Summer School Courses

Course 1

lecturer:Daniel Juteau

Title:Representations of rational Cherednik algebras


Rational Cherednik algebras were introduced by Etingof and Ginzburg in their landmark 2002 paper on symplectic reflection algebras. They are (double) degenerations of the double affine Hecke algebras (DAHAs) introduced by Cherednik in the 1990's to solve the Macdonald conjectures. While DAHAs make sense only for Weyl groups, the rational Cherednik algebras do make sense for finite Coxeter groups, and even for complex reflection group: those are generalizations of finite Coxeter groups to the complex case, with reflections allowed to have any order, rather than just order 2. Given a complex reflection group W acting on a space V, the associated rational Cherednik algebra H_c(W, V) can be defined by a faithful representation on the polynomial algebra $\mathbb{C}[V] = S(V^*)$: it is generated by polynomials acting by multiplication, the group algebra of W, and the Dunkl operators, which are a deformation of the differential operators depending on some parameters c_r (one for each conjugacy classes of reflections in W). We will review those definitions, and the key feature of Dunkl operators: they commute. We have a Poincaré-Birkhoff-Witt theorem, and hence a triangular decomposition of the algebra, analogous to that of the enveloping algebra of a semisimple Lie algebra.

Pursuing this analogy, we have a category O, which is a highest weight category, with standard modules parametrized by the irreducible representations of W. Moreover, we have an exact functor (the Khnizhnik-Zamolodchikov functor, introduced by Ginzburg-Guay-Opdam-Rouquier), from that category to the category of modules for the finite Hecke algebra; this is an analogue of Soergel's functor $\mathbb V$ in the theory of semisimple Lie algebras. I will also explain the Bezrukavnikov-Etingof induction and restriction functors and their properties. Finally, I will discuss some aspects of one of the main problems, namely the determination of the supports of simple modules, and in particular the determination of the parameters for which simple modules become finite dimensional. For example I will discuss Etingof's determination of the support of the spherical module (the unique simple quotient of the polynomial representation).

Course 2

lecturer:Weiqiang Wang

Title:Quantum symmetric pairs and applications


Course 3

lecturer:Ben Webster

Title:Categorification in representation theory: Heisenberg and Kac-Moody


This course will be an introduction to using categorifications of Lie algebras, especially sl_n and its affine generalization, in studying representation theory. Our starting point will be the notion of a categorical Heisenberg action of arbitrary level. This is, not so surprisingly, an action of a monoidal category which induces an action of the usual Heisenberg algebra on the level of Grothendieck groups.

These are easy to define and give examples of: representations of symmetric groups, general linear Lie (super)algebras and Cherednik algebras all give examples.

Heisenberg categorifications have a powerful internal structure, however: based on work of Vershik-Okounkov and Brundan-Kleshchev, one can show that they induce not just an action of the Heisenberg algebra, but of an affine Lie algebra. This will lead us to the definition of the categorified universal enveloping algebra of Khovanov, Lauda and Rouquier, which in turn supplies new insight into the representation theory of all the examples discussed above.

Notes: Heisenberg-ECNU-1