Dynamical systems with homogeneous configuration spaces

Abstract

Let G be a connected semi-simple Lie group and K a closed reductive subgroup. In this chapter we enumerate all the homogeneous spaces G/K for which arbitrary Hamiltonian systems on T*(G/K) with G-invariant Hamiltonians are integrable within the class of Noether integrals (see Section 1 for definition). It is known that all symmetric spaces G/K of semi-simple groups G possess this property (see (Timm, 1988), (Mishchenko, 1982), (Mykytiuk, 1983) and (Ii, 1982)). It will also be proved here that if, in addition, the groups G and K have a complex structure or are compact, then the following conditions are equivalent:

• (1) All G-invariant Hamiltonian systems on T*(G/K) are integrable within the class of Noether integrals.

• (2) The subgroup K of G is spherical; i.e., the quasiregular representation of G on the space C[G/K] of regular functions on the affine algebraic variety G/K has a simple spectrum if G is complex, and likewise on L ₂(G/K) if G is compact. In (Guillemin et al, 1984a) it was shown that a subgroup K of a compact Lie group G is spherical if and only if

• (3) The algebra of G-invariant functions on T*(G/K) is commutative with respect to the standard Poisson bracket.