Unimodality of the distribution of Betti numbers for some Springer fibers

In this paper, we compute the Betti numbers of the Springer fibers over nilpotent elements u∈gln(C)u∈gln(C) of Jordan forms (a,b)(a,b), (a,b,1)(a,b,1), (a,1b)(a,1b), and (22,1b)(22,1b). We show that in all these cases the distribution of Betti numbers is unimodal. In the first two cases, we also identify the type of the Springer representations on the cohomology spaces.