Isometries and extra special Sylow groups of order [formula omitted

Abstract

A new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated. Using the classifications of finite simple groups and saturated fusion systems over an extra special group P of order p 3 and exponent p, we check that the conjecture is affirmatively answered for the principal blocks of finite groups with Sylow p-subgroup P.