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A logarithmic connection determines meridian eigenvalues from its residue

Statement

Let $E$ be a finite-dimensional complex vector space, let $ζ = e^{i θ}$ be a root of unity with $0 < θ \leq 2 π$ the positive arc angle (taking $θ = 2 π$ when $ζ = 1$ ), and let $A (z)$ be a holomorphic family of endomorphisms on a disk about $0$ with $A (0) = R$ . Consider the logarithmic connection $\nabla = d - A (z) \frac{d z}{z}$ on the punctured disk. Suppose the endpoint identification $g : E_{ζ r} \to E_{r}$ is a fixed automorphism of the model fiber and satisfies $g A (ζ z) g^{- 1} = A (z) .$ At $z = 0$ this implies $g R = R g$ . Parallel transport from $r$ to $ζ r$ along the positive arc, followed by this endpoint identification $g$ , has characteristic polynomial equal to that of $g e^{i θ R}$ .

Facts & Assumptions

Given: $E$ , $ζ = e^{i θ}$ , the holomorphic family $A (z)$ , its residue $R = A (0)$ , and the compatible endpoint identification $g$ .

[F1]

A horizontal section for $\nabla = d - A (z) d z / z$ along $z (t) = r e^{i t}$ satisfies the finite-dimensional ODE $\dot{u} (t) = i A (r e^{i t}) u (t)$ .

[F2]

The coefficient identity $g A (ζ z) g^{- 1} = A (z)$ expresses equivariance under $z \mapsto ζ z$ with the stated endpoint identification from the endpoint fiber to the initial fiber.

[F3]

Linear matrix ODEs have unique global solutions on a fixed interval gives existence and uniqueness of transport along compact paths, interpreting complex matrices as real linear operators. Gronwall's integral inequality with variable and constant coefficients gives the uniform convergence estimate below.

Proof

technique · direct

Given: [F1]–[F3].

1.1F2F3givenalgebra

For $r > 0$ , let $P_{r}$ be parallel transport along the arc $z (t) = r e^{i t}$ , $0 \leq t \leq θ$ , and put $M_{r} : = g P_{r}$ , an endomorphism of the initial fiber. The connection form $Ω = A (z) d z / z$ is flat on the punctured disk: holomorphicity gives $d Ω = 0$ , and $Ω \land Ω = 0$ because both factors are multiples of $d z$ . Hence parallel transport around the boundary of an annular sector is the identity. If $r_{1}, r_{2}$ are two small radii, radial parallel transport identifies the initial fibers and the endpoint fibers; the coefficient identity in [F2] makes the resulting square commute with the endpoint identifications. Thus the two $M_{r}$ are conjugate and have the same characteristic polynomial.

1.2F1F3choosealgebra

Since $A (r e^{i t}) \to R$ uniformly for $0 \leq t \leq θ$ as $r \to 0$ , subtract the integral equations for $P_{r} (t)$ and $e^{i t R}$ from [F1]. The difference norm is bounded by $\int_{0}^{t} ∥ A (r e^{i u}) - R ∥ ∥ P_{r} (u) ∥ d u + \int_{0}^{t} ∥ R ∥ ∥ P_{r} (u) - e^{i u R} ∥ d u$ ; Grönwall's inequality gives $P_{r} (θ) \to e^{i θ R}$ . Therefore $M_{r} \to g e^{i θ R}$ .

2.1step 1.1step 1.2algebra∎

The characteristic polynomial is a polynomial in matrix entries, hence continuous. Step 1.1 makes it constant for $r > 0$ , so its common value equals the characteristic polynomial of the limit in step 1.2, namely that of $g e^{i θ R}$ .

Depends on

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Sources

Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)
Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphael Rouquier, On the category O for rational Cherednik algebras (standard reference, not scraped)