The canonical Hecke trace and symmetrizing form

Let $(S,m)$, $W$, $R$, $H$, and $\{T_w:w\in W\}$ be as in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, and The standard basis of the generic Hecke algebra and base change, and assume $W$ is finite. For the unique standard-basis expansion $h={\sum }_{w\in W}{r}_w{T}_w$, define the canonical Hecke trace

$$\tau :H\to R,\tau (h):=r_1\mathrm{.}$$

The associated $R$-bilinear form is $B_{\tau }(a,b):=\tau (ab)$. A trace is symmetric when $\tau (ab)=\tau (ba)$ for every $a,b\in H$. The form is perfect when the $R$-linear map

$$H⟶{\mathrm{Hom}}_R(H,R),a⟼[b⟼\tau (ab)]$$

is an isomorphism. A symmetrizing trace is a trace for which this form is symmetric and perfect. This is an integral definition over $R$; it assumes no field, semisimplicity, or nondegeneracy after passing to a fraction field. The theorem The canonical Hecke trace is symmetric and perfect ↗ proves these properties for $\tau$ and every commutative base change of $R$.