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| In [[mathematics]], '''Blattner's conjecture''' or '''Blattner's formula''' is a description of the [[discrete series representation]]s of a general [[semisimple group]] ''G'' in terms of their [[restricted representation]]s to a [[maximal compact subgroup]] ''K'' (their so-called ''K''-types). Harish-Chandra orally attributed the conjecture to [[Robert J Blattner]], who did not publish it. It first appeared in print in {{harvtxt|Schmid|1968|loc=theorem 2}}, though {{harvtxt|Okamoto|Ozeki|1967}} mentioned a special case of it slightly earlier. {{harvtxt|Schmid|1972}} proved Blattner's formula in some special cases, {{harvtxt|Schmid|1975a}} showed that Blattner's formula gave an upper bound for the multiplicities of ''K''-representations, {{harvtxt|Schmid|1975b}} proved Blattner's conjecture for groups whose symmetric space is Hermitian, and {{harvtxt|Hecht|Schmid|1975}} proved Blattner's conjecture for linear semisimple groups.
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| ==Statement==
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| Blattner's formula says that if a discrete series representation with infinitesimal character λ is restricted to a maximal compact subgroup ''K'', then the representation of ''K'' with highest weight μ occurs with multiplicity
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| :<math>\sum_{w\in W}\epsilon(\omega)Q(w(\mu+\rho_c)-\lambda-\rho_n)</math>
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| where
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| :''Q'' is the number of ways a vector can be written as a sum of non-compact positive roots
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| :''W'' is the Weyl group
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| :ρ<sub>c</sub> is half the sum of the compact roots
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| :ρ<sub>n</sub> is half the sum of the non-compact roots
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| :ε is the sign character of ''W''.
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| Blattner's formula is what one gets by formally restricting the [[Harish-Chandra character formula]] 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 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<sub>2</sub> 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.
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| ==References==
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| *{{Citation | last1=Hecht | first1=Henryk | last2=Schmid | first2=Wilfried | title=A proof of Blattner's conjecture | doi=10.1007/BF01404112 | mr=0396855 | year=1975 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=31 | issue=2 | pages=129–154}}
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| *{{Citation | last1=Okamoto | first1=Kiyosato | last2=Ozeki | first2=Hideki | title=On square-integrable {{overline|∂}}-cohomology spaces attached to hermitian symmetric spaces | url=http://projecteuclid.org/euclid.ojm/1200691817 | mr=0229260 | year=1967 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=4 | pages=95–110}}
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| *{{Citation | last1=Schmid | first1=Wilfried | title=Homogeneous complex manifolds and representations of semisimple Lie groups | jstor=58599 | mr=0225930 | year=1968 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=59 | pages=56–59}}
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| *{{Citation | last1=Schmid | first1=Wilfried | title=On the realization of the discrete series of a semisimple Lie group. | mr=0277668 | year=1970 | journal=Rice University Studies | issn=0035-4996 | volume=56 | issue=2 | pages=99–108 }}
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| *{{Citation | last1=Schmid | first1=Wilfried | title=Some properties of square-integrable representations of semisimple Lie groups | jstor=1971043 | mr=0579165 | year=1975a | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=102 | issue=3 | pages=535–564}}
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| *{{Citation | last1=Schmid | first1=Wilfried | title=On the characters of the discrete series. The Hermitian symmetric case | doi=10.1007/BF01389847 | mr=0396854 | year=1975b | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=30 | issue=1 | pages=47–144}}
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Conjectures]]
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