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The face monoid described in [M1] acts on the integrable highest weightmodules of a symmetrizable Kac-Moody algebra. It has similar structuralproperties as a reductive algebraic monoid whose unit group is a Kac-Moodygroup. We found in [M5] two natural extensions of the action of the Kac-Moodygroup on its building to actions of the face monoid on the building. Now wegive an algebraic geometric model of one of these actions of the face monoid.The building is obtained as a part of the spectrum of homogeneous prime idealsof the Cartan algebra of the Kac-Moody group. We describe the full spectrum ofhomogeneous prime ideals of the Cartan algebra.

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