Lusztig $\mathbf{a}$-functions for quasiparabolic sets

Kazhdan and Lusztig introduce the $W$-graphs to describe the cells and molecules corresponding to the Coxeter groups. Building on this foundation, Lusztig defines the a-funtion to classify the cells, as well as the molecules. Marberg then generalizes Kazhdan and Lusztig's $W$-graphs, using fixed-point-free involutions as their indices. The molecules of the two new $S_n$-graphs are then classified via two correspondence similar to RSK correspondence by Marberg and me. In this paper, we define an analogue of the Lusztig a-function and finish the classification of cells by proving that every molecule in the $S_n$-graphs is indeed a cell.