Exercise 10, § 4

Let G be a closed subgroup of GL(E) containing W. Assume that G is unimodular (Integration, Chap. VII, § 1, no. 3). Let D be a half-line of E^* contained in C, and let G_D be the stabiliser of D in G.

a) Let \Delta be the set of elements g \in G such that g(D) \subset C. Show that \Delta is open, stable under right multiplication by G_D, and that the composite map \Delta \to G \to W\backslash G is injective, W\backslash G denoting the homogeneous space of right cosets of G with respect to W.

b) Let \mu be a Haar measure on G. Show that, if \mu(\Delta) is finite, the subgroup G_D is compact. (Let K be a compact neighbourhood of the identity element contained in \Delta; show that there exist finitely many elements h_i \in G_D such that every set of the form Kh, with h \in G_D, meets one of the Kh_i; deduce that G_D is contained in the union of the K^{-1}.K.h_i, and hence is compact.)

c) Let \nu be a non-zero positive measure on W\backslash G invariant under G. Show that, if \nu(W\backslash G) < \infty, then G_D is compact.