Nicolas Guay (University of Amsterdam),
Rational Cherednik algebras
Charles Dunkl (University of Virginia, Charlottesville, VA, USA),
Singular polynomials and modules for the symmetric groups
Gail Letzter (Virginia Tech, Blacksburg, VA, USA),
Invariant differential operators for quantum symmetric spaces
afternoon lectures (14.30-17.00 hour): room P.018, Euclides building, Plantage Muidergracht 24
Abstract of lecture by Nicolas Guay
I will give an overview of some of the developments in the theory of rational Cherednik algebras over the past three years. I will not assume any prior knowledge of this theory, so I will start with some definitions. As time permits, I will mention interesting results concerning the category O, the functor KZ, Hecke algebras, finite dimensional representations, the spherical subalgebra and connections with the deformed Harish-Chandra homomorphism, etc.
Abstract of lecture by Charles Dunkl
For certain negative rational numbers k0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de Jeu, Opdam and Dunkl (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values -m/n with 2<=n<=N, m = 1,2... and m/n is not an integer. For each pair (m,n) satisfying these conditions there is a unique irreducible SN-module of singular polynomials for the singular value -m/n. The existence and uniqueness of these polynomials will be discussed.
By using Murphy's (J. Alg. 69 (1981), 287-297) results on the eigenvalues of the Murphy elements, the problem of existence of singular polynomials is first restricted to the isotype of a partition of N (corresponding to an irreducible representation of SN) such that (n/gcd(m,n)) divides t+1 for each part t of the partition except the last one. Arguments involving nonsymmetric Jack polynomials are used to construct singular polynomials and to show that the assumption that the second part of the partition is greater than or equal to n/gcd(m,n) leads to a contradiction. This proves the uniqueness. The polynomial ideal generated by an SN module of singular polynomials is a module of the rational Cherednik algebra.
Abstract of lecture by Gail Letzter
We prove an analog of a theorem of Harish-Chandra for quantum symmetric spaces: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a Laurent polynomial ring. A quantum version of a related theorem due to Helgason is also obtained: The image of the center under this Harish-Chandra map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not one of four exceptional types. Proofs depend on finding a particularly nice basis for the quantum invariant differential operators, which also relate to Macdonald polynomials.