Branching laws for square integrable representations. (English) Zbl 1196.22009

The authors study square integrable representations of a real reductive Lie group \(G\). They show that for a square integrable representation, its restriction to a closed connected reductive subgroup \(H\) is admissible if and only if the restriction to the intersection \(L\) of \(H\) with a maximal compact subgroup is admissible. This gives a strategy for the computation of the multiplicities in the restriction to \(H\) for the admissible case via the multiplicities in the restriction to \(L\). The authors give nice formulae (in terms of partition functions) for the multiplicities in the restriction to \(L\) under some natural additional assumptions. Finally, the authors also consider the semi-classical analogue of these results for coadjoint orbits.


22E46 Semisimple Lie groups and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text: DOI


[1] N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators , Corrected reprint of the 1992 original, Springer, Berlin, 2004. · Zbl 1037.58015
[2] M. Duflo, G. Heckman and M. Vergne, Projection d’orbites, formule de Kirillov et formule de Blattner, Mém. Soc. Math. France (N.S.) No. 15 (1984), 65-128. · Zbl 0575.22014
[3] B. Gross and N. Wallach, Restriction of small discrete series representations to symmetric subgroups, in The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) , 255-272, Proc. Sympos. Pure Math., 68, Amer. Math. Soc., Providence, RI. · Zbl 0960.22008
[4] H. Hecht and W. Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129-154. · Zbl 0319.22012
[5] G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), no. 2, 333-356. · Zbl 0497.22006
[6] T. Kobayashi, The restriction of \(A_q(\lambda)\) to reductive subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 262-267. · Zbl 0826.22014
[7] T. Kobayashi, Discrete decomposability of the restriction of \(A_{{\mathfrak{q}}(\lambda)} with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), no. 2, 181-205.\) · Zbl 0826.22015
[8] T. Kobayashi, Multiplicity free theorems in branching problems of unitary highest weight modules, Proc. of the Symposium on Repesentation Theory held at Saga, Kyushu (eds. K. Mimachi), (1997), 9-17.
[9] T. Kobayashi, Discrete decomposability of the restriction of \(A_{{\mathfrak{q}}(\lambda)} with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229-256.\) · Zbl 0907.22016
[10] T. Kobayashi, Discrete decomposability of the restriction of \(A_{{\mathfrak{q}}(\lambda)} with respect to reductive subgroups. II. micro local analysis and asymptotic K -support, Annals of Math. 147 (1998), 709-729.\) · Zbl 0910.22016
[11] T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), no. 1, 100-135. · Zbl 0937.22008
[12] T. Kobayashi, Branching problems of unitary representations, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) , 615-627, Higher Ed. Press, Beijing. · Zbl 1008.43009
[13] T. Kobayashi, Restrictions of unitary representations of real reductive groups, in Lie theory , 139-207, Progr. Math., 229, Birkhäuser, Boston, Boston, MA. · Zbl 1072.22008
[14] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 61-80. · Zbl 0219.32013
[15] A. Szenes and M. Vergne, Residue formulae for vector partitions and Euler-MacLaurin sums, Adv. in Appl. Math. 30 (2003), no. 1-2, 295-342. · Zbl 1067.52014
[16] J. Vargas, Admissible restriction of holomorphic discrete series for exceptional groups, Rev. Un. Mat. Argentina 50 (2009), no. 1, 67-80. · Zbl 1282.22009
[17] A. Weinstein, Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Lett. Math. Phys. 56 (2001), no. 1, 17-30. · Zbl 1010.53060
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.