Broué's conjecture for isolated RoCK blocks of finite odd-dimensional orthogonal groups

Abstract

In a series of papers, we shall prove that Broué's abelian defect group conjecture is true for all blocks of finite odd-dimensional orthogonal groups ${\mathrm{SO}}_{2n+1}\left(q\right)$ at linear primes with q odd. This first paper is to prove the conjecture for isolated RoCK blocks of ${\mathrm{SO}}_{2n+1}\left(q\right)$ at odd linear primes.