  In this paper we prove that Brou\'{e}'s abelian defect group conjecture is
true for the finite odd-dimensional orthogonal groups $\SO_{2n+1}(q)$ at linear
primes with $q$ odd. We first make use of the reduction theorem of
Bonnaf\'{e}-Dat-Rouquier to reduce the problem to isolated blocks. Then we
construct a categorical action of a Kac-Moody algebra on the category of
quadratic unipotent representations of the various groups $\SO_{2n+1}(q)$ in
non-defining characteristic, by generalizing the corresponding work of
Dudas-Varagnolo-Vasserot for unipotent representations. This is one of the main
ingredients of our work which may be of independent interest. To obtain derived
equivalences of blocks and their Brauer correspondents, we define and
investigate isolated RoCK blocks. Finally, we establish the desired derived
equivalence based on the work of Chuang-Rouquier that categorical actions
provide derived equivalences between certain weight spaces.
