Seminar on Special Functions
The positivity conjecture for the Opdam-Cherednik kernel
2026/08/25 (Tue.) 15:00~16:30
- Date : 2026/08/25 (Tue.) 15:00~16:30
- Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
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Speaker : Colin McSwiggen (Academia Sinica)
- Abstract :
The Opdam-Cherednik kernel, also known as Opdam’s nonsymmetric hypergeometric function, is a multivariable special function that appears in many guises throughout mathematics. It unifies Bessel and hypergeometric functions in several variables, irreducible characters of compact Lie groups, and spherical functions on Riemannian symmetric spaces, and it plays a fundamental role in the representation theory of degenerate DAHA. For nearly 30 years, a central open problem in Dunkl theory has been to prove that this function can be expressed as the Laplace transform of a positive measure. This statement can be interpreted as a simultaneous extension of the Poisson integral formula for Bessel functions, the Harish-Chandra integral formula for spherical functions, and the Kostant convexity theorem for the image of the Iwasawa projection. I will present a concise proof of this conjecture via a new formula for the Opdam-Cherednik kernel as a scaling limit of Macdonald polynomials, which is valid for all crystallographic root systems. If time allows, as an application, I will also discuss how this result gives new majorization inequalities for symmetric Heckman-Opdam hypergeometric functions in all root systems, resolving the conjecture that I stated in my job talk here at the Institute (three and a half years ago?!). The talk will not assume advanced background in any area.
This is joint work with Siddhartha Sahi.
Reference: https://arxiv.org/abs/2606.15185
