An algebraic geometric model of an action of the face monoid associated to a Kac-Moody group on its building

The face monoid Gˆ described in [11] acts on the integrable irreducible highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a symmetrizable Kac-Moody group G. We found in [15] two natural extensions of the action of the Kac-Moody group G on its building Ω to actions of the face monoid Gˆ on the building Ω. Now we give an algebraic geometric model of one of these actions of the face monoid Gˆ on Ω, where the building Ω is obtained as a part of the F-valued points of the spectrum of all homogeneous prime ideals of the Cartan algebra CA of the Kac-Moody group G. We describe the spectrum of all homogeneous prime ideals of the Cartan algebra CA and determine its F-valued points.