  We investigate the faces and the face lattices of arbitrary Coxeter group
invariant convex subcones of the Tits cone of a linear Coxeter system as
introduced by E. B. Vinberg. Particular examples are given by certain Weyl
group invariant convex cones which underlie the theory of normal reductive
linear algebraic monoids as developed by M. S. Putcha and L. E. Renner. We
determine the faces and the face lattice of the Tits cone and the imaginary
cone, generalizing some of the results obtained for linear Coxeter systems with
symmetric root bases by M. Dyer, and for linear Coxeter systems with free root
bases by E. Looijenga, P. Slodowy, and the author.
