Vogan classes in type $B_n$

Kazhdan-Lusztig left cells in type $B_n$ for intermediate parameters

“…In the present paper, we address the case s = n − 1. Using results obtained in [8], we verify the above conjectures. The key is an enhancement of D. Vogan's generalized τ -invariant [19] proposed in [3].…”

“…We show that each combinatorial left intermediate cell is either an asymptotic left cell itself or a union of two such cells. When reconciled with the results on Kazhdan-Lusztig left intermediate cells in [8], this allows us to deduce Conjecture 3.2 in this setting.…”

“…Among integral values of r, this conjecture has been verified for r = 0 in [7] and for r ≥ n − 1 in [4], the latter in particular implying that C a (w) = K a (w) for all w ∈ W n . Our work concerns r = n − 2, which we follow [8] in calling the intermediate parameter case and will refer to r-cells as intermediate cells.…”

“…For a Coxeter group W with generating reflections S, the right descent set, or the right τ -invariant, of an element w ∈ W is the set τ (w) = {s ∈ S | ℓ(ws) < ℓ(w)}. We avoid handedness and refer to it simply as the τ -invariant of w. In the type B n setting where the weight function has unequal parameters, [17] and [8] extended the τ -invariant to draw from not only simple reflections but also some of the elements of the form t j defined in Section 2.1. More precisely, It is possible read-off the enhanced τ -invariant both from the signed-permutation representation of w as well as from the domino tableaux G r (w), although in the latter case this is only straightforward for weight functions with certain parameters.…”

“…The set of non-split elements in W n admits a simple description as detailed in [8]. We reproduce it as part of the following proposition, whose proof follows directly from the definition of the map G n−1 .…”