Weight operators and group geometries (Q1801893)

For the representation theory of the Lie-type groups in the natural characteristic the most important tools are the weight operators and weight vectors [see \textit{C. Curtis}, J. Reine Angew. Math. 219, 180-199 (1965; Zbl 0132.020) and \textit{F. Richen}, Trans. Am. Math. Soc. 140, 435- 460 (1969; Zbl 0181.038)]. Using a more geometric language this has been extended by \textit{S. Smith} [J. Algebra 131, 598-625 (1990; Zbl 0699.20009)] in order to treat a more general class of groups belonging to generalizations of buildings. In the paper under review the author gives a unified approach including results of Curtis, Richen, and Smith. For this he uses the concept of a weight algebra. Let \(\Gamma\) be a geometry over some index set \(I\) with flag transitive group \(G\) (assume \(\Gamma\) to be thick). Let \(G_ i\), \(i\in I\), be the set of minimal parabolics containing a Borel subgroup \(B\). Let \(P_ i\) be a sum of coset representatives of \(B\) in \(G_ i\). The weight algebra \({\mathcal P}\) is the subalgebra of \(KG\) generated by the \(P_ i\) and the identity. It is assumed that \(B\) contains a normal Sylow \(p\)-subgroup, and so \(p\) is called the characteristic of \(G\). In the paper \({\mathcal P}\) is considered as a subalgebra of \(\text{End}(^ B\text{Ind}_ B^ G(1))\). Instead of \(B\) one can also consider \(U\), the Sylow \(p\)-subgroup of \(B\), and so \({\mathcal P}\) is a subalgebra of \(\text{End}(^ U\text{Ind}_ U^ G(1))\). A weight vector is a common eigenvector of \({\mathcal P}\) on \(^ B\text{Ind}_ B^ G(1)\) or \(^ U\text{Ind}_ U^ G(1)\), respectively. The content of the paper is to give conditions for the existence of a weight vector (for example \({\mathcal P}\) acts solvably) and prove a decomposition of the module into cycle spaces. This is also achieved in sporadic examples like \(M_{12}\) acting on the 2-transitive graph. This example is worked out in detail and very illustrative. The author proves that \({\mathcal P}\) acts solvably. Furthermore he also investigates \(^ B\text{Ind}_ B^ G(\lambda)\), where \(\lambda\) is some character of \(B\) of degree 1. Finally he looks at the example \(U_ 5(2)\) on a geometry with diagram \(\overset {c}{\circ\diagrbar\circ}\diagrBar\circ\) and on flag transitive subgroups of Lie-type groups like \(2^ 4 A_ 5\) in \(PSp(4,3)\) and \(L_ 3(4)2\) in \(U_ 6(2)\).