January 9, 2003
We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class.
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These notes are based on a course given at the EPFL in May 2005. It is concerned with the representation theory of Hecke algebras in the non-semisimple case. We explain the role that these algebras play in the modular representation theory of finite groups of Lie type and survey the recent results which complete the classification of the simple modules. These results rely on the theory of Kazhdan--Lusztig cells in finite Weyl groups (with respect to possibly unequal parameter...
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We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and Robert MacPherson.