The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method.
We also give a simple description for the top cohomology of convolution of character sheaves over ℂ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini–Procesi compactification.United States. Defense Advanced Research Projects Agency (grant HR0011-04-1-0031)National Science Foundation (U.S.) (grant DMS-0625234)National Science Foundation (U.S.) (grant DMS-0854764)AG Laboratory HSE (RF government grant, ag. 11.G34.31.0023)Russian Foundation for Basic Research (grant 09-01-00242)Ministry of Education and Science of the Russian Federation (grant No. 2010-1.3.1-111-017-029)Science Foundation of the NRU-HSE (award 11-09-0033)National Science Foundation (U.S.) (grant DMS-0602263
