arXiv subjects

Kazuki Hiroe

Publications and source records attributed to Kazuki Hiroe.

14 recordsLinked to original sources

arXiv2026

Middle convolution for Lie algebra representations

This paper introduces a Lie algebra analogue of the middle convolution functor, which is defined on the category of modules over certain Lie algebras, including, as typical motivating examples, free Lie algebras, Drinfeld-Kohno Lie algebras, and the holonomy Lie algebras of complements of hyperplane arrangements. First, we demonstrate that the middle convolution for Lie algebra representations can be regarded as a natural generalization of the infinitesimal analogue of the Long-Moody functor for Drinfeld-Kohno Lie algebras. Second, we show that our middle convolution recovers the Dettweiler-Reiter additive middle convolution for Fuchsian systems on the punctured Riemann sphere as a special case. Furthermore, we show that when applied to the holonomy Lie algebra of the complement of a hyperplane arrangement, our functor is compatible with Haraoka's middle convolution for logarithmic connections on such complements. Finally, we establish a Riemann-Hilbert correspondence between the middle convolution for the holonomy Lie algebra and the middle convolution for local systems on complements of hyperplane arrangements.

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arXiv2026

On the Riemann-Hilbert problem for hyperplane arrangements with a good line

We study a variant of the Riemann-Hilbert problem on the complements of hyperplane arrangements. This problem asks whether a given local system on the complement can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. In this paper, we generalize Katz's middle convolution as a functor for local systems on hyperplane complements and show that it preserves the solvability of this problem.

math.AG↗

arXiv2025

Unfolding of wild character varieties

In this paper, we study wild character varieties on compact Riemann surfaces and construct Poisson maps from wild to tame character varieties by unfolding irregular singularities into regular ones. Furthermore, we show that these unfolding Poisson maps induce Poisson birational equivalences between wild and tame character varieties. This result provides an affirmative answer to a conjecture posed by Klimes, Paul, and Ramis.

math.AG↗

arXiv2024

Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities

Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal $G$-bundle over $\mathbb{P}^{1}$. One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ can be deformed into a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$. Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution. We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible $G$-connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture.

math.AG↗

arXiv2023

Long-Moody construction of braid representations and Katz middle convolution

The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems on $\mathbb{C}\backslash\{n\text{-points}\}$ introduced by Katz. In this paper, we explain that these two methods are naturally unified and define a new functor which we call the Katz-Long-Moody functor. This functor extends the framework of Katz algorithm to categories of local systems on various topological spaces, for example, $B_{n}$-bundles associated with simple Weierstrass polynomials, complements of hyperplane arrangements of fiber-type, link complements in the solid torus, and so on.

math.GT↗

arXiv2022

A note on unfolding manifolds of meromorphic connections on the Riemann sphere with unramified singularities

This note explains a construction of a Poisson manifold whose symplectic foliation describes a deformation of a moduli space of meromorphic connections with unramified irregular singularities. In particular, this deformation of the moduli space corresponds to the unfolding of irregular singularities of the meromorphic connections. This is an announcement of some results in the forthcoming paper.

math.AG↗

arXiv2020

Index of rigidity of differential equations and Euler characteristic of their spectral curves

We show a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves. Also we present a comparison of local invariants, so called Milnor formula which links the Komatsu-Malgrange irregularity of differential equations and Milnor number of the spectral curves.

math.AG↗

arXiv2018

Moduli spaces of meromorphic connections, quiver varieties, and integrable deformations

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the study of the symmetries, a realization of the moduli spaces as quiver varieties is given and plays an essential role.

math.CA↗

arXiv2016

Linear differential equations on the Riemann sphere and representations of quivers

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.

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arXiv2014

Local Fourier transform and blowing up

We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connections with the fixed Puiseux characteristic by means of the structure of iterated torus knots of the plane curve germs.

math.CA↗

arXiv2014

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne-Simpson problem.

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arXiv2013

Linear differential equations on $\mathbb{P}^{1}$ and root systems

In this paper, we study the Euler transform on linear ordinary differential operators on $\mathbb{P}^{1}$. The spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms. We compute the changes of spectral types under the action of the Euler transform and show that the changes of spectral types generate a transformation group of a $\mathbb{Z}$-lattice which is isomorphic to a quotient lattice of a Kac-Moody root lattice with the Weyl group as the transformation group.

math.CA↗

arXiv2009

Twisted Euler transform of differential equations with an irregular singular point

N. Katz introduced the notion of the middle convolution on local systems. This can be seen as a generalization of the Euler transform of Fuchsian differential equations. In this paper, we consider the generalization of the Euler transform, the twisted Euler transform, and apply this to differential equations with irregular singular points. In particular, for differential equations with an irregular singular point of irregular rank 2 at $x=\infty$, we describe explicitly changes of local datum caused by twisted Euler transforms. Also we attach these differential equations to Kac-Moody Lie algebras and show that twisted Euler transforms correspond to the action of Weyl groups of these Lie algebras.

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arXiv2008

Generalized Whittaker functions for degenerate principal series of $GL(4,\R)$

We give a characterization of a generalized Whittaker model of a degenerate principal series representation of $GL(n,\R)$ as the kernel of some differential operators. By this characterization, we investigate some examples on $GL(4,\R)$. We obtain the dimensions of the generalized Whittaker models and give their basis in terms of hypergeometric functions of one and two variables. We show the multiplicity one of the generalized Whittaker models by using the theory of hypergeometric functions.

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