R440, Astronomy-Mathematics Building, NTU
Speaker(s):
Wille Shih-Wei Liu (Academia Sinica)
Organizer(s):
Chun-Ju Lai (Institute of Mathematics, Academia Sinica)
1. Course Background & Purposes
The double affine Hecke algebras (DAHA) are introduced in the ‘90s by Cherednik in his study of the integration of the so-called affine Knizhnik—Zamolodchikov (KZ) equations and their relation with Macdonald’s theory of symmetric polynomials. The main objective of this course is to discuss several topics in the representation theory of the trigonometric degeneration of the DAHA, called tDAHA.
The tDAHA possesses several faces that allow one to tackle it from different aspects, among which:
1. Analytic aspect: the tDAHA can be realised as the ring generated by the Dunkl differential-difference operators on polynomial functions on a symmetric space. This yields the KZ functor via the monodromy representation on the symmetric space.
2. Noncommutative-geometric aspect: the quotient of the cotangent bundle of the symmetric space is a Poisson variety, symplectic on the smooth locus. One can realise, via microlocalisation in some good cases, the tDAHA as quantisation of symplectic resolution of this Poisson variety.
3. Springer-theoretic aspect: the realisation of the tDAHA as cohomological convolution algebra of the affine Springer resolution and the equivariant localisation theorem allows one to employ sheaf-theoretic tools (such as six operations and perverse sheaves) . This yields geometric parametrisation of simple modules of the tDAHA.
4. Algebraic aspect: one can make abstraction of the cohomological calculus in pure algebraic terms: the divided difference calculus. This abstraction gives more freedom and uniformity in various constructions related to the representation theory of the tDAHA.
2. Course Outline & Descriptions
We hope to discuss the following topics in this course:
1. Definition of the tDAHA : as differential-difference operators or as divided difference operators.
2. Integrable modules of the tDAHA and their relation to the integration of KZ equations.
3. Finite-dimensional modules of the tDAHA.
4. Springer theory of the tDAHA: cohomological construction and geometric parametrisation of simple modules.
5. Divided difference calculus: completion of the tDAHA and the algebraic KZ functor.
6. Construction of translation functors via the divided difference calculus.
Contact:
murphyyu@ncts.ntu.edu.tw
