Discrete series representation

In

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

If G is

unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one (and hence all) matrix coefficient

\langle \rho (g)\cdot v,w\rangle \,

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

d\int \langle \rho (g)\cdot v,w\rangle {\overline {\langle \rho (g)\cdot x,y\rangle }}dg=\langle v,x\rangle {\overline {\langle w,y\rangle }}

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.

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 ρ is the

Harish-Chandra correspondence

identifying infinitesimal characters of G with points of

t ⊗ C/W_G.

So for each discrete series representation, there are exactly

|W_G|/|W_K|

discrete series representations with the same infinitesimal character.

Harish-Chandra went on to prove an analogue for these representations of the

Schwartz distribution

(represented by a locally integrable function), with singularities.

The character is given on the maximal torus T by

(-1)^{\frac {\dim(G)-\dim(K)}{2}}{\sum _{w\in W_{K}}\det(w)e^{w(v)} \over \prod _{(v,\alpha )>0}\left(e^{\frac {\alpha }{2}}-e^{-{\frac {\alpha }{2}}}\right)}

When G is compact this reduces to the Weyl character formula, with v = λ + ρ for λ the highest weight of the irreducible representation (where the product is over roots α having positive inner product with the vector 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

Points v in the coset L + ρ orthogonal to roots of G do not correspond to discrete series representations, but those not orthogonal to roots of K are related to certain irreducible representations called limit of discrete series representations. There is such a representation for every pair (v,C) where v is a vector of L + ρ orthogonal to some root of G but not orthogonal to any root of K corresponding to a wall of C, and C is a Weyl chamber of G containing v. (In the case of discrete series representations there is only one Weyl chamber containing v so it is not necessary to include it explicitly.) Two pairs (v,C) give the same limit of discrete series representation if and only if they are conjugate under the Weyl group of K. Just as for discrete series representations v gives the infinitesimal character. There are at most |W_G|/|W_K| limit of discrete series representations with any given infinitesimal character.

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

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.
L² cohomology
instead of the coherent sheaf cohomology used in the compact case.
An application of the
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
.

References

S2CID 55559836

MR 0021942

Harish-Chandra (1965), "Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions", ISSN 0001-5962
, 0219665

Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", 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.:
MR 0212135

Narasimhan, M. S.; Okamoto, Kiyosato (1970), "An analogue of the Borel-Weil-Bott theorem for hermitian symmetric pairs of non-compact type", MR 0274657

Parthasarathy, R. (1972), "Dirac operator and the discrete series", MR 0318398

Schmid, Wilfried (1976), "L²-cohomology and the discrete series", 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.:
MR 1476494

A. I. Shtern (2001) [1994], "Discrete series (of representations)", Encyclopedia of Mathematics, EMS Press

External links

Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic) (PDF)

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Discrete series representation

Properties

Semisimple groups

Limit of discrete series representations

Constructions of the discrete series

See also

References

External links