More context is needed for this question. I am going to address the case of generic representations in the spherical case.

It is an old theorem of Lusztig that over the ring $\mathbb{C}[v,v^{-1}]$, the generic Hecke algebra $H_v$ with generators $T_i$ and relations

$$T_i T_j \ldots = T_j T_i \ldots \ (m_{ij} \text{ factors}), \qquad (T_i+v^2)(T_i-v^2)=0$$

is isomorphic to the group ring of the corresponding *finite* Coxeter group $W.$ Thus every representation of $W$ can be canonically deformed to a representation of $H_v.$ In the course of developing representation theory of reductive groups over a finite field, Lusztig developed quite a bit of machinery describing these representations (fake degrees, etc). This is described in his book

G. Lusztig, *Characters of reductive groups over a finite field*. Annals of Mathematics Studies, 107. Princeton University Press, Princeton, NJ, 1984

A more recent source, in a more general situation and with improved proofs, is

G. Lusztig, *Hecke algebras with unequal parameters*. CRM Monograph Series, 18. American Mathematical Society, Providence, RI, 2003

For specific non-zero $v$ that are not roots of unity, the same result holds. The case of roots of unity and of more general Hecke algebras (in particular, for Coxeter system of affine type) has also been studied.