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Categories of highest weight modules: applications to classical Hermitian symmetric pairs. (English) Zbl 0621.17004

Mem. Am. Math. Soc. 367, 94 p. (1987).
Let \({\mathfrak g}\) be a semisimple Lie algebra over \({\mathbb{C}}\). In the first part of the memoir, the authors relate \({\mathfrak g}\)-modules with regular infinitesimal to \({\mathfrak g}\)-modules with singular infinitesimal character by translation. The latter are then analyzed in terms of \({\mathfrak g}'\)-modules of regular infinitesimal character with \({\mathfrak g}'\) of lower rank. The second part contains applications to categories of highest weight modules for classical Hermitian symmetric pairs. A more detailed statement of results requires notation: \({\mathfrak h}=Car\tan\) subalgebra of \({\mathfrak g}\), \({\mathfrak p}={\mathfrak m}+{\mathfrak u}\) parabolic, W \(= Weyl\) group, \(\Delta\) \(= roots\), \(P_ m=set\) of \(\Delta\)-integral \(\lambda\in {\mathfrak h}^*\), \(\Delta^+({\mathfrak m})\)-dominant \(\lambda\in {\mathfrak h}^*\). \({\mathcal O}(\lambda)={\mathfrak g}\)-modules (i) finitely generated over U(\({\mathfrak g})\), (ii) U(\({\mathfrak p})\)-locally finite, (iii) U(\({\mathfrak m})\)-completely reducible. P(\(\lambda)\) \(= projective\) cover of the irreducible quotient L(\(\lambda)\) of the Verma module M(\(\lambda)\).
Among the results on translations: Let \(\nu\),\(\mu\) be \(\Delta^+\)- dominant, \(\mu\) integral, \(\phi =\phi_{\nu}^{\nu +\mu}\) the translation functor \({\mathcal O}(\nu) \to {\mathcal O}(\nu +\mu)\). For \(w\in W\) and \(w\nu \in P_ m\), \(\phi\) P(w\(\nu)\) is an indecomposable projective module. Furthermore, P(w\(\nu)\) is self-dual iff \(\phi\) (w\(\nu)\) is.
Among the results on reduction to lower rank: Let \(\ell\) be the Levi factor of a maximal parabolic subalgebra not containing \({\mathfrak p}\), \(\lambda\Delta\)-integral and \(\Delta\) (\(\ell)\)-regular. Then \({\mathcal O}({\mathfrak g},{\mathfrak p},\lambda)\) is equivalent to a certain full subcategory \({\mathcal O}_ t(\ell,\ell \cap {\mathfrak p},\lambda)\) of \({\mathcal O}(\ell,\ell \cap {\mathfrak p})\) provided the shifted highest weights in these categories are the same. (Shifts by \(\rho\) and \(\rho\) (\(\ell).)\)
Among the results in the case of a Hermitian symmetric pair (G,K): a description of the composition factors of certain generalized Verma modules; results concerning Kazhdan-Lusztig-Vogan polynomials, in particular recursion relations generalizing those of Lascoux and Schützenberger for SU(p,q). - There are many other results.
Reviewer: W.Rossmann

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
22E46 Semisimple Lie groups and their representations
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