Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Summary Let F_n, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each F_n, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups D_n. For D_n, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H₃ ⊆G₃ on L²(T) that result from the irrational rotation flows on T; the representations of D_n generate infinite-dimensional simple quotients A_n,θ of the group C*-algebra C*(D_n). For n>1, there are other infinite-dimensional simple quotients of C*(D_n) arising from non-faithful representations of D_n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.