Erratum to: Geometric models for the spectra of certain Gelfand pairs associated with Heisenberg groups

Abstract

The paper contains an error which necessitates some revisions to the proof of our main result, Theorem 1.2. In fact, the polynomial functions Dα ∈ C[a∗] (α ∈ ), discussed in Section 2.2, need not, in general, be invariant under the little Weyl group W◦ as stated. One needs to introduce a “ -shift” (half the sum of the positive roots) to achieve W◦-invariance. Thismeans that although top( Dα) ∈ C[a∗]W◦ , the polynomial Dα itself need not lie in the image of the mapping ρ given in Eqs. 2.1 and 2.3. Thus, we cannot define polynomials Eα ∈ C[VR]K as in Definition 2.1 or obtain the related functions eL ∈ C(h∗V )K as claimed in Proposition 4.2. Lemma A.2 below provides a technical tool needed to revise the proof for Theorem 1.2. First, we require the following substitute for Lemma 2.7 from the paper. Lemma A.1 For a well-behaved multiplicity free action K :V and α, β ∈ one has