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On minuscule representations and the principal SL\(_2\). (English) Zbl 0986.22011

The article is devoted to minuscule representations of Lie groups over \(\mathbb{C}\), for example, \(SL_2\), \(SO(2,2n)\). Decompositions of the associated representation \(V_\lambda\) of the dual group \(\widehat G\) are considered when restricted to a principal \(SL_2\). This decomposition is given by the action of a Lefschetz \(SL_2\) on the cohomology of the flag variety \(X= G/P_\lambda\), where \(P_\lambda\) is the maximal parabolic subgroup of \(G\) associated to the co-weight \(\lambda\). Minuscule representations with a non-zero linear form \(t:V\to \mathbb{C}\) fixed by the principal \(SL_2\) are investigated, where the subgroup \(\widehat H\subset \widehat G\) fixing \(t\) acts irreducibly on the subspace \(V_0=\text{ker}(t)\). The rest of the paper studies representations \(\pi\) of \(G\), which are lifted from \(H\), in the sense of Langlands.

MSC:

22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
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