Chevalley group theory and the transfer in the homology of symmetric groups (Q1084510)

If L is a subgroup of a finite group G, let \(W_ G(L)\) denote the group \(N_ G(L)/L\) where \(N_ G(L)\) is the normalizer of L in G. There is a right action of \(W_ G(L)\) on the group cohomology \(H^*(L)\). The inclusion \(L\hookrightarrow G\) induces a map \(i^*: H^*(G)\to H^*(L)^{W_ G(L)}\) to the \(W_ G(L)\)-invariants. If K is a subgroup of G containing L, one has transfer maps \(tr^*: H^*(K)\to H^*(G)\) and \(\tau^*: H^*(L)^{W_ K(L)}\to H^*(L)^{W_ G(L)}\) induced by the inclusions \(K\hookrightarrow G\) and \(W_ K(L)\hookrightarrow W_ G(L).\) Starting with parabolic subgroups of a Chevalley group, the present paper studies cases for which one has \(i^*tr^*=\tau^*i^*\). The transfers \(\tau^*\) are studied in the context of the Hecke algebra, the particular case of the finite Lie groups theory context leads to a good understanding of the structure of natural maps between invariants of parabolic subgroups, as \(\tau^*.\) This paper, motivated by the establishment of connections between various works in stable homotopy theory, gives applications to topology, e.g. on inverse systems of Thom spectra, splittings using the Steinberg idempotent, James-Hopf invariants in homology.