The Deligne–Simpson problem for connections on $\mathbb{G}_{m}$ with a maximally ramified singularity

A fundamental concern in the study of meromorphic connections is the existence problem for connections with specified singularities. More precisely, this problem poses the question: given points $a_{1},\dots,a_{m}$ in $\mathbb{P}^{1}$ and formal connections $\widehat{\nabla}_{1},\dots,\widehat{\nabla}_{m}$, does there exist a meromorphic connection $\nabla$ which is regular away from the $a_{i}$’s and satisfies $\nabla_{a_{i}}\cong\widehat{\nabla}_{i}$ for all $i$? The classical Deligne–Simpson problem is a variant of this problem for Fuchsian connections.

From now on, we assume that the underlying vector bundles of all connections on $\mathbb{P}^{1}$ are trivializable. Without loss of generality, we assume that the collection of singular points does not include $\infty$. A Fuchsian connection with singular points $a_{1},\ldots,a_{m}$ is defined by

where $A_{i}\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ for all $i$. Note that the adjoint orbit of $A_{i}$ determines the formal isomorphism class at $a_{i}$. Since $\infty$ is not a singularity, the residue theorem forces $\sum A_{i}=0$. We say that the collection of matrices $A_{1},\dots,A_{m}$ is irreducible if they have no common invariant subspaces besides $\{0\}$ and $\mathbb{C}^{n}$. We can now state the (additive) Deligne–Simpson problem:

Given adjoint orbits $\mathscr{O}_{1},\dots,\mathscr{O}_{m}$, determine whether there exists an irreducible $m$-tuple $(A_{1},\dots,A_{m})$ with $A_{i}\in\mathscr{O}_{i}$ satisfying $\sum A_{i}=0$ [Kos03].

In other words, when is there an irreducible Fuchsian connection with residues in the given orbits? Note that the original problem considered by Deligne and Simpson was the multiplicative version, where one looks for Fuchsian connections with monodromies in specified conjugacy classes in $\operatorname{GL}_{n}(\mathbb{C})$ [Sim91]. The additive version stated above was originally formulated by Kostov, who solved it in the “generic” case. Crawley-Boevey gave a complete solution by reinterpreting the problem in terms of quiver varieties [CB03]. We remark that while there is an obvious analogue of this problem for arbitrary reductive ${G}$, little is known about the solution outside of type $A$.

In order to generalize the Deligne–Simpson problem to allow for irregular singularities, one considers connections with higher order principal parts at the singularities:

Again, we assume that $\infty$ is not a singular point, so $\sum_{i=1}^{m}A^{(i)}_{0}=0$. We now require that the singularity at each $a_{i}$ has a certain specified form called a ”formal type”.

Most previous work on the irregular Deligne–Simpson problem has restricted attention to the ”unramified case” [Kos10, Boa08, HY14, Hir17]. This means that at each singularity, the slope decomposition of the corresponding formal connection only involves integer slopes. More concretely, each such formal connection has Levelt–Turrittin (LT) normal form

where the $D_{i}$’s are diagonal, $D_{r}\neq 0$, and the residue term $R$ is upper triangular and commutes with each $D_{i}$. We view the $1$-form $(D_{r}z^{-r}+\dots+D_{1}z^{-1}+R)\frac{dz}{z}$ as an unramified formal type. In the regular singular case, one can take $R$ to be in Jordan canonical form and view the formal type as $R\frac{dz}{z}$.

For Fuchsian connections, the principal part at a singular point is just the residue. Hence, in the classical Deligne–Simpson problem, one requires that the principal part agrees with the formal type after conjugation by a constant matrix (i.e., an element of $\operatorname{GL}_{n}(\mathbb{C})$). In other words, the principal part lies in the adjoint orbit of the formal type. For unramified formal types of positive slope, one instead requires the principal part to lie in the orbit of the formal type under a certain action of the group $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])$. Let $\mathcal{B}_{r}=\{(B_{r}z^{-r}+\dots+B_{0})\frac{dz}{z}\mid B_{i}\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})\}$ denote the space of principal parts of order at most $r$. The group $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])$ acts on $\mathcal{B}_{r}$ by conjugation followed by truncation at the residue term. Note that this action factors through the finite-dimensional group $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!]/z^{r}\mathbb{C}[\![z]\!])$. If $\mathscr{A}$ is an unramified formal type of slope $r$, we call the orbit $\mathscr{O}_{\mathscr{A}}$ under this action the truncated orbit of $\mathscr{A}$. If $\mathscr{A}$ has slope $0$, $\mathscr{O}_{\mathscr{A}}$ may be identified with the usual adjoint orbit of $\mathscr{A}/\frac{dz}{z}$.

We can now state the unramified irregular Deligne–Simpson problem: Given points $a_{i}$ and unramified formal types $\mathscr{A}_{i}$ of slope $r_{i}$, determine when there exists an irreducible connection as in (1) whose principal part at each $a_{i}$ lies in $\mathscr{O}_{\mathscr{A}_{i}}$. This problem can also be restated in the language of moduli spaces. Given the formal types $\mathscr{A}_{i}$, one can consider the moduli space of “framable” connections on a rank $n$ trivial bundle whose singularities have the specified formal types [HY14]. The construction generalizes that of Boalch [Boa01], who assumes that the $\mathscr{A}_{i}$ are all nonresonant, i.e., that the leading term of each $\mathscr{A}_{i}$ is regular semisimple. This moduli space is not necessarily well-behaved, but it is a complex manifold if one restricts to the stable moduli space, i.e., the open subset consisting of irreducible connections. The unramified Deligne–Simpson problem is simply the question of when such a stable moduli space is nonempty.

This problem was solved in 2017 by Hiroe [Hir17], building on earlier work of Boalch [Boa08] and Hiroe and Yamakawa [HY14]. As in the Fuchsian case, the proof involves quiver varieties. Hiroe uses the collection of unramified formal types to define a certain quiver variety and identifies the stable moduli space with a certain open subspace of the quiver variety. He then finds necessary and sufficient conditions for this open subspace to be nonempty. As a corollary, Hiroe shows that the stable moduli space is a connected manifold as long as it is nonempty.

In this paper, we introduce the study of the ramified Deligne–Simpson problem, where ramified singularities are allowed. A singularity is called ramified if the associated formal connection can only be expressed in LT normal form after passing to a ramified cover. The LT normal form is thus no longer a suitable notion of formal type for ramified singularities. It is possible to formulate the ramified Deligne–Simpson problem by replacing the LT normal form with a “rational canonical form” for connections. Such a form may be obtained from Sabbah’s refined Levelt–Turrittin decomposition [Sab08]; we will discuss this in a future paper.

Here, we only sketch the setup of the ramified Deligne–Simpson problem for a special class of irregular connections called toral connections. Roughly speaking, a formal connection is called toral if its leading term with respect to an appropriate filtration satisfies a graded version of regular semisimplicity. (The precise definition involves the theory of fundamental and regular strata for connections introduced by Bremer and Sage [BS13b, BS18, BS13a].) The terminology reflects the fact that toral connections can be “diagonalized” into a (not necessarily split) Cartan subalgebra of the loop algebra.

First, we describe formal types for toral connections. A rank $n$ toral connection has slope $r/b$, where $b$ is a divisor of $n$ or $n-1$ and $\gcd(r,b)=1$. If $b>1$, define $\omega_{b}\in\operatorname{\mathfrak{gl}}_{b}(\mathbb{C}(\!(z)\!))$ by $\omega_{b}=\sum_{i=1}^{b-1}e_{i,i+1}+ze_{b,1}$; i.e., $\omega_{b}$ is the matrix with $1$’s in each entry of the superdiagonal, $z$ in the lower-left entry, and $0$’s elsewhere. If $b=1$, set $\omega_{b}=z$. Note that $\omega_{b}^{b}=z\mathrm{id}$. Given such a $b$ with $b\ell=n$ (resp. $b\ell=n-1$), we define a block-diagonal Cartan subalgebra $\mathfrak{s}^{b}=\mathbb{C}(\!(\omega_{b})\!)^{\ell}$ (resp. $\mathfrak{s}^{b}=\mathbb{C}(\!(\omega_{b})\!)^{\ell}\oplus\mathbb{C}(\!(z)\!)$). There is a natural $\mathbb{Z}$-filtration $\mathfrak{s}^{b}=\bigcup_{i}(\mathfrak{s}^{b})^{i}$ induced by assigning degree $i$ to $\omega_{b}^{i}$. Let $S^{{b}}$ denote the corresponding maximal torus in the loop group.

An $S^{{b}}$-formal type of slope $r/b$ (with $\gcd(r,b)=1$) is a $1$-form $A\frac{dz}{z}$, where $A\in(\mathfrak{s}^{b})^{-r}$ has regular semisimple term in degree $-r$ and no terms in positive degree. It is a fact that any toral connection of slope $r/b$ is formally isomorphic to a connection $d+A\frac{dz}{z}$ with $A\frac{dz}{z}$ an $S^{{b}}$-formal type; the formal type is unique up to an action of the relative affine Weyl group of $S^{{b}}$ [BS13b, BS13a].

In the case of unramified toral connections, $S^{{1}}=T(\mathbb{C}(\!(z)\!))$ is the usual diagonal maximal torus, and the $S^{{1}}$-formal types of slope $r$ are those connection matrices in LT normal form (2) with $D_{r}$ regular (so that $R$ is necessarily $0$). At the opposite extreme, $\mathcal{C}\coloneqq S^{{n}}=\mathbb{C}(\!(\omega_{n})\!)^{*}$ is a ”Coxeter maximal torus”.¹¹1Under the bijection between classes of maximal tori in $\operatorname{GL}_{n}(\mathbb{C}(\!(z)\!))$ and conjugacy classes in the Weyl group $\mathfrak{S}_{{n}}$ [KL88], $\mathcal{C}$ corresponds to the Coxeter class consisting of $n$-cycles. The $\mathcal{C}$-formal types of slope $r/n$ are the $1$-forms $p(\omega_{n}^{-1})\frac{dz}{z}$, where $p$ is a polynomial of degree $r$.

The unramified Deligne–Simpson problem involves global connections which satisfy a stronger condition than just having specified formal types at the singularities. One also needs the local isomorphisms transforming the matrices of the formal connections into the given formal types to satisfy a global compatibility condition called “framability”. We now explain how this condition can be generalized to toral formal types.

Recall (see, e.g., [Sag00, BS13b]) that the parahoric subgroups of $\operatorname{GL}_{n}(\mathbb{C}(\!(z)\!))$ are the local field analogues of the parabolic subgroups of $\operatorname{GL}_{n}(\mathbb{C})$. A parabolic subgroup is the stabilizer of a partial flag of subspaces in $\mathbb{C}^{n}$, and a parahoric subgroup is the stabilizer of a ”lattice chain” of $\mathbb{C}[\![z]\!]$-lattices in $\mathbb{C}(\!(z)\!)^{n}$. If $P$ is a parahoric subgroup with associated lattice chain $\{L^{j}\}_{j}$, then there is an associated ”lattice chain filtration” $\{\mathfrak{p}^{i}\}_{i\in\mathbb{Z}}$ on $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C}(\!(z)\!))$ defined by $\mathfrak{p}^{i}=\{X\mid X(L^{j})\subset L^{j+i}\text{ }\forall j\}$.

To each maximal torus $S^{{b}}$, there is a unique ”standard parahoric subgroup” $P^{b}\subset\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])$ with the property that the corresponding filtration $\{(\mathfrak{p}^{b})^{i}\}_{i}$ is compatible with the filtration on $\mathfrak{s}^{b}$, in the sense that $(\mathfrak{s}^{b})^{i}=(\mathfrak{p}^{b})^{i}\cap\mathfrak{s}^{b}$ for all $i$ [BS13b]. In the unramified case, we have $P^{1}=\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])$. For the Coxeter maximal torus $S^{{n}}$, the corresponding parahoric subgroup is the standard “Iwahori subgroup” $I\coloneqq P^{n}$; i.e., $I$ is the preimage of the upper-triangular Borel subgroup $B$ (consisting of all upper-triangular matrices in $\operatorname{GL}_{n}(\mathbb{C})$) via the map $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])\to\operatorname{GL}_{n}(\mathbb{C})$ induced by the “evaluation at zero” map $z\mapsto 0$.

The most natural way of describing framability involves coadjoint orbits. One can view the principal part at $0$ of a connection as a continuous functional on $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C}[\![z]\!])$ via $Y\mapsto\operatorname{\mathrm{Res}}(\operatorname{Tr}(YX\frac{dz}{z}))$. Similarly, an $S^{{b}}$-formal type can be viewed as a functional on $\mathfrak{p}^{b}$. The global connection $\nabla$ then is framable at $0$ with respect to the $S^{{b}}$-formal type $A\frac{dz}{z}$ at $0$ if for some global trivialization, the restriction to $\mathfrak{p}^{b}$ of the principal part at $0$ lies in the $P^{b}$-coadjoint orbit of $A\frac{dz}{z}$. (See Definition 5.1.)

However, one can also give a description more reminiscent of the definition in the unramified case.

Recall that $\operatorname{GL}_{n}(\mathbb{C})$ acts simply transitively on the space of global trivializations. If one starts with a fixed trivialization $\phi^{\prime}$, then the choice of trivialization $\phi$ in the definition above corresponds to an element $g\in\operatorname{GL}_{n}(\mathbb{C})$; i.e., there is a unique $g$ such that $\phi=g\cdot\phi^{\prime}$. This matrix $g$ is called a compatible framing (or simply, a framing) of $\nabla$ at $0$. Framability with respect to a formal type at an arbitrary point $a\in\mathbb{P}^{1}$ is defined similarly, by simply replacing $z$ by $z-a$ if $a$ is finite, and by $z^{-1}$ if $a=\infty$.

We can now state the Deligne–Simpson problem for connections whose irregular singularities are all toral. Note that the statement below can easily be extended to allow for arbitrary unramified singular points.

This problem can be restated in terms of moduli spaces of connections. Suppose that each $\mathscr{O}_{j}$ is nonresonant; i.e., suppose that no pair of the eigenvalues of the orbit differ by a nonzero integer. Further assume that $m\geq 1$, so that there is at least one irregular singular point. In [BS13b], Bremer and Sage constructed the moduli space $\operatorname{\mathcal{M}}(\mathbf{A},\mathbf{O})$ of connections satisfying all the above hypotheses except irreducibility. Let $\operatorname{\mathcal{M}}_{\mathrm{irr}}(\mathbf{A},\mathbf{O})$ be the subset of the moduli space consisting of irreducible connections. In this language, the toral Deligne–Simpson problem poses the question of when $\operatorname{\mathcal{M}}_{\mathrm{irr}}(\mathbf{A},\mathbf{O})$ is nonempty.

We now restrict attention to a simple special case: connections with a maximally ramified irregular singularity and (possibly) an additional regular singular point. Without loss of generality, we will view such connections as connections on $\mathbb{G}_{m}$ with the irregular singularity at $0$. Following [KS21b], we refer to such connections as Coxeter connections. Well-known classical examples arise from the Airy differential equation and a modified version of the Bessel equation.²²2In these connections and others described below, the irregular singularity is at $\infty$. Another important class of examples consists of the generalized Kloosterman connections studied by Katz [Kat88, Kat90]. These hypergeometric connections are the geometric incarnations of certain exponential sums called Kloosterman sums, which are of great importance in number theory.

Coxeter connections and their ${G}$-connection analogues (for ${G}$ a simple algebraic group) have played a significant role in recent work in the geometric Langlands program. For example, Frenkel and Gross [FG09], building on work of Deligne [Del70] and Katz [Kat88, Kat96], constructed a rigid ${G}$-connection of this type. This connection, which may be viewed as a ${G}$-version of a modified Bessel connection, was the first connection with irregular singularities for which the geometric Langlands correspondence was understood explicitly [HNY13, Zhu17]. This connection also arises in Lam and Templier’s proof of mirror symmetry for minuscule flag varieties [LT17]. Other examples include the Airy ${G}$-connection and more general rigid “Coxeter ${G}$-connections” constructed in [KS21b]. The Airy ${G}$-connection and its $\ell$-adic analogue have also been studied in [JKY21].

Recall that if $\widehat{\nabla}$ is a rank $n$ formal connection, then every slope of $\widehat{\nabla}$ has denominator (when the slope is expressed in lowest form) between $1$ and $n$. We say $\widehat{\nabla}$ is maximally ramified if all such denominators (or equivalently, at least one) is $n$. In this case, all the slopes are the same — say $r/n$ with $\gcd(r,n)=1$ — and equal to the slope of the connection. More concretely, the leading term of the LT normal form is of the form $D_{r}z^{-r/n}$ with $D_{r}$ a constant diagonal matrix, and is necessarily regular.

It is shown in [KS21a] that maximally ramified connections are toral connections with respect to a Coxeter maximal torus. Thus any maximally ramified connection of slope $r/n$ has a rational canonical form $d+p(\omega_{n}^{-1})\frac{dz}{z}$, where $p$ is a polynomial of degree $r$, and the set of formal types is given by $\{p(\omega_{n}^{-1})\frac{dz}{z}\mid p\in\mathbb{C}[x],\deg(p)=r\}$. Moreover, any such connection is irreducible.

In this paper, we solve the ramified Deligne–Simpson problem for Coxeter connections. More precisely, let $\mathscr{A}$ be a maximally ramified formal type, and let $\mathscr{O}$ be an adjoint orbit (which we will always assume to be nonresonant). We determine necessary and sufficient conditions for the existence of a meromorphic connection $\nabla$ on $\mathbb{P}^{1}$ which is framable at $0$ with formal type $\mathscr{A}$, is regular singular with residue in $\mathscr{O}$ at $\infty$, and is otherwise nonsingular. Note that any such connection is automatically irreducible, since its formal connection at $0$ is irreducible. Thus, in the language of moduli spaces, we determine when $\operatorname{\mathcal{M}}(\mathscr{A},\mathscr{O})$ is nonempty.

In order to state our result, we need some facts about adjoint orbits. Fix a monic polynomial $q=\prod_{i=1}^{s}(x-a_{i})^{m_{i}}$ of degree $n$ with the $a_{i}$’s distinct complex numbers. The set $\{X\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})\mid\operatorname{char}(X)=q\}$ of matrices with characteristic polynomial $q$ is a closed subset of $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ which is stable under conjugation. We denote the set of orbits with characteristic polynomial $q$ by $\pi_{q}$. This set is partially ordered under the usual Zariski closure ordering: $\mathscr{O}\preceq\mathscr{O}^{\prime}$ if and only if $\mathscr{O}\subset\overline{\mathscr{O}^{\prime}}$. The theory of the Jordan canonical form makes it clear that $\pi_{q}$ can be identified with the Cartesian product $\prod_{i=1}^{s}\mathrm{Part}(m_{i})$, where $\mathrm{Part}(m_{i})$ denotes the set of partitions of $m_{i}$. Moreover, this identification defines a poset isomorphism between the closure ordering and the direct product of the dominance orders.

Given positive integers $r$ and $m$, there exists a unique smallest partition $\lambda^{{m},{r}}\in\mathrm{Part}(m)$ with at most $r$ parts. Define $\mathscr{O}_{q}^{r}$ to be the orbit in $\pi_{q}$ corresponding to the element

This tuple of partitions is the (unique) smallest element of $\prod_{i=1}^{s}\mathrm{Part}(m_{i})$ such that each component partition has at most $r$ parts. Note that $\mathscr{O}_{q}^{1}$ is just the regular orbit in $\pi_{q}$. On the other extreme, if $r\geq m_{i}$ for all $i$ (as is the case when $r\geq n$), then $\mathscr{O}_{q}^{r}$ is the semisimple orbit, the unique minimal orbit in $\pi_{q}$. Let $\langle{\mathscr{O}_{q}^{r}}\rangle$ denote the principal filter generated by $\mathscr{O}_{q}^{r}$ in $\pi_{q}$, i.e., $\mathscr{O}\in\pi_{q}$ satisfies $\mathscr{O}\in\langle{\mathscr{O}_{q}^{r}}\rangle$ if and only if $\mathscr{O}\succeq\mathscr{O}_{q}^{r}$. This filter is proper unless $r\geq m_{i}$ for all $i$. As we will see in Theorem 2.3, the collection of orbits $\mathscr{O}_{q}^{r}$ for each fixed $r$ satisfies a generalization of one characterization of regular orbits.

We can now give the solution to the Deligne–Simpson problem for Coxeter connections. Given a rank $n$ maximally ramified formal type $\mathscr{A}$ and a monic polynomial $q$ of degree $n$ that is nonresonant (i.e., no two roots differ by a nonzero integer), let

In other words, given $\mathscr{A}=p(\omega^{-1})\frac{dz}{z}$, then $\operatorname{\mathcal{M}}(\mathscr{A},\mathscr{O})$ is nonempty if and only if $np(0)=-\operatorname{Tr}(\mathscr{O})$ and $\mathscr{O}\succeq\mathscr{O}_{\operatorname{char}(\mathscr{O})}^{r}$. Concretely, the condition $\mathscr{O}\succeq\mathscr{O}_{\operatorname{char}(\mathscr{O})}^{r}$ means that $\mathscr{O}$ has at most $r$ Jordan blocks for each eigenvalue.

Note that the solution depends only on the slope and the residue of the formal type.

There is an obvious analogue of this problem for $\operatorname{SL}_{n}$-connections (as opposed to $\operatorname{GL}_{n}$-connections). Here, maximally ramified formal types are of the form $p(\omega^{-1})\frac{dz}{z}$ with $p(0)=0$ and $\operatorname{Tr}(\mathscr{O})=0$. Thus, the trace condition becomes vacuous, and the Deligne–Simpson problem has a positive solution if and only if $\mathscr{O}\succeq\mathscr{O}_{\operatorname{char}(\mathscr{O})}^{r}$.

One can define Coxeter ${G}$-connections for any simple group ${G}$ (or for any reductive group with connected Dynkin diagram) [KS21b]. For such a ${G}$, Coxeter toral connections have slope $r/h$, where $h$ is the Coxeter number for ${G}$ and $\gcd(r,h)=1$. Moreover, there is an analogue of the Deligne–Simpson problem in this more general context. We restrict to the case where the regular singularity at $\infty$ has nilpotent residue (and thus has unipotent monodromy).

Our results have applications to the question of when Coxeter connections are rigid. Let $U\subset\mathbb{P}^{1}$ be a nonempty open set, and let $j:U\hookrightarrow\mathbb{P}^{1}$ denote the inclusion. A ${G}$-connection $\nabla$ on $U$ is called physically rigid if it is uniquely determined by the formal isomorphism class at each point of $\mathbb{P}^{1}\setminus U$. It is called cohomologically rigid if $H^{1}(\mathbb{P}^{1},j_{!*}\mathrm{ad}_{\nabla})=0$. For irreducible connections, cohomological rigidity implies that $\nabla$ has no infinitesimal deformations. For ${G}=\operatorname{GL}_{n}(\mathbb{C})$, it is a result of Bloch and Esnault that cohomological and physical rigidity are the same [BE04].

In [KS21b], Kamgarpour and Sage investigated the question of rigidity for “homogeneous” Coxeter ${G}$-connections with unipotent monodromy. A homogeneous Coxeter ${G}$-formal type of slope $r/h$ (i.e., for $\operatorname{GL}_{n}$, a formal type of the form $a\omega^{-r}_{n}\frac{dz}{z}$ with $a\neq 0$) gives rise to a Coxeter ${G}$-connection on $\mathbb{G}_{m}$ with nilpotent residue at infinity. They determined precisely when these connections are (cohomologically) rigid, thus generalizing the work of Frenkel and Gross [FG09]. For $\operatorname{GL}_{n}$, it turns out that such connections are rigid precisely when $r$ divides $n+1$ or $n-1$.

We can now generalize the results of [KS21b] to give a classification of rigid Coxeter connections in type $A$.

We expect that the analogous statement is true for Coxeter ${G}$-connections.

In §2, we discuss some facts about the poset of adjoint orbits in $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ that will be needed in our applications. In particular, we introduce and characterize a sequence of orbits which generalize regular orbits. In §3, we provide a brief review of lattice chain filtrations. In §4, we describe the role of these filtrations in studying formal connections, following earlier work of Bremer and Sage [BS13b, BS12, Sag17]. In particular, we discuss toral connections and characterize maximally ramified formal connections as Coxeter toral connections. In §5, we describe moduli spaces of connections with toral singularities and then state and prove our main result on the Deligne–Simpson problem for Coxeter connections. We conclude the paper in §6 by characterizing rigid Coxeter connections with unipotent monodromy at the regular singular point.

The authors are deeply grateful to the American Institute of Mathematics (AIM) for hosting and generously supporting their research during SQuaRE meetings in 2020 and 2021. Many of the key ideas in this paper were conceived during these meetings. The authors would also like to thank an anonymous referee for helpful comments and suggestions that served to improve the manuscript.