Brouéʹs Conjecture Holds for Principal 3-Blocks with Elementary Abelian Defect Group of Order 9

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the derived categories of the principal p-blocks of G and of the normalizer NG(P) of P in G are equivalent. We prove in this paper that BrouéʹInage-kus conjecture holds for the principal 3-block of an arbitrary finite group G with an elementary abelian Sylow 3-subgroup P of order 9, by using initiated works for the case where G is simple, which are due to Puig, Okuyama, Waki, Miyachi, and the authors. The result depends on the classification of finite simple groups.