Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group.
Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense. Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction.
Over finite fields, the unipotent representations are those that occur in the decomposition of the Deligne–Lusztig characters R1
T of the trivial representation 1 of a torus T . They were classified by Lusztig (1978, 1979). Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the Steinberg representation, and θ10.
Non-archimedean local fields
Lusztig (1995) classified the unipotent characters over non-archimedean local fields.
Archimedean local fields
Vogan (1987) discusses several different possible definitions of unipotent representations of real Lie groups.
- Barbasch, Dan (1991), "Unipotent representations for real reductive groups", in Satake, Ichirô (ed.), Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990), Tokyo: Math. Soc. Japan, pp. 769–777, ISBN 978-4-431-70047-0, MR 1159263
- Lusztig, George (1979), "Unipotent representations of a finite Chevalley group of type E8", The Quarterly Journal of Mathematics. Oxford. Second Series, 30 (3): 315–338, doi:10.1093/qmath/30.3.315, ISSN 0033-5606, MR 0545068
- Lusztig, George (1978), Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1689-9, MR 0518617
- Lusztig, George (1995), "Classification of unipotent representations of simple p-adic groups", International Mathematics Research Notices (11): 517–589, arXiv:math/0111248, doi:10.1155/S1073792895000353, ISSN 1073-7928, MR 1369407
- Vogan, David A. (1987), Unitary representations of reductive Lie groups, Annals of Mathematics Studies, 118, Princeton University Press, ISBN 978-0-691-08482-4