231:
for a discrete series representation to the maximal torus of a maximal compact group. The problem in proving the
Blattner formula is that this only gives the character on the regular elements of the maximal torus, and one also needs to control its behavior on the singular elements. For non-discrete
236:
the character is identically zero on the non-singular elements of the maximal compact subgroup, but the representation is not zero on this subgroup. In this case the character is a distribution on the maximal compact subgroup with support on the singular elements.
253:, theorem 2), where it was first referred to as "Blattner's Conjecture," despite the results of that paper having been obtained without knowledge of Blattner's question and notwithstanding Blattner's not having made such a conjecture.
180:
232:
irreducible representations the formal restriction of Harish-Chandra's character formula need not give the decomposition under the maximal compact subgroup: for example, for the principal series representations of SL
279:
by infinitesimal methods which were totally new and completely different from those of Hecht and Schmid (1975). Part of the impetus for
Enright’s paper (1979) came from several sources: from
761:
80:
437:
Enright, Thomas J (1979), "On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae",
559:
381:
66:
Blattner's formula says that if a discrete series representation with infinitesimal character λ is restricted to a maximal compact subgroup
800:
249:
as a question
Blattner raised, not a conjecture made by Blattner. Blattner did not publish it in any form. It first appeared in print in
347:
Enright, Thomas J; Wallach, Nolan R (1978), "The fundamental series of representations of a real semisimple Lie algebra",
228:
291:. In Enright (1979) multiplicity formulae are given for the so-called mock-discrete series representations also.
29:
309:
Enright, Thomas J; Varadarajan, V. S. (1975), "On an infinitesimal characterization of the discrete series.",
275:
proved
Blattner's conjecture for linear semisimple groups. Blattner's conjecture (formula) was also proved by
659:
Schmid, Wilfried (1975a), "Some properties of square-integrable representations of semisimple Lie groups",
379:
Enright, Thomas J (1978), "On the algebraic construction and classification of Harish-Chandra modules",
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Schmid, Wilfried (1968), "Homogeneous complex manifolds and representations of semisimple Lie groups",
759:
Wallach, Nolan R (1976), "On the
Enright-Varadarajan modules: a construction of the discrete series",
40:
705:
Schmid, Wilfried (1975b), "On the characters of the discrete series. The
Hermitian symmetric case",
707:
477:
44:
631:
Schmid, Wilfried (1970), "On the realization of the discrete series of a semisimple Lie group.",
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55:
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175:{\displaystyle \sum _{w\in W_{K}}\epsilon (\omega )Q(w(\mu +\rho _{c})-\lambda -\rho _{n})}
8:
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used his ideas to obtain results on the construction and classification of irreducible
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is the number of ways a vector can be written as a sum of non-compact positive roots
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proved
Blattner's conjecture for groups whose symmetric space is Hermitian, and
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Proceedings of the
National Academy of Sciences of the United States of America
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Proceedings of the
National Academy of Sciences of the United States of America
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showed that
Blattner's formula gave an upper bound for the multiplicities of
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Hecht, Henryk; Schmid, Wilfried (1975), "A proof of Blattner's conjecture",
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Blattner's formula is what one gets by formally restricting the
527:-cohomology spaces attached to hermitian symmetric spaces"
58:, despite not being formulated as a conjecture by him.
83:
762:
Annales Scientifiques de l'École Normale Supérieure
245:Harish-Chandra orally attributed the conjecture to
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257:mentioned a special case of it slightly earlier.
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74:with highest weight μ occurs with multiplicity
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521:Okamoto, Kiyosato; Ozeki, Hideki (1967),
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216:is half the sum of the non-compact roots
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299:of any real semisimple Lie algebra.
209:is half the sum of the compact roots
801:Representation theory of Lie groups
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281:Enright & Varadarajan (1975)
229:Harish-Chandra character formula
30:discrete series representations
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70:, then the representation of
531:Osaka Journal of Mathematics
289:Enright & Wallach (1978)
219:ε is the sign character of W
61:
7:
54:-types). It is named after
10:
822:
255:Okamoto & Ozeki (1967)
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41:restricted representations
273:Hecht & Schmid (1975)
708:Inventiones Mathematicae
478:Inventiones Mathematicae
45:maximal compact subgroup
28:is a description of the
633:Rice University Studies
523:"On square-integrable
404:10.1073/pnas.75.3.1063
297:Harish-Chandra modules
176:
662:Annals of Mathematics
440:Annals of Mathematics
312:Annals of Mathematics
247:Robert James Blattner
199:is the Weyl group of
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56:Robert James Blattner
22:Blattner's conjecture
582:10.1073/pnas.59.1.56
81:
776:10.24033/asens.1304
721:1975InMat..30...47S
573:1968PNAS...59...56S
395:1978PNAS...75.1063E
729:10.1007/BF01389847
491:10.1007/BF01404112
364:10.1007/bf02392301
267:-representations,
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39:in terms of their
26:Blattner's formula
665:, Second Series,
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50:(their so-called
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389:(3): 1063–1065,
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32:of a general
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806:Conjectures
447:(1): 1–82,
319:(1): 1–15,
18:mathematics
795:Categories
537:: 95–110,
303:References
753:120935812
737:0020-9910
683:0003-486X
645:0035-4996
591:0027-8424
543:0030-6126
515:123048659
499:0020-9910
161:ρ
157:−
154:λ
151:−
139:ρ
132:μ
114:ω
108:ϵ
93:∈
86:∑
62:Statement
625:16591593
431:16592507
785:0422518
745:0396854
717:Bibcode
699:0579165
691:1971043
653:0277668
607:0225930
569:Bibcode
551:0229260
507:0396855
469:0541329
461:1971244
413:0480871
391:Bibcode
373:0476814
341:0476921
333:1970970
241:History
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599:58599
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457:JSTOR
329:JSTOR
43:to a
733:ISSN
679:ISSN
641:ISSN
621:PMID
587:ISSN
539:ISSN
495:ISSN
427:PMID
771:doi
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671:doi
667:102
611:PMC
577:doi
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417:PMC
399:doi
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321:doi
317:102
24:or
16:In
797::
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773::
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535:4
525:∂
489::
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393::
361::
323::
265:K
234:2
223:.
221:K
214:n
212:ρ
207:c
205:ρ
201:K
197:K
195:W
190:Q
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165:n
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143:c
135:+
129:(
126:w
123:(
120:Q
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111:(
101:K
97:W
90:w
72:K
68:K
52:K
48:K
37:G
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