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:
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:
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.