Generalised Lamé operators and double affine Hecke algebras

Research Project, 2025 – 2028

We aim to introduce and study new and far-reaching integrable systems of (relativistic) Calogero-Moser type. Given by highly symmetric differential- and difference operators, the intended systems describe three groups of particles moving on the circle under the influence of particular elliptic potentials. By solving associated joint eigenvalue problems, we expect to obtain novel generalisations of the celebrated Jack- and Macdonald polynomials, featuring remarkable orthogonality properties, as well as natural higher-dimensional generalisations of meromorphic Lamé functions.

Key ingredients will be Cherednik's double affine Hecke algebras, as well as Rains' elliptic generalisation thereof, and quasi-invariant functions, which are invariant with respect to a given reflection group up to a specified order. Moreover, we anticipate that our results will shed new light on a 20 year old open conjecture of Chalykh, Etingof and Oblomkov that promises a striking classification of integrable systems of elliptic Calogero-Moser type in terms of quasi-invariance of their potentials.

Our aim is further motivated by very recent developments in both mathematics and theoretical physics. Specifically, we plan to explore potential applications to to the theory of coincident root loci, multiple elliptic hypergeometric series, exceptional orthogonal polynomials and gauge/string theories.