiltonian reduction. Addendum on injective rational representations.
  • Nov 9, Lecture 20: TDO vs quantum Hamiltonian reduction. Quantization of induced varieties.
  • Nov 14, Lecture 21: Quantizations of induced varieties, cont'd.
  • Nov 16, Lecture 22: Classification of quantizations. Addendum on quantizations of affine schemes.
  • Nov 28, Lecture 23: Recap and goals. Automorphisms and isomorphisms.
  • Nov 30, Lecture 24: Automorphisms and isomorphisms, continued.
  • Dec 5, Lecture 25: Automorphisms and isomorphisms, finished. Action on quantizations.
  • Dec 7, Lecture 26 (the last!): Automorphisms of quantizations.

    References:

    [Be]: A. Beauville, Symplectic singularities. Invent. Math. 139 (2000), no. 3, 541–549. arXiv.

    [BK]: R. Bezrukavnikov, D. Kaledin, Fedosov quantization in algebraic context. Mosc. Math. J. 4 (2004), no. 3, 559–592, 782.

    [BCHM]: C. Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468

    [B]: N. Bourbaki, Lie groups and Lie algebras. Chapters 1-3 (1998), 4-6 (2002), 7-9 (2005). Elements of Mathematics (Berlin). Springer-Verlag

    [BPW]: T. Braden, N. Proudfoot, B. Webster, Quantizations of conical symplectic resolutions I: local and global structure. Astérisque No. 384 (2016), 1–73.

    [CdS]: Ana Cannas da Silva, Lectures on Symplectic geometry. Available here.

    [CG]: N. Chriss, V. Ginzburg, Complex geometry and Representation theory. Birkhauser, 2010.

    [CM]: D. Collingwood, W. McGovern, Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold, 1993.

    [D]: I. Dolgachev, Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003.

    [E]: D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. GTM 150, Springer.

    [G]: V. Ginzburg, Lectures on D-modules, available here.

    [J]: J.C. Jantzen, Nilpotent orbits in Representation theory. In Progress in Mathematics 228.

    [K]: B. Kostant, Lie group representations on polynomial rings. Amer. J. Math. 85 (1963), 327–404.

    [KP1]: H. Kraft, C. Procesi, Closures of conjugacy classes of matrices are normal. Invent. Math. 53 (1979), no. 3, 227–247.

    [KP2]: H. Kraft, C. Procesi, On the geometry of conjugacy classes in classical groups. Comment. Math. Helv. 57 (1982), no. 4, 539–602.

    [L1]: I. Losev, Deformations of symplectic singularities and Orbit method for semisimple Lie algebras. Selecta Math, 28 (2022), N2, paper N30, 52 pages.

    [L2]: I. Losev, Harish-Chandra bimodules over quantized symplectic singularities. Transform. Groups 26 (2021), no. 2, 565–600.

    [L3]: I. Losev, Derived equivalences for symplectic reflection algebras. Int. Math. Res. Not. IMRN 2021, no. 1, 444–474.

    [L4]: I. Losev, Isomorphisms of quantizations via quantization of resolutions. Adv. Math. 231(2012), 1216-1270.

    [N2a] Y. Namikawa, Poisson deformations of affine symplectic varieties. Duke Math. J. 156 (2011), no. 1, 51–85.

    [N2b] Y. Namikawa, Poisson deformations of affine symplectic varieties, II. Kyoto J. Math. 50 (2010), no. 4, 727–752.

    [LMBM]: I. Losev, L. Mason-Brown, D. Matvieievskyi, Unipotent Ideals and Harish-Chandra Bimodules. arXiv:2108.03453.

    [OV]: A. Onishchik, E. Vinberg, Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990.

    [PV]: V. Popov, E. Vinberg, Invariant theory in Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences (EMS, volume 55).

    [S]: R. Stanley, Invariants of finite groups and their applications to Combinatorics, available here.