Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

TL;DR

This paper establishes a new integral identity for Baker-Akhiezer functions linked to hyperplane arrangements, generalizing Gaussian self-duality, and computes related Macdonald-Mehta integrals for various arrangements and root systems.

Contribution

It introduces a generalized integral identity for Baker-Akhiezer functions and computes explicit integrals for all known arrangements and deformed root systems.

Findings

Derived integral identities for Baker-Akhiezer functions

Explicit computation of Macdonald-Mehta type integrals

Extended calculations to deformed root systems using Dotsenko-Fateev integrals

Abstract

For the rational Baker-Akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the self-duality property of the usual Gaussian function with respect to the Fourier transformation. We show that the value of properly normalised Baker-Akhiezer function at the origin can be given by an integral of Macdonald-Mehta type and explicitly compute these integrals for all known Baker-Akhiezer arrangements. We use the Dotsenko-Fateev integrals to extend this calculation to all deformed root systems, related to the non-exceptional basic classical Lie superalgebras.