Spin Calogero-Moser models on symmetric spaces

Abstract

In this paper we construct and prove superintegrability of spin Calogero-Moser type systems on symplectic leaves of K 1 ∖ T ∗ G / K 2 K_1\backslash T^*G/K_2 where K 1 ,   K 2 ⊂ G K_1,\space K_2\subset G are subgroups. We call them two-sided spin Calogero-Moser systems. One important type of such systems correspond to K 1 = K 2 = K K_1=K_2=K where K K is a subgroup of fixed points of Chevalley involution θ : G → G \theta : G\to G . The other important series of examples come from pair G ⊂ G × G G\subset G\times G with the diagonal embedding. We explicitly describe examples of such systems corresponding to symplectic leaves of rank one when G = S L n G=SL_n .