Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $g\in G$. Let $S$ be the set of reflection hypersurfaces. We denote by $c$ the vector space of
$G$-equivariant maps from $S$ to $\mathbb{C}$. In page 8 of the paper Cherednik and Hecke algebras of varieties with a finite group action, the Cherednik algebra $H_{t,c,\omega}(X,G)$ is defined as the subalgebra of $G\ltimes \mathcal{D}_{\omega/t}^{r}(X)[c]$ generated by $G$, $O(X)$ and the Dunkl-Opdam operators. The algebra $\mathcal{D}_{\omega/t}^{r}(X)$ is the algebra of twisted differential operators associated with $\omega /t$ with rational coefficients, where $\omega$ is a $G$-invariant closed 2-form and $t$ is a non-zero complex number. The following formula defines the Dunkl-Opdam operators:
$$D := t\mathbb{L}_{v} + \sum_{(Y,g)} \frac{2c(Y,g)}{1-\lambda_{Y,g}}f_{Y}(x)(1-g)$$
where $v\in \Gamma(X,TX)$, and $f_{Y}(x)$ is a representative of the coset of $\xi_{Y}(x)\in \Gamma (X,O_{Z}(X)/O(X))$ defined in pages 5-6 of the aforementioned paper. The element $\lambda_{Y,g}$ is defined as the eigenvalue of $g$ on the conormal bundle of $Y$. I have several questions regarding this last part of the definition. In particular, $\lambda_{Y,g}$ is a regular function on $Y$. However, in this definition it is regarded as an element of $O(X)$. In the case where $Y$ is the vanishing locus of a function $f$, it follows that $g$ induces an automorphism $g:f\cdot O(X)\rightarrow f\cdot O(X)$ of the ideal defining $Y$. Thus, we have $g(f)=\lambda_{Y,g}f$, and it follows that the class of $\lambda_{Y,g}$ in $O(Y)$ is the desired eigenvalue. However, in the general case where $Y$ is not given by the vanishing locus of a single regular function I don't see a way to extend $\lambda_{Y,g}$ to a regular function on $X$. My first question regarding this operator can be summarized as follows:
- What is the definition of $\lambda_{Y,g}$ in the general case?
Later on in Theorem 2.17, using the case of rational Cherednik algebras and some local linearization arguments, Etingof shows the PBW theorem for Cherednik algebras. The proof seems to rely on one of the statements below being true. However, I have not been able to show any of them.
- Let $Y=\mathbb{V}(f)$ and $h\in O(X)$. Is it true that $\frac{h-g(h)}{(1-\lambda_{Y,g})f}$ is a regular function on $X$?
This is equivalent to:
- Is $1-\lambda_{Y,g}$ invertible?
Notice that both are clearly true in the case of rational Cherednik algebras.
On another note, Cherednik algebras can also be defined by considering the terms $t,c$ and $\omega$ as variables. In the paper, the case $t=0$ is not excluded. The Dunkl-Opdam operators make sense for $t=0$, but the ring $\mathcal{D}_{\omega/t}^{r}(X)$ a priori does not. Some clarification to this regard would be appreciated.
Best Answer
Questions 2. and 3. are correct. By Cartan's Lemma $Y$ is smooth. For a point $p\in Y$ one has $$T_{p}Y=(T_{p}X)^{g}\text{.}$$ If $1-\lambda_{Y,g}(p)=0$ then the action of $g$ on $T_{p}X$ is trivial. Hence $T_{p}Y$ has the same dimension as $X$. But this can't be possible, for $Y$ is a smooth hypersurface. Therefore, you can take a $G$-invariant open containing $p$ in which $1-\lambda_{Y,g}$ is invertible.