On domino insertion and Kazhdan--Lusztig cells in type B_n

Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnaf\'e and Iancu obtained earlier in the so-called ``asymptotic case''. We give some partial results in support of our conjectures.