  A canonical basis in the sense of Lusztig is a basis of a free module over a
ring of Laurent polynomials that is invariant under a certain semilinear
involution and is obtained from a fixed "standard basis" through a triangular
base change matrix with polynomial entries whose constant terms equal the
identity matrix.
  Among the better known examples of canonical bases are the Kazhdan-Lusztig
basis of Iwahori-Hecke algebras, Lusztig's canonical basis of quantum groups
and the Howlett-Yin basis of induced $W$-graph modules.
  This paper has two major theoretical goals: First to show that having bases
is superfluous in the sense that canonicalisation can be generalized to
non-free modules. This construction is functorial in the appropriate sense. The
second goal is to show that Howlett-Yin induction of $W$-graphs is well-behaved
a functor between module categories of $W$-graph-algebras that satisfies
various properties one hopes for when a functor is called "induction", for
example transitivity and a Mackey theorem.
