There are known to be integrable Sutherland models associated to every real root system -- or, which is almost equivalent, to every real reflection group. Real reflection groups are special cases of complex reflection groups. In this paper we associate certain integrable Sutherland models to the classical family of complex reflection groups. Internal degrees of freedom are introduced, defining dynamical spin chains, and the freezing limit taken to obtain static chains of Haldane-Shastry type. By considering the relation of these models to the usual BC_N case, we are led to systems with both real and complex reflection groups as symmetries. We demonstrate their integrability by means of new Dunkl operators, associated to wreath products of dihedral groups.
对任意有限复反射群W引入了拟不变多项式。与Coxeter的情形不同，给定重数的拟不变量空间Q_k一般不是代数，而是某个（奇异）仿射簇X_k的坐标环上的模。本文推广了Etingof，Ginzburg和第一作者（见[BEG]）的主要结果：特别地，我们证明了簇X_k和模Q_k是Cohen-Macaulay，X_k和Q_k上的微分算子环是单环，Morita等价于Weyl代数A_n（C），其中n = dim X_k .我们的方法依赖于复杂的Cherednik代数的表示论，是平行的[BEG]。作为一个副产品，我们证明了存在的移位算子的任意复杂的反射群，证实了一个猜想的Dunkl和Opdam。另一个结果是一个猜想的奥普丹的证明，关于某些操作（KZ扭曲）的不可约表示的W。
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 gi…
