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 where 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, 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. mentioned a special case of it slightly earlier. proved Blattner's formula in some special cases. showed that Blattner's formula gave an upper bound for the multiplicities of K-representations, proved Blattner's conjecture for groups whose symmetric space is Hermitian, and proved Blattner's conjecture for linear semisimple groups. Blattner's conjecture was also proved by by infinitesimal methods which were totally new and completely different from those of Hecht and Schmid. Part of the impetus for Enright’s paper came from several sources: from,,. In Enright multiplicity formulae are given for the so-called mock-discrete series representations also. used his ideas to obtain results on the construction and classification of irreducible Harish-Chandra modules of any real semisimple Lie algebra.
