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

I will talk about root system generalizations of the quantum Bose gas on the circle with delta function interactions. By considering Dunkl-type differential-reflection operators associated to the integrable system, it is possible to show that the fundamental object controlling the algebraic relations between the Dunkl-type operators and the natural Weyl group action is a suitable graded algebra of Cherednik's degenerate double affine Hecke algebra. The allowed spectral parameters are controlled by certain transcendental equations called the Bethe ansatz equations.

After a short survey of all this, we discuss briefly some algebraic and analytic aspects: Pauli principle for the spectral parameters, completeness of the Bethe ansatz eigenfunctions. If time permits, some conjectures will be made on orthogonality and norms of the eigenfunctions.

*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 S_{N} 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
S_{N}-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 S_{N}) 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 S_{N}
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.

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