Blattner's conjecture

In mathematics, Blattner's conjecture or Blattner's formula is a description of the discrete series representations of a general semisimple group G in terms of their restricted representations to a maximal compact subgroup K (their so-called K-types). It is named after Robert James Blattner, despite not being formulated as a conjecture by him.

Statement

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

$\sum _{w\in W_{K}}\epsilon (\omega )Q(w(\mu +\rho _{c})-\lambda -\rho _{n})$

where

Q is the number of ways a vector can be written as a sum of non-compact positive roots

W_K is the Weyl group of K

ρ_c is half the sum of the compact roots

ρ_n is half the sum of the non-compact roots

ε is the sign character of W_K.

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 SL₂ 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.

History

Harish-Chandra orally attributed the conjecture to Robert James Blattner 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 Schmid (1968, 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. Okamoto & Ozeki (1967) mentioned a special case of it slightly earlier.

Schmid (1972) proved Blattner's formula in some special cases. Schmid (1975a) showed that Blattner's formula gave an upper bound for the multiplicities of K-representations, Schmid (1975b) proved Blattner's conjecture for groups whose symmetric space is Hermitian, and Hecht & Schmid (1975) proved Blattner's conjecture for linear semisimple groups. Blattner's conjecture (formula) was also proved by Enright (1979) 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 Enright & Varadarajan (1975), Wallach (1976), Enright & Wallach (1978). In Enright (1979) multiplicity formulae are given for the so-called mock-discrete series representations also. Enright (1978) used his ideas to obtain results on the construction and classification of irreducible Harish-Chandra modules of any real semisimple Lie algebra.

References

• Enright, Thomas J; Varadarajan, V. S. (1975), "On an infinitesimal characterization of the discrete series.", Annals of Mathematics, 102 (1): 1–15, doi:10.2307/1970970, JSTOR 1970970, MR 0476921
• Enright, Thomas J; Wallach, Nolan R (1978), "The fundamental series of representations of a real semisimple Lie algebra", Acta Mathematica, 140 (1–2): 1–32, doi:10.1007/bf02392301, MR 0476814
• Enright, Thomas J (1978), "On the algebraic construction and classification of Harish-Chandra modules", Proceedings of the National Academy of Sciences of the United States of America, 75 (3): 1063–1065, Bibcode:1978PNAS...75.1063E, doi:10.1073/pnas.75.3.1063, MR 0480871, PMC 411407, PMID 16592507
• Enright, Thomas J (1979), "On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae", Annals of Mathematics, 110 (1): 1–82, doi:10.2307/1971244, JSTOR 1971244, MR 0541329
• Hecht, Henryk; Schmid, Wilfried (1975), "A proof of Blattner's conjecture", Inventiones Mathematicae, 31 (2): 129–154, doi:10.1007/BF01404112, ISSN 0020-9910, MR 0396855, S2CID 123048659
• Okamoto, Kiyosato; Ozeki, Hideki (1967), "On square-integrable ∂-cohomology spaces attached to hermitian symmetric spaces", Osaka Journal of Mathematics, 4: 95–110, ISSN 0030-6126, MR 0229260
• Schmid, Wilfried (1968), "Homogeneous complex manifolds and representations of semisimple Lie groups", Proceedings of the National Academy of Sciences of the United States of America, 59 (1): 56–59, Bibcode:1968PNAS...59...56S, doi:10.1073/pnas.59.1.56, ISSN 0027-8424, JSTOR 58599, MR 0225930, PMC 286000, PMID 16591593
• Schmid, Wilfried (1970), "On the realization of the discrete series of a semisimple Lie group.", Rice University Studies, 56 (2): 99–108, ISSN 0035-4996, MR 0277668
• Schmid, Wilfried (1975a), "Some properties of square-integrable representations of semisimple Lie groups", Annals of Mathematics, Second Series, 102 (3): 535–564, doi:10.2307/1971043, ISSN 0003-486X, JSTOR 1971043, MR 0579165
• Schmid, Wilfried (1975b), "On the characters of the discrete series. The Hermitian symmetric case", Inventiones Mathematicae, 30 (1): 47–144, Bibcode:1975InMat..30...47S, doi:10.1007/BF01389847, ISSN 0020-9910, MR 0396854, S2CID 120935812
• Wallach, Nolan R (1976), "On the Enright-Varadarajan modules: a construction of the discrete series", Annales Scientifiques de l'École Normale Supérieure, 4 (1): 81–101, doi:10.24033/asens.1304, MR 0422518