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.


22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
Full Text: DOI


[1] James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13 – 71. Orbites unipotentes et représentations, II. · Zbl 0728.22014
[2] N. Bourbaki, Groupes et algèbres de Lie. Hermann, Paris, 1982. · Zbl 0505.22006
[3] Pierre Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247 – 289 (French). · Zbl 0437.14012
[4] J. de Siebenthal, Sur certains sous-groupes de rang un des groupes de Lie clos. Comptes Rendus 230 (1950), 910-912. · Zbl 0036.15602
[5] Jiri Dadok and Victor Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504 – 524. · Zbl 0611.22009
[6] Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. · Zbl 0901.22001
[7] Benedict H. Gross, On the motive of \? and the principal homomorphism \?\?\(_{2}\)\to \Hat \?, Asian J. Math. 1 (1997), no. 1, 208 – 213. · Zbl 0942.20031
[8] Benedict Gross and Gordan Savin, Motives with Galois group of type \(G_2\): An exceptional theta correspondence, Compositio Math. 114 (1998), 153-217. · Zbl 0931.11015
[9] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[10] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190 – 213. · Zbl 0431.17007
[11] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1 – 155. · Zbl 0321.14030
[12] David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51 – 90. · Zbl 0692.22008
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.