A structure theorem for Harish-Chandra bimodules via coinvariants and Golod rings
This paper contains two main results. The first one establishes the double centralizer property with respect to the antidominant projective module for the associative algebras, describing the blocks \({}_{\lambda}\mathcal{H}_{\mu}^n\) of truncated categories of Harish-Chandra bimodules, associated with a complex semisimple Lie algebra. This generalizes Theorem 12 in [\textit{W. Soergel}, ``The combinatorics of Harish-Chandra bimodules''. J. Reine Angew. Math. 429, 49--74 (1992; Zbl 0745.22014)] and Theorem 1 in [\textit{V. Futorny, S. König} and \textit{V. Mazorchuk}, ``\(S\)-subcategories in \(O\)''. Manuscr. Math. 102, No. 4, 487--503 (2000; Zbl 1018.17005)]. The second result explicitly describes the extension algebra of certain simple modules in some categories of Harish-Chandra bimodules. The description is very explicit, but explains only the homological grading on the extension space. Another natural grading on the extension spaces, which comes from the natural grading on the category of Harish-Chandra bimodules, still remains a complete mystery. Some interesting smaller results are obtained on the way. In particular, it is shown that the category \({}_{0}\mathcal{H}_{-\rho}^n\) is not equivalent to the corresponding thick category \(\mathcal{O}\) in the general case; some combinatorial statements for Harish-Chandra bimodules are also proved.
scientific article; zbMATH DE number 2119567
scientific article