In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.
Properties edit
If G is unimodular, an irreducible unitary representation
with v, w non-zero vectors is square-integrable on G, with respect to Haar measure.
When G is unimodular, the discrete series representation has a formal dimension d, with the property that
for v, w, x, y in the representation. When G is compact this coincides with the dimension when the Haar measure on G is normalized so that G has measure 1.
Semisimple groups edit
Harish-Chandra (1965, 1966) classified the discrete series representations of connected semisimple groups G. In particular, such a group has discrete series representations if and only if it has the same rank as a maximal compact subgroup K. In other words, a maximal torus T in K must be a Cartan subgroup in G. (This result required that the center of G be finite, ruling out groups such as the simply connected cover of SL(2,R).) It applies in particular to special linear groups; of these only SL(2,R) has a discrete series (for this, see the representation theory of SL(2,R)).
Harish-Chandra's classification of the discrete series representations of a semisimple connected Lie group is given as follows. If L is the weight lattice of the maximal torus T, a sublattice of it where t is the Lie algebra of T, then there is a discrete series representation for every vector v of
- L +
ρ ,
where
- t ⊗ C/WG.
So for each discrete series representation, there are exactly
- |WG|/|WK|
discrete series representations with the same infinitesimal character.
Harish-Chandra went on to prove an analogue for these representations of the Weyl character formula. In the case where G is not compact, the representations have infinite dimension, and the notion of character is therefore more subtle to define since it is a Schwartz distribution (represented by a locally integrable function), with singularities.
The character is given on the maximal torus T by
When G is compact this reduces to the Weyl character formula, with v =
Harish-Chandra's regularity theorem implies that the character of a discrete series representation is a locally integrable function on the group.
Limit of discrete series representations edit
Points v in the coset L +
Limit of discrete series representations are tempered representations, which means roughly that they only just fail to be discrete series representations.
Constructions of the discrete series edit
Harish-Chandra's original construction of the discrete series was not very explicit. Several authors later found more explicit realizations of the discrete series.
- Narasimhan & Okamoto (1970) constructed most of the discrete series representations in the case when the symmetric space of G is hermitian.
- Parthasarathy (1972) constructed many of the discrete series representations for arbitrary G.
- Langlands (1966) conjectured, and Schmid (1976) proved, a geometric analogue of the Borel–Bott–Weil theorem, for the discrete series, using L2 cohomology instead of the coherent sheaf cohomology used in the compact case.
- An application of the index theorem, Atiyah & Schmid (1977) constructed all the discrete series representations in spaces of harmonic spinors. Unlike most of the previous constructions of representations, the work of Atiyah and Schmid did not use Harish-Chandra's existence results in their proofs.
- Discrete series representations can also be constructed by cohomological parabolic induction using Zuckerman functors.
See also edit
References edit
- Atiyah, Michael; Schmid, Wilfried (1977), "A geometric construction of the discrete series for semisimple Lie groups", Inventiones Mathematicae, 42: 1–62, doi:10.1007/BF01389783, ISSN 0020-9910, MR 0463358, S2CID 55559836
- Bargmann, V (1947), "Irreducible unitary representations of the Lorentz group", Annals of Mathematics, Second Series, 48 (3): 568–640, doi:10.2307/1969129, ISSN 0003-486X, JSTOR 1969129, MR 0021942
- Harish-Chandra (1965), "Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions", Acta Mathematica, 113: 241–318, doi:10.1007/BF02391779, ISSN 0001-5962, 0219665
- Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", Acta Mathematica, 116: 1–111, doi:10.1007/BF02392813, ISSN 0001-5962, MR 0219666, S2CID 125806386
- Langlands, R. P. (1966), "Dimension of spaces of automorphic forms", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 253–257, MR 0212135
- Narasimhan, M. S.; Okamoto, Kiyosato (1970), "An analogue of the Borel-Weil-Bott theorem for hermitian symmetric pairs of non-compact type", Annals of Mathematics, Second Series, 91 (3): 486–511, doi:10.2307/1970635, ISSN 0003-486X, JSTOR 1970635, MR 0274657
- Parthasarathy, R. (1972), "Dirac operator and the discrete series", Annals of Mathematics, Second Series, 96 (1): 1–30, doi:10.2307/1970892, ISSN 0003-486X, JSTOR 1970892, MR 0318398
- Schmid, Wilfried (1976), "L²-cohomology and the discrete series", Annals of Mathematics, Second Series, 103 (2): 375–394, doi:10.2307/1970944, ISSN 0003-486X, JSTOR 1970944, MR 0396856
- Schmid, Wilfried (1997), "Discrete series", in Bailey, T. N.; Knapp, Anthony W. (eds.), Representation theory and automorphic forms (Edinburgh, 1996), Proc. Sympos. Pure Math., vol. 61, Providence, R.I.: American Mathematical Society, pp. 83–113, doi:10.1090/pspum/061/1476494, ISBN 978-0-8218-0609-8, MR 1476494
- A. I. Shtern (2001) [1994], "Discrete series (of representations)", Encyclopedia of Mathematics, EMS Press
External links edit
- Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic) (PDF)