On decomposition of modular representations from Cohen-Macaulay geometries (Q912975)

Let G be a group of Lie type and B be the Borel subgroup. Then the splitting of \(Ind^ G_ B(1)\) in the natural characteristic is well known. This corresponds to the various truncations of the corresponding building. The author gives an elementary proof of this result just using the underlying geometry. The basic property of the geometry used is the Cohen-Macaulay property. So one can obtain results of this type for more general geometries than buildings. He in fact applies his treatment to so called p-local group geometries. In particular he investigates carefully the sporadic \(C_ 3\)-geometry for \(A_ 7\).