Exercise 1, § 5

Assume that G is simply-connected; the alcoves of G are the subsets of G of the form exp(A), where A is an alcove of a Cartan subalgebra of $\mathfrak{g}$.

a) Show that the alcoves of G form a partition of $G_r$.

b) Every alcove of G is contained in a unique maximal torus.

c) Show that the alcoves of G contained in T form a partition of $T_r$, and that the set of such alcoves is a principal homogeneous space under W.

d) Every conjugacy class of regular elements of G has a unique point in each alcove.

e) Let E be an alcove of G. Show that E is a contractible space; if $\mathfrak{g}$ is simple, E is homeomorphic to an open simplex in a euclidean space.