Sur l'homologie des groupes orthogonaux et symplectiques \`a coefficients tordus

Abstract

We compute the stable homology of orthogonal and symplectic groups over a finite field k with coefficients coming from an usual endofunctor F of k-vector spaces (exterior, symmetric, divided powers...), that is, for all natural integer i, we compute the colimits of the vector spaces $H_i(O_{n,n}(k) ; F(k^{2n}))$ and $H_i(Sp_{2n}(k) ; F(k^{2n}))$. In this situation, the stabilization is a classical result of Charney. We give a formal framework to connect stable homology of some families of groups and homology of suitable small categories thanks to a spectral sequence which collapses in several cases. By our purely algebraic methods (i.e. without stable K-theory) we obtain again results of Betley for stable homology of linear groups and symmetric groups. For orthogonal and symplectic groups over a field we prove a categorical result for vector spaces equipped with quadratic or alternating forms and use powerful cancellation results known in homology of functors (Suslin, Scorichenko, Djament) to deduce a spectacular simplification of the second sheet of our general spectral sequence. When we consider the orthogonal and symplectic groups over a finite field and we take coefficients with values in vector spaces over the same field, we can compute the second sheet of the spectral sequence thanks to classical results: homological cancellation with trivial coefficients (Quillen, Fiedorowicz-Priddy) and calculation of torsion groups between usual functors (Franjou-Friedlander-Scorichenko-Suslin, Chalupnik).