On Blattner's formula for the discrete series representations of SO(2n, 1)

Blattner's conjecture gives a formula for the multiplicity with which a unitary irreducible representation of the maximal compact subgroup K appears in any discrete series representation of a semisimple Lie group G. We give an elementary derivation of this formula from Harish Chandra's character formula for G ～- SO_e(2n, 1) (n ? 2). The idea is to regularize the character (on the Cartan subgroup), to show that the regularization is unique, and to derive the multiplicities by expanding the resulting distribution in a Fourier series. To prove uniqueness of the regularization one uses a priori constraints on the multiplicities (=Fourier coefficients) that follow from the subquotient theorem.