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). Harish-Chandra orally attributed the conjecture to Robert J 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 and Varadarajan (1975), Wallach (1976), Enright and 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 deep results on the construction and classification of irreducible Harish-Chandra modules of any real semisimple Lie algebra.

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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