On the range of non-vanishing \(p\)-torsion cohomology for \(\text{GL}_n(\mathbb{F}_p)\). (Q1882980)

For \(p\geq n\) the author constructs a nontrivial cohomology class in the cohomology group \(H^{2p-2}(\text{GL}_n(\mathbb{F}_p),\mathbb{F}_p)\). This is a remarkably low degree, so that these classes must be new. The method relies heavily on the Hecke algebra \({\mathcal H}(G//U)\) of \(G=\text{GL}_n(\mathbb{F}_p)\) with respect to its standard \(p\)-Sylow subgroup \(U\). This Hecke algebra is used to construct the classes over \(U\) and also to show that they extend to \(G\), by establishing that they are appropriate simultaneous eigenvectors for the action of \({\mathcal H}(G//U)\). Once the classes are constructed it is easy enough to show them nontrivial.