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The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a"standard basis" indexed by the elements of $W$ and a "bar involution" given bya certain antilinear map. Together, these form an example of what Webster callsa pre-canonical structure, relative to which the well-known Kazhdan-Lusztigbasis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined arepresentation of a modified Iwahori-Hecke algebra on the free$\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in$W$, and shown that this module has a unique pre-canonical structure satisfyinga certain compatibility condition, which admits its own canonical basis whichcan be viewed as a generalization of the Kazhdan-Lusztig basis. One can modifythe parameters defining Lusztig and Vogan's module to obtain otherpre-canonical structures, each of which admits a unique canonical basis indexedby twisted involutions. We classify all of the pre-canonical structures whicharise in this fashion, and explain the relationships between their resultingcanonical bases. While some of these canonical bases are related in a trivialfashion to Lusztig and Vogan's construction, others appear to have no simplerelation to what has been previously studied. Along the way, we also clarifythe differences between Webster's notion of a canonical basis and the relatedconcepts of an IC basis and a $P$-kernel.

Comparing and characterizing some constructions of canonical bases from Coxeter systems