Comparing and characterizing some constructions of canonical bases from Coxeter systems

Abstract

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 by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan–Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan defined a representation 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 showed that this module has a unique pre-canonical structure compatible with the $\mathcal{H}$-module structure, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan–Lusztig basis. One can modify the definition of Lusztig and Vogan’s module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this manner, and explain the relationships between their resulting canonical bases. Some of these canonical bases are equivalent in a trivial fashion to Lusztig and Vogan’s construction, while others appear to be unrelated. Along the way, we also clarify the differences between Webster’s notion of a canonical basis and the related concepts of an IC basis and a P-kernel.

Dedicated to David Vogan on the occasion of his 60th birthday