Affine Springer Theory reading group

Spring 2022

Tuesdays, 15:00 - 16:30, JCMB 5323 (seminar room) + online.

Adam House, 5-6 Chambers street, Edinburgh

• Feb 1 Kostya Tolmachov Introduction. references. notes.
• Feb 15 Karim Réga Classical Springer theory. notes.
• Feb 22 Sarunas Kaubrys Affine Springer theory, definitions and examples. notes.
• Mar 15 Minh-Tâm Trinh Global Springer action. notes.
• Mar 22 David Jordan Finite, affine, double affine Hecke algebras. notes.
• Mar 29 Lucien Hennecart Equivariant (co)homology of affine Springer fibers. notes.
• Apr 5 Sasha Minets Representations of Cherednik algebras and affine Springer theory. notes.
• Apr 12 Kostya Tolmachov Hilbert schemes and Procesi bundles. notes.
• Apr 19 15:40 Iain Gordon Hilbert schemes and Cherednik algebras. notes.
• Apr 22 12:00 Oscar Kivinen Hilbert schemes and affine Springer fibers. notes. abstract   
  Starting from a conjugacy class in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses 1) Coulomb branch technology, as mathematically developed by Braverman-Finkelberg-Nakajima and many others 2) a version of affine Springer theory generalizing that developed by Garner-Kivinen and Hilburn-Kamnitzer-Weekes, as well as 3) a Z-algebra construction similar to the one used by Gordon-Stafford for rational Cherednik algebras. When G=GL_n the partial resolution we construct is the Hilbert scheme of n points on T*C*. For other groups, the partial resolution is an analog of the Hilbert scheme and there are many open questions regarding its geometry. We will explain some examples of the construction, recovering known sheaves such as the Procesi bundle, and conjecture the sheaves we get are coherent in general. Based on joint work with Gorsky and Oblomkov.