Erratum and Addendum to: Invariant Differential Operators and Eigenspace Representations on an Affine Symmetric Space

TL;DR

This paper provides corrections and updates to previous work on invariant differential operators and eigenspace representations on affine symmetric spaces, ensuring the main results remain valid while introducing new proofs and definitions.

Contribution

It introduces a new proof of existence, a new bijection, and revised proofs for key theorems, correcting earlier errors without altering the main results.

Findings

Main results of the original paper remain valid.

New proof of the existence of λ(ξ) without Proposition 7.2.

A new bijection and proof technique for Theorem 5.2.

Abstract

The purpose of this erratum and addendum is to correct the errors in [1]. It consists of five components: 1. Lemma 7.1 and Proposition 7.2 are wrong and discarded; 2. A new proof of existence $\lambda(\xi)$ in (7.1) without Proposition 7.2; 3. Definition of a new bijection in Theorem 5.2 and a proof by a new technique; 4. A new proof of Theorem 5.5 based on the new bijection in Theorem 5.2; 5. Correction to the list of exceptional simple pairs in Proposition 3.1. The main results of [1] remain true as stated. We also add a final remark on generalization.