Casian, Luis G. Graded characters of induced representations. II: Classification of principal series modules for complex groups. (English) Zbl 0734.22003 J. Algebra 137, No. 2, 369-387 (1991). Let G be a connected complex reductive algebraic group which is viewed as a real Lie group by taking its complex points and then ignoring its complex structure. The corresponding real Lie group is denoted by \(G_{{\mathbb{R}}}\) and its complexification by \(G_{{\mathbb{C}}}\). A similar convention is used for Lie algebras. Let W be the Weyl group of \({\mathfrak g}=Lie G\) and S a set of simple reflections in W. The pair (W,S) generates a braid group B. The Weyl group \(W_{{\mathbb{C}}}\) of \({\mathfrak g}_{{\mathbb{C}}}\) is isomorphic to \(W\times W.\) Let \({\mathfrak h}_{{\mathbb{C}}}\) be a Cartan subalgebra of \({\mathfrak g}_{{\mathbb{C}}}\) and \(\rho\) equals half the sum of the positive roots of (\({\mathfrak g}_{{\mathbb{C}}},{\mathfrak h}_{{\mathbb{C}}})\). Let P be a Borel subgroup of G. The paper reviewed treats the induced Harish-Chandra modules \(Ind^ G_ P(-w\rho +\rho)\), \(w\in W_{{\mathbb{C}}}\). The main result is a criterion for two such modules being isomorphic. The criterion consists in coincidence of some two elements in B. It can be also formulated in terms of the graded characters [see part I; J. Algebra 123, 289-326 (1989; Zbl 0688.22002)]. This result gives a classification of principal series modules for \(G_{{\mathbb{R}}}\) with integral regular infinitesimal characters (up to the isomorphism of Harish-Chandra modules). Reviewer: V.F.Molchanov (Tambov) Cited in 1 ReviewCited in 1 Document MSC: 22E30 Analysis on real and complex Lie groups 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 20D30 Series and lattices of subgroups 22E46 Semisimple Lie groups and their representations 22E15 General properties and structure of real Lie groups Keywords:connected complex reductive algebraic group; real Lie group; Weyl group; simple reflections; Cartan subalgebra; positive roots; induced Harish- Chandra modules; graded characters; principal series modules; integral regular infinitesimal characters Citations:Zbl 0688.22002 PDFBibTeX XMLCite \textit{L. G. Casian}, J. Algebra 137, No. 2, 369--387 (1991; Zbl 0734.22003) Full Text: DOI References: [1] Beilinson, A.; Bernstein, J., Localisation de g-modules, C.R. Acad. Sci. Paris, 292, 15-18 (1981) · Zbl 0476.14019 [2] Beilinson, A.; Bernstein, J.; Deligne, P., Faisceaux perverse, Ast., 100 (1982) · Zbl 0536.14011 [3] Bien, F., Spherical \(D\)-Modules and Representations of Reductive Lie Groups, (Thesis (1986), MIT: MIT Cambridge, MA) [4] Casian, L., Graded characters of induced representations for real reductive Lie groups, I, J. Algebra, 123, 289-326 (1989) · Zbl 0688.22002 [6] Casian, L.; Collingwood, D., Weight filtrations for induced representations of real reductive Lie groups, Adv. in Math., 73, No. 1, 79-146 (Jan. 1989) [7] Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 434 (1983) [8] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184 (1979) · Zbl 0499.20035 [9] Lusztig, G.; Vogan, D., Singularities of closures of \(K\)-orbits on flag manifolds, Invent. Math., 71, 365-379 (1983) · Zbl 0544.14035 [10] Vogan, D., Irreducible characters of semisimple Lie groups, III, Proof of the Kazhdan-Lusztig conjectures in the integral case, Invent. Math., 71, 381-417 (1983) · Zbl 0505.22016 [11] Schmid, W., Vanishing theorems for Lie algebra cohomology and the cohomology of discrete subgroups of semisimple Lie groups, Adv. in Math., 41, 78-113 (1981) · Zbl 0472.22003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.