The second lowest two-sided cell in the affine Weyl group (B)over-tilden

作者：Shi, Jian-yi^*

来源：Journal of Combinatorial Theory - Series A, 2023, 196: 105736.

DOI：10.1016/j.jcta.2023.105736

• The affine Weyl group of type (B)over-tilde(n)
• The second lowest two-sided cell
• Sign types
• Left-connectedness

Let (W-a, S-a) be an irreducible affine Weyl group with W-0 the associated Weyl group. In my previous paper [24], I proved that the number n(qr) of left cells of W-a in the second lowest two-sided cell Omega(qr) satisfies the inequality n(qr) <= 1/2 |W-0| and conjectured that the equality should actually hold. In the present paper, we verify the conjecture when W-a = (B) over tilde (n); we prove that any left cell of (B) over tilde (n) in Omega(qr) is left-connected, verifying a conjecture of Lusztig in our case; we show that Omega(qr) consists of all the extensions of w(J) in the set is the lowest two-sided cell of (B) over tilde (n) - W-(nu), where W-(nu) is the lowest two-sided cell of (B) over tilde (n) and J subset of S-a is such that w(J) is the longest element in the subgroup W-J of (B) over tilde (n) of type D-n.