Actions of the face monoid associated to a Kac–Moody group on its building

We described in [C. Mokler, An analogue of a reductive algebraic monoid whose unit group is a Kac–Moody group, Mem. Amer. Math. Soc. 823 (2005), 90 pp.] a monoid acting on the integrable highest weight modules of a symmetrizable Kac–Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac–Moody group G. Now we find natural extensions of the action of the Kac–Moody group G on its building Ω to actions of the monoid on Ω. These extensions are partly motivated by representation theory and the combinatorics of the faces of the Tits cone.