Аннотация:
To every irreducible finite crystallographic reflection group
(i.e., an irreducible finite reflection group G acting
faithfully on an abelian variety X), we attach a family of
classical and quantum integrable systems on X (with
meromorphic coefficients). These families are parametrized by
G-invariant functions of pairs (T,s), where T is a hypertorus
in X (of codimension 1), and s in G is a reflection acting
trivially on T. If G is a real reflection group, these
families reduce to the known generalizations of elliptic
Calogero-Moser systems, but in the non-real case they appear
to be new. We give two constructions of the integrals of
these systems - an explicit construction as limits of
classical Calogero-Moser Hamiltonians of elliptic Dunkl
operators as the dynamical parameter goes to 0 (implementing
an idea of Buchshtaber, Felder, and Veselov from 1994), and a geometric
construction as global sections of sheaves of elliptic
Cherednik algebras for the critical value of the twisting
parameter. We also prove algebraic integrability of these
systems for values of parameters satisfying certain
integrality conditions. This is joint work with G. Felder,
X. Ma, and A. Veselov.