Before revision:  Based on this SLP, we propose a self-index data structure of O(g) words of space that can be built from a string T in O(n + g \lg g) time, retrieving the starting positions of all occurrences of a pattern P of length m in O(m + \lg m \lg n + occ \lg g) time, where n is the length of T, g is the size of the Lyndon SLP for T, and occ is the number of occurrences of P in T.
After revision:  Based on this SLP, we propose a self-index data structure of O(g) words of space that can be built from a string T in O(n \lg n) expected time, retrieving the starting positions of all occurrences of a pattern P of length m in O(m + \lg m \lg n + occ \lg g) time, where n is the length of T, g is the size of the Lyndon SLP for T, and occ is the number of occurrences of P in T.