On Certain Distinguished Involutions in the Weyl Group of Type Bn

Let (W, S) be a Weyl group of type B-n. In this paper, we find certain nontrivial distinguished involutions in the two-sided cell Omega(t) of W with a-value 1/2 (n(2) - n + 4t(2) + 2t) for 1 <= 2t <= n and n even (resp., 1/2 (n(2) - n + 4t(2) - 2t) for 1 <= 2t < n and n odd). Besides, for 1 <= i(1) < ... < i(t) <= n - 1, let L-i1 ... (it) be a left cell of W with a-value t(2) and R(L-i1 ... (it)) = {s(i1), ... , s(it)}: There are only two involutions y(i(1), i(1)) ... y(i(t-1), i(t-1))s(it) and y(i(1), i(1)) ... y(i(t), i(t)) in the left cell L-i1 ... (it). We prove that the former (resp., latter) is a distinguished involution in the left cell L-it ... (it) for t odd (resp., even).