The KZ monodromy functor and the reflection Hecke algebra

Fix $(W,\mathfrak{h},c)$ as above and use ${\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and $\delta$ from Dunkl localization gives flat connections on the regular locus. For $b\in {\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$, define the reflection braid group $B_W$ to be the orbifold fundamental group of $W\curvearrowright {\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$. Equivalently, its arrows at $b$ are homotopy classes of pairs $(w,\gamma )$ with $w\in W$ and $\gamma :b\to wb$; composition concatenates paths after applying the first endpoint group element, with the corresponding operator order. A $W$-equivariant local system on ${\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ therefore gives a representation of $B_W$.

For $M\in {\mathcal{O}}_c(W,\mathfrak{h})$, localize to $M_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and restrict along the Dunkl localization isomorphism to the differential-operator algebra. By Dunkl localization gives flat connections on the regular locus, this is a finite-rank vector bundle with integrable $W$-equivariant connection. Its fiber at $b$, with parallel transport, defines the KZ monodromy functor $\mathrm{KZ}:{\mathcal{O}}_c\to \mathbb{C}[B_W]\text{-mod}$. The convention here is monodromy of horizontal sections for the left differential-module connection.

For each reflecting hyperplane $H$, the pointwise stabilizer $W_H$ is cyclic: it acts faithfully on the one-dimensional quotient $\mathfrak{h}\mathrm{/}H$. Put $e_H=\mathrm{\mid }{W}_H\mathrm{\mid }$ and ${\zeta }_H=e^{2\pi i\mathrm{/}{e}_H}$. Choose a generator $s_H$ whose eigenvalue on the conormal line $\mathbb{C}{\alpha }_H\subset {\mathfrak{h}}^{\ast }$ is ${\zeta }_H$, and write $c_{H,\mathrm{\ell }}=c_{s_H^{\mathrm{\ell }}}$ for $1\le \mathrm{\ell }<e_H$. Then $s_H^{-1}$ acts on the normal coordinate by ${\zeta }_H$, so a positive meridian lifts through angle $2\pi \mathrm{/}{e}_H$ and ends at the $s_H^{-1}$-translate; the endpoint identification is the action of $s_H$. On a $W_H$-character where $s_H$ acts by ${\zeta }_H^j$, put $r_{H,j}={\sum }_{\mathrm{\ell }=1}^{e_H-1}{c}_{H,\mathrm{\ell }}(1-{\zeta }_H^{j\mathrm{\ell }}),{\mu }_{H,j}={\zeta }_H^j\exp \text{\hspace{0.17em}⁣}(-\frac{2\pi i}{e_H}{r}_{H,j})(0\le j<e_H)\mathrm{.}$ The residue on standard modules is computed in The standard-module KZ connection has the reflection residue, and the horizontal transport with endpoint identification is computed in A logarithmic connection determines meridian eigenvalues from its residue; together they give the displayed scalars on each $W_H$-character. The annihilating-polynomial assertion for standards is a separate theorem, KZ monodromy of standard modules satisfies the reflection Hecke relations. To compare residues with GGOR, let $s_H^{\mathrm{G}}$ denote their reflection with eigenvalue ${\zeta }_H$ on the normal line $\mathfrak{h}\mathrm{/}H$. Under the present convention, $s_H^{\mathrm{G}}=s_H^{-1}$. Their character ${\det }^{-j}{\mathrm{\mid }}_{W_H}$ therefore has eigenvalue ${\zeta }_H^j$ for the present generator $s_H$. GGOR's residue $-e_H{k}_{H,j}$ on that character matches the locally computed $r_{H,j}$, giving $k_{H,j}=-r_{H,j}\mathrm{/}{e}_H$ with the same character index. Their displayed Hecke root uses their reflection and braid-generator convention; this definition derives ${\mu }_{H,j}$ from the positive lift and endpoint action fixed here, and does not identify the two root lists without translating the braid generator as well.

The finite reflection Hecke algebra ${\mathcal{H}}_{q(c)}(W)$ is the quotient of $\mathbb{C}[B_W]$ by ${\prod }_{j=0}^{e_H-1}(T_H-{\mu }_{H,j})=0(H\in \mathcal{A}),$ where $T_H$ is the positive meridian class in the quotient, computed by parallel transport on its positive lift and then identifying the endpoint by $s_H$. Thus both the loop orientation and endpoint action are fixed. In particular ${\mu }_{H,0}=1$. This definition makes no dimension or freeness claim for ${\mathcal{H}}_{q(c)}(W)$.

The braid-group representation and quotient Hecke algebra are defined here. The standard-module residue calculation is proved locally; factorization for every object of $\mathcal{O}$ and the quotient equivalence are retained as separate claims with explicit proof status in KZ monodromy on category O factors through the reflection Hecke algebra and KZ identifies the torsion quotient with Hecke modules under the Hecke-rank hypothesis.