Generalized Macdonald-Mehta integrals and Baker-Akhiezer functions
Mikhail Feigin (University of Glasgow)
Tuesday 2nd October, 2012 15:00-16:00 325
We consider Baker-Akhiezer functions associated with the special arrangements of hyperplanes with multiplicities; these are particular eigenfunctions for the generalized Calogero-Moser operators. We establish an integral identity for the Baker-Akhiezer functions that may be viewed as a generalization of the self-duality property of the usual Gaussian with respect to the Fourier transformation. We derive that the value of the Baker-Akhiezer function at the origin is given by the integral of Macdonald-Mehta type. In contrast to the standard Macdonald-Mehta integrals arising in random matrix theory our integrand has singularities on the arrangement so a regularisation is needed. We explicitly compute these regularised integrals for known Coxeter and non-Coxeter arrangements admitting the Baker-Akhiezer functions. In the Coxeter cases we use analytic continuation of the usual Macdonald-Mehta integral. In the two-dimensional examples we use the explicit form of the Baker-Akhiezer functions found by Berest, Cramer and Eshmatov. In the higher-dimensional non-Coxeter examples we evaluate these integrals by taking a special limit of the Dotsenko-Fateev integral arising in conformal field theory. We also discuss some properties of the algebras of quasi-invariant polynomials associated with these arrangements, and in particular we express a bilinear form on these algebras through a further generalisation of the Macdonald-Mehta integral. The talk is based on joint works with M.A. Hallnas and A.P.Veselov, and with D. Johnston.