Course on Geometric Satake isomorphism, spring 2014

Last modified by tixue@helsinki_fi on 2024/03/27 10:20

Course on Geometric Satake isomorphism, spring 2014


Xue Ting


5 sp.


Advanced studies


Basic algebra. 


This course aims to give an introduction to the geometric Satake isomorphism, which

is a first instance of geometric Langlands duality. The tentative topics of the course will

include the following: 

  • Basic introduction to affine algebraic groups, root datum, Borel subgroups, flag variety
  • Structure properties of reductive algebraic groups
  • Tannakian formalism, recovering a group from its representations 
  • Introduction to derived categories, perverse sheaves 
  • Affine Grassmannian 
  • Geometric Satake isomorphism

The topics might vary depending on the interest of the audience. 



Weeks 3-9, Tuesday 12-14 and Friday 14-16 in room B321


Weekly homework

Exercise Set 1 ES1.pdf

Exercise Set 2 ES2.pdf

Exercise Set 3 ES3.pdf

Exercise Set 4 ES4.pdf


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

Borel, A. Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991.

Humphreys, J. E. Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.

Springer, T. A. Linear algebraic groups. Second edition. Progress in Mathematics, 9. Birkhäuser Boston, Inc., Boston, MA, 1998.

Milne, J.S. Basic theory of affine group schemes.

Conrad, B. Reductive group schemes.

Waterhouse, William C. Introduction to affine group schemes. Graduate Texts in Mathematics, 66. Springer-Verlag, New York-Berlin, 1979.

Deligne, P; Milne, J.S.; Ogus, A.; Shih, KHodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982.


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