Let H_n be the Hecke algebra of type B_n over Z[Q,Q^-1,q,q^-1] with parameters given by Q = q - q - q - ... - q - q (Dynkin diagram of type B_n). We describe the left cells of W(B_n) whenever Q >> q and prove that cell representations are irreducible. We also discuss some of Lusztig's conjectures on the a-function.