Summer School Courses

Zhiwei Yun 恽之玮 (MIT)

Hitchin moduli spaces and ramified geometric Langlands (week 1)

I will start with a general introduction of Hitchin moduli space and its role in geometric representation theory. Then I will focus on the specific Hitchin moduli spaces for Higgs bundles on $P^{1}$ with level structures that were introduced and studied in joint work with Bezrukavnikov, Boixeda-Alvarez and McBreen. I will explain their role in formulating special cases of the ramified geometric Langlands conjecture, and give evidence for the conjecture. (recommended reading)

Copilot: Zeyu Wang 王泽宇

Z. Yun, Lectures on Springer theories and orbital integrals

P. Etingof, H. Liu, Hitchin Systems and their quantization (+ video )

Tsao-Hsien Chen 陈朝铣 (U. of Minnesota)

Langlands Duality and Symmetric Varieties (week 1)

In this course I will discuss recent progress on Langlands duality for symmetric varieties. I will review the structure theory for symmetric varieties $X = G / K$ for a complex connected reductive group $G$ and explain combinatorial constructions of dual groups $G_{X}$ of symmetric varieties. Then I will explain the work of Gaitsgory-Nadler on relative geometric Satake equivalence which provides a geometric construction $G_{X}$ using perverse sheaves on the loop space of $X$ . Finally, I will discuss the derived version of the story and its connections to Coulomb branches and geometric Langlands for real groups. (recommended reading)

Copilots: Ruotao Yang 杨若涛 and Lingfei Yi 易灵飞

Symmetric varieties and real groups:

A. Onishchik, E. Vinberg, Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, Springer, 1994.

S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, Journal of Mathematics Osaka City University 13(1); 1—34

T. Springer, The classification of involutions of simple algebraic groups, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 1987.

S. Takeda, On dual groups of symmetric varieties and distinguished representations of p-adic groups

F. Knop, B. Schalke, The dual group of a spherical variety

Langlands duality for reductive groups:

I. Mirkovic, K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95—143.

X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence

P. Baumann, S. Riche, Notes on the geometric Satake equivalence

Langlands duality for symmetric varieties, real groups, and spherical varieties:

D. Gaitsgory, D. Nadler, Spherical varieties and Langlands duality, Moscow Math. J. 10 (2010), no. 1, 65—137.

D. Nadler, Perverse sheaves on real loop Grassmannians, Inventiones Math. 159 (2005), 1—73.

T.-H. Chen, D. Nadler, Real groups, symmetric varieties and Langlands duality

T.-H. Chen, L. Yi, Singularities of orbit closures in loop spaces of symmetric varieties

D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh, Relative Langlands duality

R. Travkin, R. Yang, Twisted Gaiotto equivalence for $G L (M | N)$

Sam Raskin (Yale)

Geometric Langlands for projective curves (week 2)

The geometric Langlands equivalence asserts that (suitable) sheaves on the space of $G$ -bundles on $X$ and on the space of $\overset{ˇ}{G}$ -local systems are equivalent categories. I will provide some background on the conjecture(s) and highlight some aspects of its proof. (recommended reading)

Copilot: Lin Chen 陈麟

E. Frenkel, Recent advances in the Langlands program

D. Gaitsgory, N. Rozenblyum A study in derived algebraic geometry, Part I

D. Gaitsgory, Outline of the proof of geometric Langlands conjecture for $G L_{2}$

P1

Roman Bezrukavnikov (MIT)

Some applications of affine Springer theory (week 2)

Springer fibers are subvarieties in a flag variety of a reductive group, playing an important role in some questions of representation theory. Their loop group counterparts known as affine Springer fibers are more complicated geometrically, they turn out to be connected to problems about characters of $p$ -adic groups, to quantum groups etc. I will survey some of these connections. (recommended reading)

Copilots: Henry Liu 刘华昕 and Michael McBreen

Z. Yun, Lectures on Springer theories and orbital integrals

X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence

P1