Here is the webpage for my introductory course to geometric representation theory at YMSC in 2022 fall.

Lecture 1 Plan and Grothendieck/Springer resolution in Type A

Lecture 2 Grothendieck/Springer resolution in general and some symplectic geometry

Lecture 3 Steinberg variety and Borel--Moore homology

Lecture 4 Convolution in Borel--Moore homology

Lecture 5 Springer theory for the Weyl groups

Lecture 6 Springer theory continued and constructible sheaves

Lecture 7 Constructible sheaves and revisiting BM homology

Lecture 8 t-structures

Lecture 9 Perverse t-structure

Lecture 10 minimal extensions

Lecture 11 Decomposition theorem

Lecture 12 Sheaf-theoretic analysis of convolution algebras

Lecture 13 Revisiting the Springer theory

Lecture 14 Springer theory for U(sl_n): statement of the results

Lecture 15 Springer theory for U(sl_n): example of sl_2 and checking relations

Lecture 16 sheaf-theoretic approach, Howe duality, and Schur--Weyl duality

Lecture 17 Equivariant K-theory I: basic constructions

Lecture 18 Equivariant K-theory II: Koszul complex and Beilinson resolution

Lecture 19 Equivariant K-theory III: localiztion

Lecture 20 Equivariant K-theory IV: flag variety

Lecture 21 Deligne--Langlands conjecture, affine Hecke algebra and Equivariant K-theory

Lecture 22 Proof of the Kazhdan--Lusztig, Ginzburg isomorphism

Lecture 23 Classification of irreps of the affine Hecke algebra

Lecture 24 Proof of the classification

References:

• Chriss--Ginzburg, Representation theory and complex geometry

• Ginzburg, Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups

• Nakajiama's course notes

• Braverman--Gaitsgory, On Ginzburg's Lagrangian construction of representations of GL(n)

• Wang, Lagrangian construction of the (gln,glm)-duality

• Hotta--Takeuchi--Tanisaki, D-Modules, Perverse Sheaves, and Representation Theory

• Okounkov, Lectures on K-theoretic computations in enumerative geometry

• Williamson's course notes Langlands correspondence and Bezrukavnikov's equivalence