# Representation Theory Seminar

Organizers: Eric Opdam and Jasper Stokman
Address: Science Park 105-107 (entrance at Nikhef, 3rd floor) Amsterdam, Rooms F3.04 and F3.03.
Emails: e.m.opdam (AT) uva.nl and j.v.stokman (AT) uva.nl
Tel.: 020-5255205 and 020-5255202

The representation theory seminar meets on an irregular 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.

Schedule

Thursday October 9 (2014), 15:00-16:00: John Duncan (Case Western Reserve University, USA).
Location: G5.29.
Title: Vertex Algebra and Umbral Moonshine.
Abstract: Umbral moonshine attaches distinguished vector-valued mock modular forms---the umbral McKay--Thompson series---to automorphisms of Niemeier lattices. Conjecturally, these series may be recovered as the trace functions arising from actions of the umbral groups on bi-graded modules. It is natural to regard the mock modular forms arising as theta-coefficients of meromorphic Jacobi forms. We will report on joint works with Miranda Cheng and Andrew O'Desky that use certain super vertex algebras to realize these umbral meromorphic Jacobi forms explicitly as bi-graded traces, for a number (about a third) of the 23 Niemeier lattices.
Thursday October 16 (2014), 15:00-16:00: Wellington Galleas (UU).
Location: C0.110.
Title: Twists vs. Domain-walls: Relating boundary conditions for the 6V model.
Abstract: The six-vertex model with domain-wall boundary conditions is a protagonist in the theory of exactly solvable models. This model not only plays a fundamental role for the computation of correlation functions but it also exhibits interesting connections with the theory of classical integrable systems, enumerative combinatorics and special functions. On the other hand, for this particular boundary conditions we do not have a solution in terms of Bethe ansatz equations as we have for the cases with periodic and open boundary conditions. In this talk I will demonstrate a relation between the six-vertex model with domain-wall boundaries and the case with twisted boundary conditions. The latter then admits a Bethe ansatz like solution and is related to an one-dimensional spin chain hamiltonian.

Thursday October 30 (2014), 15:00-16:00: Tom Koornwinder (UvA).
Location: D1.112.
Title: Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit.
Abstract: Okounkov [1] defined BC-type interpolation Macdonald polynomials, which generalize shifted Macdonald polynomials and shifted Jack polynomials. Among others, he derived a combinatorial formula for these polynomials, and he derived a binomial formula which gives Koornwinder polynomials $P_\lambda(x;q,t;a_1,a_2,a_3,a_4)$ (a 5-parameter extension of BC-type Macdonald polynomials) as a sum of products of two BC-type interpolation Macdonald polynomials, one factor depending on $q^\lambda$ and one factor depending on $x$. From this formula the duality property of Koornwinder polynomials is clear. Rains [2] used this binomial formula as definition for the Koornwinder polynomials and he derived all their other properties from this formula. In later work he defined elliptic analogues in a similar way.

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.

Thursday November 13 (2014), 15:00-16:00: Gert Heckman (IMAPP, RU).
Location: F3.20 (Science Park 105-107, entrance at Nikhef, 3rd floor).
Title: An odd presentation for W(E_6) and its geometric origin.
Abstract: The bimonster has an odd presentation, due to Ivanov and Norton, as factor group of a particular hyperbolic Coxeter group. Christopher Simons described similar odd presentations for a hand full of other groups, such as W(E_6). We intend to explain a natural geometric setting for this presentation, coming for the period map of Allcock--Carlson--Toledo for the moduli space of cubic surfaces. If time permits, we indicate a similar story for W(E_7).

Thursday December 11 (2014), 15:00-16:00: Volker Heiermann (Universite d'Aix-Marseille, France).
Location: A1.04.
Title: Local Langland correspondence for classical groups and representations of affine Hecke algebras.
Abstract: We will describe the local Langlands correspondence for classical groups, which is in good shape now, at the example of the odd orthogonal group, and translate it into representation theory for affine Hecke algebras.

Thursday April 16 (2015), 15:00-16:00: Tamas F. Gorbe (University of Szeged, Hungary).
Location: F1.15.
Title: The trigonometric BC(n) Sutherland system: action-angle duality and applications (Joint work with Laszlo Feher).
Abstract: We report our recent results on action-angle duality for the trigonometric BC(n) Sutherland system. Two Liouville integrable many-body systems (X and Y) are said to be dual to each other if there exists a global symplectomorphism between their phase spaces that converts the particle position variables and the action variables of system X into the action variables and the particle positions of system Y, respectively. This sort of duality was originally explored by Ruijsenaars between 1988 and 1995, dealing with Calogero-Sutherland and Toda systems associated with root systems of type A [1]. More recently, it turned out (see e.g. [2,3]) that Ruijsenaars' action-angle dualities admit interesting group-theoretic interpretations based on Hamiltonian reduction. The reduction method is also applicable for finding new dualities, as is exemplified by the paper [4], where duality between the hyperbolic BC(n) Sutherland and the rational Ruijsenaars-Schneider-van Diejen systems was established. The talk expounds the generalization of this result to the (topologically) considerably more complicated case of the trigonometric BC(n) Sutherland system [5,6], applications of this duality and the connection [7] to the family of commuting Hamiltonians found by van Diejen [8].
References:
[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.

Thursday April 23 (2015), 14:00-15:00: Dan Ciubotaru (University of Oxford, UK).
Location: F3.20.
Title: Dirac cohomology for symplectic reflection algebras.
Abstract:TBA.

Thursday April 30 (2015), 15:00-16:00: Kei Yuen Chan (Universiteit van Amsterdam).
Location: F3.20.
Title: Extensions of modules for graded Hecke algebras.
Abstract: Graded Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. The representation theory of graded Hecke algebra has close connections to local Langlands correspondence and Springer correspondence. We are interested in the extensions between modules of graded Hecke algebras. In this talk, I shall discuss some recent study on the Ext-groups for graded Hecke algebras.

Tuesday June 16 (2015), 16:00-17:00: Nicolai Reshetikhin (University of California, Berkeley, USA; University of Amsterdam).
Location: F3.20.
Title: Degenerate, or super-integrability.
Abstract: The notion of Liouville integrability is well known in Hamiltonian mechanics and symplectic geometry. The main theme o this talk is the degenerate integrable systems. On a symplectic manifold of dimension 2n such system has k Poisson commuting integrals and 2n-k integrals of motion which generate a Poisson subalgebra in the algebra of functions on the symplectic manifold. Example related to Poisson Lie groups and representation theory will be given.

Friday June 26 (2015), 15:00-16:00: Simon Gindikin (Rutgers University, USA).
Location: F3.20.
Title: Horospherical transform for discrete series of representations.
Abstract: On the example of pseudohyperbolic spaces I will discuss the problem to connect a version of horospherical transform with discrete series of representations. It is an answer on the old question of I.Gelfand. For non holomorpfic discrete series this transform takes values at cohomology Cauchy- Riemann. I believe that it explains their appearance at discrete series of real groups.

Thursday July 2 (2015), 14:00-15:00: Jan de Gier (University of Melbourne, Australia).
Location: F3.20.
Title: A matrix product formula for Macdonald polynomials.
Abstract: I will discuss how an inhomogeneous version of the multi-species asymmetric simple exclusion process gives rise to a matrix product formula for Macdonald polynomials in terms of $t$-deformed oscillators. This result is derived by exploiting the underlying Yang-Baxter integrability based on $U_\sqrt{t}(\hat(sl)(n))$. This is work in collaboration with Luigi Cantini and Michael Wheeler.

Friday October 16 (2015), 15:00-16:00: Jules Lamers (University of Utrecht).
Location: F3.20.
Title: Functional equations for elliptic SOS models with domain walls and a reflecting end.
Abstract: Solid-on-solid (SOS) models in statistical physics describe the growth of crystal surfaces. In the special case in which these models are quantum integrable they can equivalently be viewed as generalized six-vertex models. This talk is about my recent work [arxiv:1510.00342] on such models with certain integrability-preserving boundary conditions: domain walls and one reflecting end. After introducing these models we explain how the underlying algebraic structure -- the so-called dynamical reflection algebra -- can be used as a source of functional equations characterizing the partition function of that model, and what can be learned from this approach.

Friday November 20 (2015), 14:30-15:30: Olivier Taibi (Imperial College London, UK).
Location: F3.20.
Title: Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups.
Abstract: I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of certain special orthogonal groups and certain inner forms of symplectic groups G over a number field F. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set S of real places of F such that G has discrete series at places in S and is quasi-split at places outside S, and by restricting to automorphic representations of G(AA_F) which have algebraic regular infinitesimal character at all places in S. In particular, I prove the general multiplicity formula for groups G such that F is totally real, G is compact at all real places of F and quasi-split at all finite places of F. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

Monday May 30 (2016), 13:00-14:00: Marcelo Goncalves de Martino (UvA).
Location: F3.20.
Title: On the Unramified spherical automorphic spectrum.
Abstract: Let G be a connected split reductive group over a number field. In this talk, I will report on how part of the spectral decomposition of the space of square integrable functions on the automorphic quotient can be related to the spectral decomposition of the spherical subalgebra of a graded affine Hecke algebra. This is a joint work with V. Heiermann and E. Opdam.

Friday June 17 (2016), 15:00-17:00: Pavel Etingof (MIT, USA).
Location: F3.20.
Title: Minimal support modules over Cherednik algebras and applications.
Abstract: I will consider modules over the rational Cherednik algebra from category O, i.e., finitely generated over the subalgebra of polynomials and with locally nilpotent action of Dunkl operators. With such a module one can associate its support, a closed subvariety of the affine space. A simple module is said to have minimal support if there are no nonzero modules with smaller support.
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.

Wednesday June 22 (2016), 13:30-14:30: Dan Ciubotaru (University of Oxford, UK).
Location: F3.20.
Title: Cocenters of affine Hecke algebras and p-adic groups.
Abstract: The cocenter of an associative algebra is the quotient of the algebra by the subspace spanned by all the commutators. This is the same s the zero-th Hochschild homology of the algebra and it plays an important role in representation theory in relation to questions regarding the density of irreducible characters. For a reductive p-adic group, the algebra that governs the smooth representation theory is the convolution Hecke algebra H(G) of locally constant compactly-supported functions and the relations between the cocenter, the K_0 group, and the space of admissible characters has been the subject of much study beginning with the work of Bernstein, Deligne, and Kazhdan 30 years ago. In this talk, I will present an explicit description of the cocenter of certain subalgebras of H(G), the Iwahori-Hecke algebras, and talk about work in progress regarding a new, more general description of the cocenter of H(G) and its duality with group representations.

Friday March 10 (2017), 15:00-16:00: Jan Felipe van Diejen (Universidad de Talca, Chile).
Location: F3.20.
Title: Bethe Ansatz for a finite q-boson system with boundary interactions.
Abstract: The q-boson system is a collection of q-oscillators on a one-dimensional integral lattice. We construct an orthogonal basis of algebraic Bethe Ansatz eigenfunctions for a finite q-boson system with diagonal open-end boundary interactions at the lattice ends. Via a continuum limit, this allows to verify the orthogonality of the Bethe Ansatz eigenfunctions for the Laplacian on a classical Weyl alcove with repulsive Robin boundary conditions.