## Discrete series representation

Type of group representation for locally compact groups

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 ρ of *G* is in the discrete series if and only if one (and hence all) matrix coefficient

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 ρ is the Weyl vector of *G*, that is not orthogonal to any root of *G*. Every discrete series representation occurs in this way. Two such vectors *v* correspond to the same discrete series representation if and only if they are conjugate under the Weyl group*W*_{K} of the maximal compact subgroup *K*. If we fix a fundamental chamber for the Weyl group of *K*, then the discrete series representation are in 1:1 correspondence with the vectors of *L* + ρ in this Weyl chamber that are not orthogonal to any root of *G*. The infinitesimal character of the highest weight representation is given by *v* (mod the Weyl group *W*_{G} of *G*) under 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 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* = *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[edit]

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[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.

### 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 - Bargmann, V (1947), "Irreducible unitary representations of the Lorentz group",
*Annals of Mathematics*, Second Series,**48**: 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 - 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**: 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**: 1–30, 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.,**61**, Providence, R.I.: American Mathematical Society, pp. 83–113, doi:10.1090/pspum/061/1476494, ISBN , MR 1476494 - A.I. Shtern (2001) [1994], "Discrete series of representation",
*Encyclopedia of Mathematics*, EMS Press

### External links[edit]

## Restriction of discrete series representations

I'll give some fairly general remarks - too long for a comment.

I assume $G$ is reductive and $P$ parabolic. The Mackey Restriction Induction formula yields $$ Res_{H} Ind_{P}^G \pi = \int\limits_{\gamma \in H \backslash G / P} Ind_{H \cap \gamma P \gamma^{-1}}^H \pi^\gamma \; d(\gamma)$$ Since $G/P$ is compact, the integral is over a compact space.

The Casselman submodule theorem tells you that all irreducible smooth admissible representations are obtained as submodule of $Ind_P^G \pi$. When you apply the Mackey induction restriction formula for representation for the compact subgroups, you at least obtain partial information by substraction of the $K$-types.

If you want to know whether $Ind_{H \cap \gamma P \gamma^{-1}}^H \pi^\gamma$ is irreducible, you can use Frobenius reciprocity, Mackey restriction formula and so on.

For normal subgroups, the Mackey machine describes the full unitary dual. Here you need that $H$ is reductive (or more exactly type I).

Other than that, I recall a book called "Dirac Operators in Representation Theory", where the author describes a bunch of "branching laws" (=key word). From my experience with small rank classical (mostly compact) groups, you can read it off sometimes from the Weyl character formula.

Also you might want to think about finite groups, which makes it clear that no nice things should be expected in general.

answered Jul 18 '12 at 16:54

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## Discrete series (of representations)

The family of continuous irreducible unitary representations of a locally compact group $ G $ which are equivalent to the subrepresentations of the regular representation of this group. If the group $ G $ is unimodular, then a continuous irreducible unitary representation $ \pi $ of $ G $ belongs to the discrete series if and only if the matrix entries of $ \pi $ lie in $ L _ {2} ( G) $. In such a case there exists a positive number $ d _ \pi $, known as the formal degree of the representation $ \pi $, such that the relations

$$ \tag{1 } \int\limits _ { G } ( \pi ( g) \xi , \eta ) ( {\pi ( g) \xi ^ \prime , \eta ^ \prime } bar ) dg = d _ \pi ^ {-} 1 ( \xi , \xi ^ \prime ) ( {\eta , \eta ^ \prime } bar ); $$

$$ \tag{2 } ( \pi ( g) \xi , \eta ) * ( \pi ( g) \xi ^ \prime , \eta ^ \prime ) = d _ \pi ^ {-} 1 ( \xi , \eta ^ \prime ) ( \pi ( g) \xi ^ \prime , \eta ) , $$

are satisfied for all vectors $ \xi , \eta , \xi ^ \prime , \eta ^ \prime $ of the space $ H _ \pi $ of the representation $ \pi $. If $ \pi _ {1} $ and $ \pi _ {2} $ are two non-equivalent representations of $ G $ in the spaces $ H _ {1} $ and $ H _ {2} $, respectively, which belong to the discrete series, then the relations

$$ \tag{3 } \int\limits _ { G } ( \pi _ {1} ( g) \xi _ {1} \eta _ {1} ) ( {\pi _ {2} ( g) \xi _ {2} \eta _ {2} } bar ) dg = 0 , $$

$$ \tag{4 } ( \pi _ {1} ( g) \xi _ {1} , \eta _ {1} ) * ( \pi _ {2} ( g) \xi _ {2} , \eta _ {2} ) = 0 , $$

are valid for all $ \xi _ {1} , \eta _ {1} \in H _ {1} $, $ \xi _ {2} , \eta _ {2} \in H _ {2} $. The relations (1)–(4) are generalizations of the orthogonality relations for the matrix entries of representations of compact topological groups (cf. Representation of a compact group); the group $ G $ is compact if and only if all continuous irreducible unitary representations of $ G $ belong to the discrete series, and if $ G $ is compact and the Haar measure $ dg $ satisfies the condition $ \int _ {G} dg = 1 $, then the number $ d _ \pi $ coincides with the dimension of the representation $ \pi $. Simply-connected nilpotent real Lie groups and complex semi-simple Lie groups have no discrete series.

The equivalence class of a representation $ \pi $ forming part of the discrete series is a closed point in the dual space $ \widehat{G} $ of the group $ G $, and the Plancherel measure of this point coincides with the formal degree $ d _ \pi $; if, in addition, some non-zero matrix entry of the representation $ \pi $ is summable, the representation $ \pi $ is an open point in the support of the regular representation of $ G $, but open points in $ \widehat{G} $ need not correspond to representations of the discrete series. The properties of discrete series representations may be partly extended to the case of non-unimodular locally compact groups.

#### References

[1] | J. Dixmier, " algebras" , North-Holland (1977) (Translated from French) |

[2a] | Harish-Chandra, "Discrete series for semisimple Lie groups I" Acta Math. , 113 (1965) pp. 241–318 |

[2b] | Harish-Chandra, "Discrete series for semisimple Lie groups II" Acta Math. , 116 (1966) pp. 1–111 |

[3] | W. Schmid, "-cohomology and the discrete series" Ann. of Math. , 103 (1976) pp. 375–394 |

[4a] | A. Kleppner, R. Lipsman, "The Plancherel formula for group extensions" Ann. Sci. Ecole Norm. Sup. , 5 (1972) pp. 459–516 |

[4b] | A. Kleppner, R. Lipsman, "The Plancherel formula for group extensions II" Ann. Sci. Ecole Norm. Sup. , 6 (1973) pp. 103–132 |

#### Comments

Especially for a semi-simple Lie group the representations belonging to the discrete series of the group or of some of its subgroups play an essential role in the harmonic analysis on the group.

#### References

[a1] | V.S. Varadarajan, "Harmonic analysis on real reductive groups" , Springer (1977) |

**How to Cite This Entry:**

Discrete series (of representations).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Discrete_series_(of_representations)&oldid=46736

## Holomorphic discrete series representation

Representation of semisimple Lie groups

In mathematics, a **holomorphic discrete series representation** is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian. Holomorphic discrete series representations are the easiest discrete series representations to study because they have highest or lowest weights, which makes their behavior similar to that of finite-dimensional representations of compact Lie groups.

Bargmann (1947) found the first examples of holomorphic discrete series representations, and Harish-Chandra (1954, 1955a, 1955c, 1956a, 1956b) classified them for all semisimple Lie groups.

Martens (1975) and Hecht (1976) described the characters of holomorphic discrete series representations.

### See also[edit]

### References[edit]

- 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 (1954), "Representations of semisimple Lie groups. VI",
*Proceedings of the National Academy of Sciences of the United States of America*,**40**(11): 1078–1080, doi:10.1073/pnas.40.11.1078, ISSN 0027-8424, JSTOR 89268, MR 0064780, PMC 1063968, PMID 16578441 - Harish-Chandra (1955a), "Integrable and square-integrable representations of a semisimple Lie group"(PDF),
*Proceedings of the National Academy of Sciences of the United States of America*,**41**(5): 314–317, doi:10.1073/pnas.41.5.314, ISSN 0027-8424, JSTOR 89123, MR 0070957, PMC 528085, PMID 16589671 - Harish-Chandra (1955c), "Representations of semisimple Lie groups. IV",
*American Journal of Mathematics*,**77**(4): 743–777, doi:10.2307/2372596, ISSN 0002-9327, JSTOR 2372596, MR 0072427 - Harish-Chandra (1956a), "Representations of semisimple Lie groups. V",
*American Journal of Mathematics*,**78**(11): 1–41, doi:10.2307/2372481, ISSN 0002-9327, JSTOR 2372481, MR 0082055, PMC 1063967, PMID 16578440 - Harish-Chandra (1956b), "Representations of semisimple Lie groups. VI. Integrable and square-integrable representations",
*American Journal of Mathematics*,**78**(3): 564–628, doi:10.2307/2372674, ISSN 0002-9327, JSTOR 2372674, MR 0082056 - Hecht, Henryk (1976), "The characters of some representations of Harish-Chandra",
*Mathematische Annalen*,**219**(3): 213–226, doi:10.1007/BF01354284, ISSN 0025-5831, MR 0427542, S2CID 120850258 - Martens, Susan (1975), "The characters of the holomorphic discrete series",
*Proceedings of the National Academy of Sciences of the United States of America*,**72**(9): 3275–3276, doi:10.1073/pnas.72.9.3275, ISSN 0027-8424, JSTOR 65377, MR 0419687, PMC 432971, PMID 16592271

### External links[edit]

## Representation discrete series

.

DSP Lecture-19: Introduction to Discrete Fourier Series (DFS).

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