Hecke relations specialize in the generalized-standard family

Write $X=S({\mathfrak{h}}^{\ast })$, $Y=S(\mathfrak{h})$, and $B=Y⋊\mathbb{C}W$. Let ${\mathbb{C}}^m$ be the parameter space of conjugacy-invariant $c$, and fix $n$. Use the same finite vector space $U_n=\mathbb{C}W\otimes (Y\mathrm{/}{Y}_{>n})$ at every parameter. PBW identifies $G_{n,c}=H_{1,c}{\otimes }_B{U}_n$ with $X\otimes U_n$, with all commutation matrices polynomial in $c$. This holds over $\mathbb{C}[c]$ as well: the overlap and Jacobi proof of PBW consists of polynomial identities and uses no division; alternatively a polynomial relation among normal words vanishing in the universal quotient specializes at each $c$, where pointwise PBW forces every coefficient to vanish. Hence the normal words are independent over $\mathbb{C}[c]$. The family is therefore free over the parameter ring and finite free over $X[c]$.

Let $Y_y$ be multiplication by $y$ on $U_n$, and let $s$ act diagonally on it. The localized derivation from [F1] is ${\mathrm{\partial }}_y=y+{\sum }_s{c}_s{\alpha }_s(y){\alpha }_s^{-1}(1-s)$. Its action on a constant generator $1\otimes u$ has matrix ${\mathrm{\Gamma }}_y(c,z)=Y_y+{\sum }_s{c}_s{\alpha }_s(y){\alpha }_s(z{)}^{-1}(1-s{\mathrm{\mid }}_{U_n})$; on nonconstant sections the ordinary derivative of the coefficient is added. These are finite matrices rational in $z$ and polynomial in $c$, without poles on fixed paths in ${\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$. Flatness is the localized Cherednik relation. On a normal disk to a generic point of a reflecting hyperplane $H$, the residue is $R_H(c)={\sum }_{s\text{ fixes }H}{c}_s(1-s{\mathrm{\mid }}_{U_n})$, diagonal in the fixed cyclic $W_H$-decomposition; the term $Y_y$ is holomorphic. The horizontal equation has residue $A_0(c)=-R_H(c)$ in the stated convention.

We prove an actual nonresonant gauge normal form. For $z{U}^{\mathrm{\prime }}=A(z)U$ with $A(z)={\sum }_{j\ge 0}{A}_j{z}^j$ holomorphic, assume $A_0$ is diagonalizable and no difference of two of its eigenvalues is a nonzero integer. A normalized gauge $G(z)=I+{\sum }_{r\ge 1}{G}_r{z}^r$ solving $z{G}^{\mathrm{\prime }}=A(z)G-G{A}_0$ must satisfy $(rI-\mathrm{ad}{A}_0)G_r={\sum }_{j=1}^r{A}_j{G}_{r-j}$. Each operator is invertible. Its inverse norm is at most $C\mathrm{/}r$: for $r>2\mathrm{\parallel }\mathrm{ad}{A}_0\mathrm{\parallel }$, the Neumann series gives $2\mathrm{/}r$, and increasing $C$ handles the finitely many remaining indices. If $\mathrm{\parallel }{A}_j\mathrm{\parallel }\le M{R}^{-j}$, the recursion gives $\mathrm{\parallel }{G}_r\mathrm{\parallel }\le (CM\mathrm{/}r){\sum }_{j=1}^r{R}^{-j}\mathrm{\parallel }{G}_{r-j}\mathrm{\parallel }$. The scalar majorant with $g_0\ge \mathrm{\parallel }I\mathrm{\parallel }$ solves $g^{\mathrm{\prime }}(z)=CM(R-z{)}^{-1}g(z)$, so $g(z)=g_0(1-z\mathrm{/}R{)}^{-CM}$ has positive Taylor coefficients dominating the matrix series. Thus $G$ converges and is invertible near zero. It gauges the equation to $z{V}^{\mathrm{\prime }}=A_{0}V$, proving analytic conjugacy rather than merely equality of characteristic polynomials.

The fractional-meridian endpoint is respected by this gauge. Choose the normal disk at a generic point fixed by $W_H$. Its cyclic action rotates the coordinate; the endpoint acts on the fiber by the fixed matrix $g=s_H$, which commutes with $A_0$. Equivariance of the connection transforms the normalized gauge to another solution of the same equation with constant term $I$. Uniqueness of the recursion in step 3.1 forces the equivariance identity for $G$. Hence endpoint-adjusted transport through angle $\vartheta =2\pi \mathrm{/}{e}_H$ is genuinely conjugate to $g\exp (i\vartheta A_0)$. On the $j$th cyclic eigenspace this is the defined scalar ${\mu }_{H,j}$. The commuting matrices $g,A_0$ are simultaneously diagonal on that decomposition, so ${\prod }_j(T_H-{\mu }_{H,j})$ annihilates this nonresonant monodromy, even if two resulting roots coincide. Parallel transport along a basepoint path conjugates the operator and preserves this relation. No diagonalizability assertion is made at resonant parameters.

Exclude the countable collection of conditions $a_E(c)-a_F(c)\in {\mathbb{Z}}_{>0}$ and the nonzero integral differences of normal residue eigenvalues at all hyperplanes. Every relevant difference is linear in $c$ and is zero at $c=0$, so each nonzero-integer condition is a proper affine hyperplane or empty. Their complement is dense in every parameter neighborhood: choose a line with direction avoiding the finitely many nonzero linear forms; each excluded condition removes at most one point from that line. The zero-dimensional parameter space or the absence of such forms gives no excluded condition. Steps 3.1–4.1 therefore apply on a dense set. For clarity, category ${\mathcal{O}}_c$ is also semisimple there: a proper submodule of a standard with irreducible label has a singular simple label $F$ at degree $d>0$, forcing $a_F-a_E=d$, which is excluded. Such standards are therefore simple. The standard-extension criterion in [F2] makes Ext${}^1$ between these standards vanish, and the finite-length result then splits every object into irreducible standards. A standard induced from a reducible finite-dimensional $W$-module is the direct sum of the standards for its irreducible constituents. The generic relation for $G_{n,c}$ follows already from step 4.1 applied directly to $U_n$; this argument does not require the special-parameter standard-factorization assertion.

The monodromy matrices of this fixed family are holomorphic at every parameter. Choose fixed piecewise smooth compact paths in ${\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ for the based meridians. Their horizontal ODE coefficients are continuous on each segment and holomorphic in $c$. Picard iteration is a normally convergent sum of iterated integrals on every compact parameter set: its $r$th term has norm at most $L^r\mathrm{/}r!$, where $L$ bounds the integral of the coefficient norm there. Each term is holomorphic, so the sum is holomorphic in $c$; reversed paths give holomorphic inverses, and the endpoint matrix is constant. Each root ${\mu }_{H,j}(c)$ is an exponential of a linear function and is entire. Thus the matrix $Q_H(c)={\prod }_j(T_H(c)-{\mu }_{H,j}(c)I)$ is holomorphic everywhere. It vanishes on the dense set from step 5.1, so continuity gives $Q_H(c)=0$ for all $c$. This is an exact polynomial identity on every $\mathrm{KZ}(G_{n,c})$; it does not infer a relation on an extension merely from its subquotients.

For fixed $c$, choose $n$ by the uniform Euler bound in [F2]. The projective generator is a direct summand of $G_{n,c}$, and every object $M$ is therefore a quotient of a finite sum of these generalized standards. Exact KZ sends that surjection to a surjection of braid representations. The relation proved in step 6.1 passes to finite sums and quotients, so all the defining Hecke polynomials annihilate $\mathrm{KZ}(M)$. This proves arbitrary-object factorization for every finite complex reflection group and every $c$. Taking $M=\mathrm{\Delta }(E)$ also proves standard factorization, with no inference of diagonalizability at a resonant parameter.