arXiv subjects

Masoud Kamgarpour

Publications and source records attributed to Masoud Kamgarpour.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic loops, braids, and tropicalization

Let $Y$ be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in $Y^{\mathrm{an}}$ associated with formal loops $\mathrm{Spec}\, \mathbb{C}(\!(t)\!)\to Y$. When $Y$ is very affine, every formal loop determines an integral point of $\mathrm{Trop}(Y)$. For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when $Y$ is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type $A$, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.

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arXiv2026

Counting points on braid varieties and the Deligne--Simpson problem

We solve the isoclinic Deligne--Simpson problem for exceptional groups, completing a program initiated by Sage et al. and Jakob--Yun. As a by-product, we obtain new examples of physically rigid irregular connections on the projective line. Our approach uses the Riemann--Hilbert correspondence to reduce the problem to determining the non-emptiness of certain braid varieties associated to periodic braids. We show that this can be achieved by counting points over finite fields. Our approach is inspired by Lusztig's construction of a map from conjugacy classes in the Weyl group to unipotent classes.

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arXiv2026

Differential Galois groups of $G$-connections with Coxeter singularities

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank $n$ connections on algebraic curves with slope $r/n$ at a singularity, where $\gcd(r,n)=1$. We extend this result to $G$-connections, where $G$ is a simple algebraic group and the slope is $r/h$, with $h$ the Coxeter number of $G$ and $\gcd(r,h)=1$. This allows us to compute the differential Galois groups of a broad class of $G$-connections that have been central to recent advances in the geometric Langlands program and the Deligne--Simpson problem -- namely, Coxeter connections, generalised Frenkel--Gross connections, and Airy connections. We apply our results to inverse differential Galois theory by giving uniform and explicit constructions of $G$-connections whose differential Galois groups realise all reductive subgroups of maximal degree.

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arXiv2026

Counting points on generic character varieties

We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers.

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arXiv2025

Arithmetic geometry of character varieties with regular monodromy

We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.

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arXiv2022

Character stacks are PORC count

We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a Polynomial On Residue Classes (PORC). The key ingredients in the proof are Lusztig's Jordan decomposition of complex characters and Deriziotis's results on genus numbers of finite reductive groups. As a consequence of our main theorem, we obtain an expression for the $E$-polynomial of the character stack.

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arXiv2022

Hypergeometric sheaves for classical groups via geometric Langlands

In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as $\check{G}$-local systems, for a classical group $\check{G}$. This article aims to realize the geometric Langlands correspondence for these $\check{G}$-local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group $G$ in the framework of Yun, one of whose local components is a new class of euphotic representations in the sense of Jakob-Yun. We prove the rigidity of hypergeometric automorphic data under natural assumptions, which allows us to define $\check{G}$-local systems $\mathcal{E}_{\check{G}}$ on $\mathbb{G}_m$ as Hecke eigenvalues (in both $\ell$-adic and de Rham setting). In the second approach (which works only in the de Rham setting), we quantize an enhanced ramified Hitchin system, following Beilinson-Drinfeld and Zhu, and identify $\mathcal{E}_{\check{G}}$ with certain $\check{G}$-opers on $\mathbb{G}_m$. Finally, we compare these $\check{G}$-opers with hypergeometric local systems.

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arXiv2021

Airy sheaves for reductive groups

We construct a class of $\ell$-adic local systems on $\mathbb{A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation $y''(z)=zy(z)$. We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For $\mathrm{GL}_n$, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behaviour of the local systems at $\infty$. These conjectures in particular imply cohomological rigidity of Airy sheaves.

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arXiv2020

Geometric Langlands for hypergeometric sheaves

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

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arXiv2020

Rigid connections on $\mathbb{P}^1$ via the Bruhat-Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

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arXiv2019

A geometric analogue of a conjecture of Gross and Reeder

Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.

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arXiv2019

Jordan Decomposition for Formal G-Connections

A theorem of Hukuhara, Levelt, and Turrittin states that every formal differential operator has a Jordan decomposition. This theorem was generalised by Babbit and Varadarajan to the case of formal $G$-connections where $G$ is a semisimple group. In this paper, we provide straightforward proofs of these facts, highlighting the analogy between the linear and differential settings.

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arXiv2017

On the image of the parabolic Hitchin map

We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and $G_2$. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type $D$, where the image can be singular.

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arXiv2017

Complete integrability of the parahoric Hitchin system

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm{Bun}_{\mathcal{G}}$ is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.

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arXiv2016

On the notion of conductor in the local geometric Langlands correspondence

Under the local Langlands correspondence, the conductor of an irreducible representation of $\Gl_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

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arXiv2015

Stabilisers of eigenvectors of finite reflection groups

Let $x$ be an eigenvector for an element of a finite irreducible reflection group $W$. Let $W_x$ denote the subgroup of $W$ which stabilises $x$. We provide an upper bound for the number of roots in the root system of $W_x$ . This generalises a result of Kostant, who showed that every eigenvector with eigenvalue a primitive $h^\mathrm{th}$ root of unity is regular, where $h$ is the Coxeter number of $W$. We also give a Lie-theoretic interpretation of our result in the study of semisimple conjugacy classes over Laurent series. In a forthcoming paper, we use this result to establish a geometric analogue of a conjecture of Gross and Reeder.

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arXiv2014

Preservation of depth in local geometric Langlands correspondence

It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. A key ingredient of our proof is a new definition of the slope of a meromorphic connection, a definition which uses opers. In the appendix, we consider a relationship between our conjecture and Zhu's conjecture on non-vanishing of the Hecke eigensheaves produced by Beilinson and Drinfeld's quantisation of Hitchin's fibration for non-constant groups.

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arXiv2013

Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for jet algebras

Let $\fg$ be a simple finite-dimensional complex Lie algebra with a Cartan subalgebra $\fh$ and Weyl group $W$. Let $\fg_n$ denote the Lie algebra of $n$-jets on $\fg$. A theorem of Rais and Tauvel and Geoffriau identifies the centre of the category of $\fg_n$-modules with the algebra of functions on the variety of $n$-jets on the affine space $\fh^*/W$. On the other hand, a theorem of Feigin and Frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine Kac-Moody algebra with the algebra of functions on the ind-scheme of opers for the Langlands dual group. We prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities. We also define generalized Verma and Wakimoto modules and relate them by a nontrivial morphism.

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