The representation theory seminar will meet on a regular basis. The talks will be announced on this website and by email. If you are not yet on the mailing list but would like to receive the email announcements, please send an email to j.v.stokman AT uva.nl.

What seems to have remained unobserved in the literature is that a straightforward q=1 limit of BC-type interpolation Macdonald polynomials leads to the definition of what we may call BC-type interpolation Jack polynomials. The corresponding limit of the binomial formula for Koornwinder polynomials gives BC-type Jacobi polynomials $P_\lambda(x;\tau;\alpha,\beta)$ as a sum of products of a BC-type interpolation Jack polynomial depending on $\lambda$ and a Jack polynomial depending on $x$. This formula was already given by Macdonald [3, (9.15)], but his combinatorial formula [3, p.58] for the first factor in the sum of products is different from the combinatorial formula which follows as a limit of the combinatorial formula for the BC-type interpolation Macdonald polynomials. It is this last formula which specializes in the rank 2 case to a balanced ${}_4F_3(1)$ expression in Koornwinder & Sprinkhuizen [4, Cor. 6.6].

The lecture will present these old and new results following the recent preprint [5]. Some attention will also be paid to the recent work by van Diejen & Emsiz [6], who give a combinatorial formula for Koornwinder polynomials.

[1] A. Okounkov, BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181-207.

[2] E.M. Rains, $BC_n$-symmetric polynomials, Transform. Groups 10 (2005), 63-132.

[3] I.G. Macdonald, Hypergeometric functions I, Unpublished manuscript, 1987; arXiv:1309.4568v1.

[4] T.H. Koornwinder and I.G. Sprinkhuizen, Generalized power series expansions for a class of orthogonal polynomials in two variables, SIAM J. Math. Anal. 9 (1978), 457-483.

[5] T.H. Koornwinder, Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit, arXiv:1408.5993v1.

[6] J.F. van Diejen and E. Emsiz, Branching formula for Macdonald-Koornwinder polynomials, arXiv:1408.2280v1.

[1] S.N.M. Ruijsenaars, Systems of Calogero-Moser type, in: Proceedings of the 1994 Banff summer school "Particles and fields", G. Semenoff, L. Vinet, Eds., 251-352, Springer, 1999; and references therein.

[2] V. Fock, A. Gorsky, N. Nekrasov, V. Rubtsov, Duality in integrable systems and gauge theories, JHEP 0007, 028, 2000.

[3] L. Feher, C. Klimcik, Poisson-Lie interpretation of trigonometric Ruijsenaars duality, Commun. Math. Phys. 301, 55-104, 2011.

[4] B.G. Pusztai, The hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality, Nucl. Phys. B 856, 528-551, 2012.

[5] L. Feher, T.F. Gorbe, Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system, J. Math. Phys. 55, 102704, 2014.

[6] T.F. Gorbe, On the derivation of Darboux form for the action-angle dual of trigonometric BC(n) Sutherland system, J. Phys.: Conf. Ser. 563, 012012, 2014.

[7] T.F. Gorbe, L. Feher, Equivalence of two sets of deformed Calogero-Moser Hamiltonians, arXiv:1503.01303 [math-ph]; submitted, 2015.

[8] J.F. van Diejen, Commuting difference operators with polynomial eigenfunctions, Compos. Math. 95, 183-233, 1995.

Modules with minimal support are interesting from many points of view. First of all, they are Cohen-Macaulay modules over the polynomial subalgebra. This allows one to prove the Cohen-Macaulay property of many rings of quasi-invariants - specifically, rings of quantum integrals of deformed Calogero-Moser systems considered by O. Chalykh, M. Feigin, A. Sergeev, and A. Veselov, as well as some generalizations. Also, characters of minimally supported modules in type A have nice explicit formulas, which allows one to compute Hilbert series of quasi-invariant rings. The same character formulas can be used to compute Hilbert series of multiplicity spaces in equivariant D-modules on the nilpotent cone for the Lie algebra sl(m). Finally, these modules arise in the computation of the HOMFLY polynomial of the torus knot, and allow one to prove its positivity properties.

This is based on joint work with I. Losev, E. Gorsky, and E. Rains.

The talk will consist of two parts.

Part 1. Minimally supported modules and their characters, with application to D-modules and knot invariants.

Part 2. Minimally supported modules and Cohen-Macaulay properties of quasiinvariant rings.