Теория Гурвица в вещественном случае и для систем корней типа B и DHurwitz theory in the real case and for root systems of type B and D

Classical Hurwitz numbers have many definitions, among them there is a topological one as thenumber of ways to glue ribbons to a collection of disks so as to obtain a given surface with boundary. A purely algebraic one as the number of sequences of transpositions with a product of a specified cyclic structure;and an algebro-geometric one as the number of ramified coverings of the sphere with a prescribed number of simple critical values and a given type of ramification over the infinity. Furthermore the generating function of the Hurwitz numbers satisfies a PDE parabolic equation called the cut-and-join equation, and as a result can be expressed via Schur polynomials. Finally the generating function of Hurwitz numbers is a tau-function for the KP hierarchy. We will present new types of Hurwitz numbers: in the first part we will describe "twisted" versions of all this: ribbons are allowed to twist (literally), transpositions double, and the ramified coverings become conjugation-equivariant. The cut-and-join is replaced by the Laplace--Beltrami equation for the parameter b=1, and an explicit formula for the twisted Hurwitz numbers uses zonal polynomials instead of Schur. In the second part we will show that there is a parallel theory of Hurwitz numbers for the reflection groups of series B and D : transpositions are replaced by reflections, the cut-and-join for the group B is actually a tensor square of the classical cut-and-join, and the cut-and-join for the D group is a direct sum of the cut-and-join for B and the classical cut-and-join rescaled. This leads to an expression of the Hurwitz numbers for B and D via Schur polynomials. The generating function for the Hurwitz numbers for the reflection group B is a 2-parameter family of tau-functions of the KP hierarchy, independently in 2 variables, and we have a similar result for D. Finally Hurwitz numbers for the group B involve ribbon decomposition of surfaces equipped with an involution.