Reflection data and parameters for a rational Cherednik algebra

Let $\mathfrak{h}$ be a finite-dimensional complex vector space and let $W\subseteq \mathrm{GL}(\mathfrak{h})$ be a finite subgroup generated by complex reflections, meaning elements $s\mathrm{\ne }1$ for which $\mathrm{rank}(1-s\mathrm{\mid }\mathfrak{h})=1$. Let $S$ be the set of all such elements in $W$. The dual space is ${\mathfrak{h}}^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(\mathfrak{h},\mathbb{C})$, with the contragredient $W$-action $(wx)(y)=x(w^{-1}y)$.

For each $s\in S$, choose a nonzero ${\alpha }_s\in {\mathfrak{h}}^{\ast }$ spanning the one-dimensional image of $1-s$ on ${\mathfrak{h}}^{\ast }$. Since that image has dimension one, there is a unique ${\alpha }_s^{\vee }\in \mathfrak{h}$ such that $sx=x-{\alpha }_s^{\vee }(x){\alpha }_s(x\in {\mathfrak{h}}^{\ast })\mathrm{.}$ Replacing ${\alpha }_s$ by $a{\alpha }_s$ replaces ${\alpha }_s^{\vee }$ by $a^{-1}{\alpha }_s^{\vee }$, so the product ${\alpha }_s(y){\alpha }_s^{\vee }(x)$ is independent of this choice.

Put $\mathfrak{V}=\mathfrak{h}\oplus {\mathfrak{h}}^{\ast }$. Define an alternating form ${\omega }_s$ on $\mathfrak{V}$ by ${\omega }_s(y+x,y^{\mathrm{\prime }}+x^{\mathrm{\prime }})={\alpha }_s(y){\alpha }_s^{\vee }(x^{\mathrm{\prime }})-{\alpha }_s(y^{\mathrm{\prime }}){\alpha }_s^{\vee }(x),$ for $y,y^{\mathrm{\prime }}\in \mathfrak{h}$ and $x,x^{\mathrm{\prime }}\in {\mathfrak{h}}^{\ast }$. Its radical is $\ker (1-s\mathrm{\mid }\mathfrak{V})$, and its restriction to $\mathrm{im}(1-s\mathrm{\mid }\mathfrak{V})$ is nondegenerate; that image has dimension two. The form $\omega (y+x,y^{\mathrm{\prime }}+x^{\mathrm{\prime }})=x^{\mathrm{\prime }}(y)-x(y^{\mathrm{\prime }})$ is the nondegenerate $W$-invariant alternating form on $\mathfrak{V}$.

A reflection parameter is a function $c:S\to \mathbb{C}$ constant on $W$-conjugacy classes. The pair $(W,\mathfrak{h})$ and such a function $c$ are the reflection data and parameters used below.

Only the products ${\alpha }_s(y){\alpha }_s^{\vee }(x)$ enter the algebra presentation, so no normalization of an individual eigenvector is part of the parameter data.