Daniel Baker

Abstract: This note discusses a class of foliations and a technique for evaluating the generalized Godbillon-Vey invariants on these foliations. The information obtained yields information about the cohomology of the Haefliger spaces H*(Brn, R) and H*(FT r n, R), r > 2. The class of foliations contains examples which have been studied by others as well. In particular, the foliations examined in [KT2] and in [Y] are of this type. Let G be a complex semisimple Lie group. There is a class of subgroups of G called parabolic subgroups, and the conjugacy classes of these subgroups are in 1-1 correspondence with subsets of the Dynkin diagram for 3 e , the Lie algebra of G (see [S] for a more detailed exposition). If P° is a parabolic subgroup then the Lie algebra P c of P^ can be written in the form V ~ Gf{ ® Tf ® N°. Here Ĝ is semisimple and has a Dynkin diagram obtained by removing the subset of vertices mentioned above from the Dynkin diagram for G. T? is an abelian subalgebra of G9G^ ® T*{ contains a Cartan subalgebra of G , and W is a nilpotent subalgebra. In fact, G = G? ® T? ® W ® W~ where M~ is a nilpotent subalgebra isomorphic to N, and [G^, T^] = 0 , [G^ ® TÇ, N C ] CW, [G?©Tf ,Wc ] c r c . Now let G be a real form of (f such that G = Gt ® Tt ® M ® N~ where Gt = Gf n G, etc. Then G has a subalgebra P = Gx ® Tx ® W. If G has Lie algebra G, then there is a discrete subgroup, F C G, with T\G a compact manifold (see [R]\ and the left translates of P determine a foliation on T\G. This is the foliation we study. Let Wn = Pn [cx, . . . , cn] ® A*(MX , . •. , un) be the cochain complex with deg ct = 2i, deg u( = 2/ 1, dct = 0, dui = cv Pn [ct, ... , cn] is the polynomial algebra in cx, . . . , cn, truncated above deg 2n where n is the codimension of the above foliation. There is a map H*(F\G, R) giving characteristic classes for the foliation (see [BT] for the construction of y). We analyse this map