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We present an algorithm to compute the Hecke operators on the equivariantcohomology of an arithmetic subgroup $\Gamma$ of the general linear group$\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or afinite-dimensional division algebra. As coefficients, we may use anyfinite-dimensional local coefficient system. Unlike earlier methods, thealgorithm works for the cohomology $H^i$ in all degrees $i$. It starts from thewell-rounded retract $\tilde{W}$, a $\Gamma$-invariant cell complex whichcomputes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex$\tilde{W}^+$ of one higher real dimension, using a real parameter called thetemperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$for $n=2,3,4$; we present some results for congruence subgroups of$\mathrm{SL}_3(\mathbb{Z})$.

Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups