On Springer's representations of Weyl groups containing -1
Let G be a connected reductive complex algebraic group. For \(u\in G\) unipotent, let \({\mathcal B}_ u\) be the variety of all Borel subgroups of G containing x, \(\beta\) (u) its dimension. Springer has defined a natural representation of the Weyl group W of G in \(H^*({\mathcal B}_ u;{\mathbb{C}})\) which commutes with the natural action of the finite group \(A(u)=C_ G(u)/C^ 0_ G(u)\). Let \(\rho\),\(\phi\) be irreducible representations of W, A(u) respectively. In the case where -1\(\in W\), the paper gives a necessary condition mod 2 on \(i\in {\mathbb{N}}\) for \(\rho \otimes\phi \) to occur in \(H^{2i}({\mathcal B}_ u;{\mathbb{C}})\) (it is known that \(H^{2i- 1}({\mathcal B}_ u;{\mathbb{C}})=0)\). The proof uses results of the representation theory of finite Chevalley groups, in particular an algorithm discovered by Shoji to compute the Green functions of finite Chevalley groups.
connected reductive complex algebraic group
Green functions of finite Chevalley groups
scientific article; zbMATH DE number 3873593
scientific article