The unequal-parameter Coxeter Hecke algebra

Let $(W,S)$, $\mathrm{\Gamma }$, $A=\mathbb{Z}[\mathrm{\Gamma }]$, and $L:W\to \mathrm{\Gamma }$ be as in Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order and Weight functions and ordered group-algebra coefficients. For $s\in S$, write $q_s=v^{L(s)}\in A^{\times }$. Let $F=A⟨T_s:s\in S⟩$ be the free associative $A$-algebra and let $I$ be the two-sided ideal generated by

$$(T_s-q_s)(T_s+q_s^{-1})(s\in S)$$

and, for each distinct $s,t\in S$ with finite $m(s,t)$, the equality of the two alternating products of $T_s,T_t$ having $m(s,t)$ factors. Define the unequal-parameter Coxeter Hecke algebra by $H_A(W,L)=F\mathrm{/}I$, and use $T_s$ for the image of a generator. For $m(s,t)=\mathrm{\infty }$ there is no braid relation between $T_s$ and $T_t$.

The quadratic relation is equivalently

$$T_s^2=(q_s-q_s^{-1})T_s+1,$$

so each $T_s$ is invertible with $T_s^{-1}=T_s-(q_s-q_s^{-1})$. The basis, bar, and Bruhat support properties are not part of this definition; they are proved in The arbitrary-rank weighted Hecke basis and Bruhat-triangular bar ↗.