DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS
Forum of Mathematics, Pi, Tome 7 (2019)

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We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\unicode[STIX]{x1D6E4}$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra.From this construction we extract an action of certain $p$-adic Galois cohomology groups on $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$, and formulate the central conjecture: the motivic $\mathbf{Q}$-lattice inside these Galois cohomology groups preserves $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$.
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     title = {DERIVED {HECKE} {ALGEBRA} {AND} {COHOMOLOGY} {OF} {ARITHMETIC} {GROUPS}},
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AKSHAY VENKATESH. DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS. Forum of Mathematics, Pi, Tome 7 (2019). doi: 10.1017/fmp.2019.6

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