The Primitive Spectrum and Category ${\mathcal{O}}$ for the Periplectic Lie Superalgebra

Chen, Chih-Whi ; Coulembier, Kevin

Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 625-655

Résumé

We solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

Détail

DOI : 10.4153/S0008414X18000081

Mots-clés : Harish-Chandra bimodules, periplectic Lie superalgebra, twisting functors, completion functors, primitive spectrum, category O, block decomposition

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