12.15-13.00:
Nicolas Guay (University of Amsterdam),
Rational Cherednik algebras
14.30-15.30:
Charles Dunkl (University of Virginia, Charlottesville, VA, USA),
Singular polynomials and modules for the symmetric groups
16.00-17.00:
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.