The (cyclic) enhanced nilpotent cone via quiver representations

The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.


Introduction
Let V be a complex finite-dimensional vector space and N (V ) ⊂ End(V ) the nilpotent cone of V i.e. the variety of nilpotent endomorphisms of V . The enhanced nilpotent cone is defined as V × N (V ); it admits a diagonal action of GL(V ). This action has been examined in detail by several authors including Achar-Henderson [1], Travkin [22], Mautner [16] and Sun [21]. In particular the GL(V )-orbits in V × N (V ) are enumerated; it was shown by Achar-Henderson and Travkin that the orbits are naturally in bijection with bi-partitions of dim V . The orbit closure relations are also described combinatorially.
This classification of GL(V )-orbits in V × N (V ) was extended to the enhanced cyclic nilpotent cone by Johnson [14]. Let Q( ) be the cyclic quiver with vertices, and Q ∞ ( ) the framing ∞ → 0 of this quiver at the vertex 0. The enhanced cyclic nilpotent cone N ∞ ( , x) is the space of representations of Q ∞ ( ) with dimension one at the framing vertex ∞, and the endomorphism obtained by going once around the cycle having nilpotency ≤ x. The group G = −1 i=0 GL n acts on N ∞ ( , x) with finitely many orbits.
In this article, we return to the question of parameterizing the G-orbits in N ∞ ( , x). Our motivation comes from two quite different sources. Firstly, this is a generalization of the problem of studying parabolic conjugacy classes in the space of nilpotent endomorphisms; see section 1.2. Secondly, via the Fourier transform, the enhanced cyclic nilpotent cone plays a key role in the theory of admissible D-modules on the space of representations of Q ∞ ( ). Applications of our parameterization of orbits to the representation theory of admissible D-modules are explored in the sister article [3].

A representation theoretic parameterization
There are (essentially) two different approaches to the classification of G-orbits in the (cyclic) enhanced nilpotent cone. The first is to consider it as a problem in linear algebra, that of classifying pairs (v, X), of a nilpotent endomorphism X and a vector v ∈ V , up to change of basis ("coloured endomorphisms and coloured vectors" in the case of the enhanced cyclic nilpotent cone). This is the approach taken in [1], [22] and [14]. Secondly, one can consider it as a problem in representation theory. Namely, it is clear that the enhanced cyclic nilpotent cone parameterizes representations of a particular algebra A ∞ ( , x) (realized as an admissible quotient of the corresponding path algebra), with the appropriate dimension vector. The usual enhanced nilpotent cone corresponds to = 1. Then it is a matter of classifying the isomorphism classes of representations of this algebra. It is this latter approach that we take here. One natural way to try and classify the isomorphism classes of these algebras is to consider their universal covering algebras. This works well only if the representation type of the algebras A ∞ ( , x) is finite. Unfortunately, we show: Moreover, we show that if > 1 or x > 3, then for ( , x) = (4, 1), (2,2) the algebra A ∞ ( , x) is tame, and it is wild in all other case; see section 5.1. Fortunately, the algebra A( , x), whose representations correspond to the (non-enhanced) cyclic nilpotent cone, has finite representation type by Kempken [15]. Hence we deduce its indecomposable representations from the universal covering algebra. From this we can read off the isomorphism classes of indecomposable representations M of A ∞ ( , x) with (dim M ) ∞ = 1, even though A ∞ ( , x) does not have finite representation type in general. We introduce the set of Frobenius circle diagrams C F ( ); from each Frobenius circle diagram one can easily reconstruct an indecomposable nilpotent representation of Q ∞ ( ). Each Frobenius circle diagram C has a weight wt (C). We also introduce the weight wt (λ) of a partition λ; see section 3 for the definition of these combinatorial objects.

There are canonical bijections between:
• The set of isomorphism classes of indecomposable nilpotent representations M of Q ∞ ( ) with (dim M ) ∞ = 1.
• The set C F ( ) of Frobenius circle diagrams.
• The set of all partitions.

These bijections restrict to bijections between:
• The set of isomorphism classes of indecomposable representations M of A ∞ ( , x) with (dim M ) ∞ = 1.
• The set {C ∈ C F ( ) | wt (C) ≤ x} of Frobenius circle diagrams of weight at most x.
• The set {λ ∈ P | wt (λ) ≤ x} of partitions of weight at most x.
In the case of most interest to us, dim M = (1, n, . . . , n) = ε ∞ + nδ where δ is the minimal imaginary root for the cyclic quiver, the classification can be interpreted combinatorially. If P denotes the set of all partitions, P the set of all -multipartitions, then we show that: Here res (λ) and sres (ν) are the (shifted) -residues of the corresponding partitions; see section 5.3 for details. We note that our parameterization is clearly different from the parameterization given in [1], [22] and [14]. In the case of the usual enhanced nilpotent cone ( = 1), we explain in subsection 4.3 how to go between the two parameterizations (this is also explained in [16,Lemma 2.4]). When > 1, we relate our parameterization to the one given by Johnson [14] in subsection 5.

Parabolic conjugacy classes in the nilpotent cone
In [8], the second author considered the adjoint action of parabolic subgroups P ⊆ GL(V ) on the varieties N (x) of x-nilpotent endomorphisms of V . In particular, the question of which pairs (P, x) have the property that there are finitely many P -orbits on N (x) is addressed. The methods used in loc. cit. are mainly representation-theoretic: the algebraic group action is translated, via an associated fibre bundle, to a setup of representations of a certain finite quiver with relations. In all those cases (excluding the enhanced nilpotent cone) where there are finitely many P -orbits, the orbits are enumerated and the orbit closures are described in detail.
In this article, we describe how GL(V )-orbits in the enhanced nilpotent cone relate to the P -orbits of a particular parabolic (the "mirabolic") subgroup on the nilpotent cone N . More generally, for any dimension vector d of the cyclic quiver, there is a certain parabolic P ⊂ GL d such that GL d -orbits in the cyclic enhanced nilpotent cone N ∞ ( , x, d) are in bijection with P -orbits in the cyclic nilpotent cone N ( , x, d). This can be seen as a first step in the generalization of the above question to the case of parabolic conjugacy classes in the nilcone of Vinberg's θ-representations. See remark 5.2 for more details.

Admissible D-modules
As mentioned previously, one motivation for developing a quiver-theoretic approach to the G-orbits in the enhanced cyclic nilpotent cone is that one gets in this way immediate results regarding the category of admissible D-modules on the space X = Rep(Q ∞ ( ); v) of representations of the framed cyclic quiver. Fix a character χ of the Lie algebra g of G. The category C χ of admissible D-modules on X is the category of all smooth (G, χ)-monodromic D-modules on X, whose singular support lies in a certain Lagrangian Λ. Essentially those modules whose singular support is nilpotent in the conormal direction; see [3] for details. Admissible D-modules are always regular holonomic, and it is easily shown (since N ∞ ( , n) has finitely many G-orbits) that there are only finitely many simple objects in C χ . The behaviour of the category C χ depends heavily on the parameter χ.
Using the results of this article, we are able to describe precisely, in [3], the locus where C χ is semi-simple. It is shown that this is the complement to countably many (explicit) affine hyperplanes. In [3], we are able to list 10 other properties of the category C χ are equivalent to "C χ is semi-simple". The reason why our new parametrization of the G-orbits in the cyclic enhanced nilpotent cone is so useful in this context is because it allows us to easily compute the fundamental group of the orbits.

Outline of the article
In section two the required background results in representation theory are given. Section three introduces the combinatorial notions that we will use, in particular the notion of Frobenius circle diagrams. In section four we consider parabolic conjugacy classes in the nilpotent cone. Section five deals with the parameterization of orbits in the enhanced nilpotent cone (i.e. for = 1). Then the enhanced cyclic nilpotent cone is considered in section six. In particular, we prove Proposition 1.1, Theorem 1.2 and Corollary 1.3.

Outlook
Our representation-theoretic approach makes it possible to apply techniques from representation theory to better understand the geometry of the enhanced cyclic nilpotent cone. For example, there are several techniques available to calculate degenerations; that is, orbit closure relations. Namely, the results of Zwara [25,26] and Bongartz [6,7] are applicable. By making use of these, we hope to define combinatorially the closure ordering on the set Q(n, ) in the near future.

Quiver representations
The concepts in this subsection are explained in detail in [2]. A (finite) quiver Q is a direct graph Q = (Q 0 , Q 1 , s, t), with Q 0 a finite set of vertices, and Q 1 a finite set of arrows, with α : s(α) → t(α). The path algebra CQ is the C-vector space with basis consisting of all paths in Q, that is, sequences of arrows ω = α s . . . α 1 , such that t(α k ) = s(α k+1 ) for all k ∈ {1, . . . , s − 1}; we formally include a path ε i of length zero for each i ∈ Q 0 starting and ending in i. The multiplication ω · ω of two paths ω = α s ...α 1 and ω = β t ...β 1 is by concatenation if t(β t ) = s(α 1 ), and is zero otherwise. This way, CQ becomes an associative C-algebra. The path ideal I(CQ) of CQ is the (two-sided) ideal generated by all paths of positive length; then an arbitrary ideal I of CQ is called admissible if there exists an integer s with I(CQ) s ⊂ I ⊂ I(CQ) 2 .
A finite-dimensional C-representation of Q is a tuple of C-vector spaces M i and C-linear maps M α . There is the natural notion of a morphism For a representation M and a path ω in Q as above, we denote M ω = M αs ·. . .·M α1 . A representation M is said to be bound by I if ω λ ω M ω = 0 whenever ω λ ω ω ∈ I. Thus, we obtain certain categories: the abelian C-linear category rep C CQ of all representations of Q and the category rep C CQ/I of representations of Q bound by I; the latter is equivalent to the category of finite-dimensional A-representations, where A CQ/I is the quotient algebra. The Krull-Remak-Schmidt Theorem says that every finite-dimensional A-representation decomposes into a direct sum of indecomposable representations. We denote by Γ A = Γ(Q, I) the Auslander-Reiten quiver of rep C A.

Representation types
Consider a finite-dimensional, basic C-algebra A := CQ/I. The algebra A is said to be of finite representation type if there are only finitely many isomorphism classes of indecomposable representations. If it is not of finite representation type, the algebra is of infinite representation type. The Dichotomy Theorem of Drozd [11] says that if A is of infinite type, then A is one of two type: • tame representation type (or is tame) if, for every integer n, there is an integer k and finitely generated C[x]-A-bimodules M 1 , . . . , M k which are free over C[x], such that for all but finitely many isomorphism classes of indecomposable right A-modules M of dimension n, there are elements i ∈ {1, . . . , k} and λ ∈ C, such that M • wild representation type (or is wild) if there is a finitely generated C X, Y -A-bimodule M which is free over C X, Y and sends non-isomorphic finite-dimensional indecomposable C X, Y -modules via the functor _ ⊗ C X,Y M to non-isomorphic indecomposable A-modules.
If A is a tame algebra then there are at most one-parameter families of pairwise nonisomorphic indecomposable A-modules; in the wild case there are families of representations of arbitrary dimension.
Several different criteria are available to determine the representation type of an algebra. We say that an algebra B = CQ /I is a full subcategory of A = CQ/I, if Q is a convex subquiver of Q (that is, a path closed full subquiver) and I is the restriction of I to CQ .
An indecomposable projective P has separated radical if, for any two non-isomorphic direct summands of its radical, their supports (as subsets of Q) are disjoint. We say that A fulfills the separation condition if every projective indecomposable has a separated radical.
In general, the definition of a strongly simply connected algebra is quite involved. However, in case of a triangular algebra A (meaning that the corresponding quiver Q has no oriented cycles) there is an equivalent description: A is strongly simply connected if and only if every convex subcategory of A satisfies the separation condition [19]. For a triangular algebra A = CQ/I, the Tits form q A : Z Q0 → Z is the integral quadratic form defined by for v = (v i ) i ∈ Z Q0 ; here r(i, j) := dim ε i Rε j , for any minimal generating subspace R of I.
The quadratic form q A is called weakly positive, if q A (v) > 0 for every v ∈ N Q0 ; and (weakly) non-negative, if q A (v) ≥ 0 for every v ∈ Z Q0 (or v ∈ N Q0 , respectively). These concepts are closely related to the representation type of A and many results are, for example, summarized by De la Peña and Skowroński in [10]. There are many necessary and sufficient criteria for finite, tame and wild types available, for example by Bongartz [5] and Brüstle, De la Peña and Skowroński [9]. For our purposes, however, the following statement, which follows from these results, suffices.

Group actions
If the algebraic group G acts on an affine variety X, then X/G denotes the set of orbits and X//G := Spec C[X] G is the categorical quotient. The following is a well-known fact on associated fibre bundles [18], which will help translating certain group actions.

Lemma 2.2. Let G be an algebraic group, let X and Y be G-varieties, and let
Let H be the stabilizer of y 0 and set F := π −1 (y 0 ). Then X is isomorphic to the associated fibre bundle G × H F , and the embedding φ : F → X induces a bijection between H-orbits in F and G-orbits in X preserving orbit closures.
For an element x ∈ g := Lie G, its centralizer in G is denoted Z G (x), and its centralizer in g is Z g (x).

Combinatorial objects
In this subsection, we define the combinatorial objects we use later.

(Frobenius) Partitions and Young diagrams
The set of all weakly decreasing partitions is denoted P and P denotes the set of allmultipartitions. The subset of P, resp. of P , consisting of all partitions of n ∈ N, resp. of all -multipartitions of n, is denoted P(n), resp. P (n). Then P 2 (n) is the set of bipartitions of n, that is, of tuples of partitions (λ, µ), such that λ m and µ n − m for some integer m ≤ n.Given a partition λ, its Young diagram is denoted by Y (λ).
The transpose of the partition λ is denoted λ t and we define s(λ) to be the number of diagonal Definition 3.1. We denote by P F (n) the set of Frobenius partitions of n. That is, the set pairs of tuples of strictly decreasing integers (a 1 > · · · > a k ≥ 0) and We call k the length of (a, b). It is a classical result that the set of Frobenius partitions P F (n) can be naturally identified with the set P(n) of partitions of n. To be explicit, this is a bijection ϕ : P(n) → P F (n) defined as follows: . Graphically speaking, the Frobenius partition can be read off the Young diagram Y (λ): a i is the number of boxes below the ith diagonal and b i the number of boxes to the right of the ith diagonal.
We also associate to a Frobenius partition (a, b) the strictly decreasing partition One cannot recover (a, b) from P (a, b) in general. Moreover, ϕ(7, 5, 3, 2, 1) = ((4, 2, 0), (6, 3, 0)) which we naturally get by pigeon-holing the Frobenius partition into the partition: Pictorially, one simply counts the number of boxes of content 0 mod in the first Frobenius hook of λ.

The affine root system of type A
Throughout the article, Q( ) will denote the cyclic quiver with vertices, whose underlying graph is the Dynkin diagram of type A −1 . Then, as explained in the introduction, Q ∞ ( ) is the framed cyclic quiver. We denote by R ⊂ ZQ( ) 0 the set of roots and R + = R ∩ NQ( ) 0 the subset of positive roots. If δ = (1, . . . , 1) denotes the minimal imaginary root and Φ := {α ∈ R | ε 0 · α = 0} is the finite root system of type A −1 , then Let λ be a partition. Recall that the content ct( ) of the box ∈ Y (λ) in position (i, j) is the integer j − i. We fix a generator σ of the cyclic group Z . Given a partition λ, the -residue of λ is defined to be the element res (λ) := ∈λ σ ct( ) in the group algebra Z[Z ]. Similarly, given an -multipartition ν, the shifted -residue of ν is defined to be

Matrices
We fix the nilpotent cone N of nilpotent matrices in gl(V ). The GL(V )-conjugacy classes in N are labeled by their Jordan normal forms. In order to make use of these concepts later, we fix some notation here. We denote by J k the nilpotent Jordan block of size k; and by J λ the nilpotent matrix J λ1 ⊕ · · · ⊕ J λ k in Jordan normal form of block sizes λ 1 ≥ · · · ≥ λ k .

Circle diagrams
Given a positive integer > 0, we define a circle diagram of type to be a quiver C, whose set C 0 of vertices is partitioned into blocks b 0 , ..., b −1 , such that each vertex has at most one incoming arrow and one outgoing arrow; and an arrow can only be drawn from a vertex in block b i to a vertex in block b i+1 ; or from a vertex in block b −1 to a vertex in block b 0 , and there are no oriented cycles. We say that the vertices in block b i are in position i and call the vector d = (d 0 , . . . , d −1 ), where d i is the number of vertices in position i, the dimension vector of C. Given a circle diagram, each complete connected path of arrows is called a circle. The number of arrows in a circle is its length.
This circle diagram has one circle of length 1, one circle of length 2 and one circle of length 4.
The set of all circle diagrams of type , modulo permutation of vertices in the same position, is denoted C( ). The subset consisting of all circle diagrams, whose circles have length at most x, is denoted C (x) ( ). Furthermore, we denote by (C) the length of a circle diagram C, that is, the number of circles in the diagram.
A Frobenius circle diagram is a circle diagram C of type , with t circles, such that: 1. Each circle C(i) contains a distingushed (or marked) vertex s i in position 0.
2. If a i is the number of vertices following s i in the circle, and b i the number of vertices preceeding s i in the circle, then, after possibly relabelling circles, determine a Frobenius partition.
The set of Frobenius circle diagrams is denoted C F ( ).

Example 3.4.
A Frobenius circle diagram of (4, 5, 4, 3) is given by Then there are three circles: One circle of length 8 with mark s 1 = 4, one circle of length 4 with mark s 2 = 3 and one circle of length 1 with mark s 3 = 1. A Frobenius partition arises as ((3, 2, 0), (5, 2, 1)) which can be visualized by the partition (6,4,4,2); and by the diagrams Clearly, not every circle diagram with arbitrary marks yields a Frobenius circle diagram, as the following counterexample shows.
Counter-example 3.5. Consider the circle diagram with marks s 1 , s 2 and s 3 :

This does not correspond to any Frobenius partition.
By definition, each Frobenius circle diagram gives rise to a partition. Conversely, if a partition (a, b) is given in Frobenius form, then for each Frobenius hook (a i , b i ), we construct a circle C(i) whose vertices are in bijection with the boxes of the hook, a vertex u being in position i if the content of the corresponding box equals i modulo . Then there is an arrow from vertex u to vertex v if the box of v is above, or to the right of u, in the hook. Finally, the vertex s i corresponding to the hinge of the hook will be in position 0. We mark this vertex. In this way, we get a Frobenius circle diagram. It is straight-forward to check that this defines a bijection between the set of all Frobenius circle diagrams and the set of all partitions.
The weight of a circle is simply the number of vertices in block zero (or the number of times the circle passes through zero). The weight wt (C) of a Frobenius circle diagram is the weight of the longest circle. This notion is defined so that the weight of a Frobenius circle diagram equals the weight of the corresponding partition.

The enhanced nilpotent cone
Let V be an n-dimensional complex vector space, and N (V ) ⊂ End(V ) the nilpotent cone. We denote by . This action has been studied in [8]. In particular, the main result of loc. cit., together with Theorem 4.2 below, implies that: Cases 1. and 2. are described in detail in [8]. Let Q ∞ be the framed Jordan quiver: Part (2). There is a well-defined group homomorphism η : P → C × given by projection onto Thus, there are bijections between the sets of orbits These bijections preserve orbit closure relations, dimensions of stabilizers (of single points) and codimension of orbits. Therefore the closure order relation, orbit dimensions and singularity type of the orbits in V × N (V ) (x) / GL(V ), that were obtained in [1], can be translated into the corresponding information for orbits in N (x) /P .

Representation types
We begin to examine the representation theory of the algebra A ∞ (x) by figuring out, if there are infinitely many representations of a fixed dimension vector before discussing the representation type of A ∞ (x).

Lemma 4.3. There are only finitely many isomorphism classes of
Proof. Finiteness for d ∞ = 1 follows from [1], finiteness for x ≤ 3 follows from [8]. The fact that A ∞ (x) has infinite representation type in all other cases was shown in [8], where explicit one parameter families were constructed.
In order to decide whether the algebra A ∞ (x) is of finite representation type, tame or wild, we look at the universal covering quiver Γ ∞ of Q ∞ [13]: . Together with the admissible ideal (ϕ x ) (this notation means that the ideal is generated by every vertical path of length x), we obtain the covering algebra Γ ∞ (x) := CΓ ∞ /(ϕ x ). If Γ ∞ (x) is of wild representation type, then via the covering functor [13], the algebra A ∞ (x) is of wild representation type, as well. Proof. Finiteness for x ≤ 3 follows from Lemma 4.3. The algebra Γ(x) is strongly simply connected, since every convex subcategory is triangular and fulfills the separation condition: the radicals of all projective indecomposables are indecomposable. Thus, Corollary 2.1 implies that Γ ∞ (x) has wild representation type if and only if there is a dimension vector d ∈ N(Γ ∞ ) 0 , such that q Γ∞(x) (d) ≤ −1. If x ≥ 4, one such dimension vector is:

The indecomposable representations
In this section, we classify all indecomposable representations M of A ∞ (x) that have the property that (dim M ) ∞ ≤ 1.

Classification of indecomposables of dimension vector (0, n)
The Jordan normal form implies that there is (up to isomorphism) exactly one indecomposable representation with dimension vector (0, n), which is given by the Jordan block of size n. We denote the natural indecomposable representative by

Classification of indecomposables of dimension vector (1, n)
Some additional work is required to understand the indecomposable representations with dimension vector (1, n). We recall some notions from [1]. First, given a nilpotent matrix X of type λ n, a Jordan basis where µ ⊂ λ is a partition such that ν i = λ i − µ i also defines a partition. Thus, given a bipartition (µ; ν) of n, by choosing a normal basis, one gets an element (v, X) ∈ N ∞ (n). By [1,Proposition 2.3], the orbit of (v, X) does not depend on the choice of normal basis and the rule (µ; ν) → G · (v, X) =: Ξ(µ; ν) is a bijection Ξ : P 2 (n) → N ∞ (n)/G.
(a) We assume that (µ, ν) P 2,F (n). Let λ = µ+ν. If (λ) = 1, then (µ; ν) = ((n−k), (k)) for some k ≤ n. These all belong to P 2,F (n), except when k = n. This corresponds to X a single Jordan block and v = 0, which is clearly decomposable. Also, we note that if µ k = 0 and ν k 0, then let V 1 be the span of {v i,j | i < k} and V 2 the span of the {v k,j }. Then v ∈ V 1 and X(V i ) ⊂ V i (with X| V2 a Jordan block of length ν k ). Thus, (v, X) is decomposable. Therefore, we may assume that k > 1 and µ k 0.
(b) Take (µ; ν) ∈ P 2,F (n), and assume that the corresponding representation is decomposable i.e. V = V 1 ⊕ V 2 with v ∈ V 1 and X(V i ) ⊂ V i . By [1, Corollary 2.9], the Jordan type of X| Zg(X)·v is µ and the type of X| V /Zg(X)·v is ν. If the Jordan type of X| V1 is η and the type of X| V2 is ζ, then the fact that g X · v ⊂ V 1 implies that µ ⊆ η and ζ ⊆ ν.
Here λ = η ζ. The fact that V 2 0 implies that there exists some i such that µ i = 0 but ν i 0. But this contradicts the fact that (µ; ν) ∈ P 2,F (n).

Translating between the different parameterizations
The GL(V )-orbits in V × N (V ) were first studied by Bernstein [4]. It particular, it was noted there that there are only finitely many orbits. Explicit representatives of these orbits were independently given by Achar-Henderson [1, Proposition 2.3] and Travkin [22,Theorem 1].
Recall that the parameterization in loc. cit. is given by Ξ : P 2 (n) → N ∞ (n)/G.

The enhanced cyclic nilpotent cone
The results described above for the enhanced nilpotent cone all have analogues for the enhanced cyclic nilpotent cone. As one might expect, this situation is combinatorially more involved, but the approach is similar.
Let Q ∞ ( ) be the enhanced cyclic quiver with + 1 vertices.
We define the cyclic enhanced algebra to be Analogous to Theorem 4.2, we have Theorem 5.1. There is an isomorphism of GL d -varieties (resp. of GL d∞ -varieties): Rep(A( , x), d). Rep(A( , x), d).
There is an automorphism θ of g := gl N such that d). The space g 1 is a representation of GL θ N = GL d , and is an example of a θ-representation as introduced and studied by Vinberg [24]. Under the above identification, the cyclic nilpotent cone Rep(A( , x), d) is precisely the nilcone in the θ-representation g 1 . Therefore one can view Theorem 5.1 as a first step in a programme to study parabolic conjugacy classes in the nilcone of θ-representations. In particular, it raises the following problem: Classify all triples (G, θ, P ), where G is a reductive group over K, θ is a finite automorphism of g = Lie G and P ⊂ G θ is a parabolic subgroup such that the number of P -orbits in the nilcone N (g 1 ) is finite.

Representation types
We begin by classifying the representation type of the algebra A ∞ ( , x). The universal covering quiver Γ ∞ ( ) is given by . . . . . .
0 is a path of length . The quotient of this path algebra by the relations ϕ (i) • · · · • ϕ (i+x−1) • ϕ (i+x) | i ∈ Z gives the covering algebra Γ ∞ ( , x) := CΓ ∞ ( )/(ϕ x ). If Γ ∞ ( , x) is of wild representation type, then via the covering functor [13], the algebra A ∞ ( , x) is of wild representation type as well. Since the covering algebra is strongly simply connected (as every projective indecomposable admits at every vertex a vector space of dimension at most 1), we can make use of the results of subsection 2.2; in particular of Lemma 2.1. Furthermore, since Γ ∞ ( , x) is locally bounded (our ideal cancels infinite paths) and Z acts freely by shifts, we know by [13]: If Γ ∞ ( , x) is locally of finite representation type, then A ∞ ( , x) is of finite representation type and every indecomposable representation is obtained from an indecomposable Γ ∞ ( , x)-representation (via the obvious functor which builds direct sums of vector spaces in the same "column" and linear maps accordingly). In every remaining case, the algebra A ∞ ( , x) is of wild representation type.
Proof. This is a case by case analysis.
• Firstly, let = 1, that is, we are in the situation of the enhanced nilpotent cone. Then every case follows from Lemma 4.4.
-For x = 1, by knitting, we compute the Auslander-Reiten quiver of Γ ∞ (2, 1), which is finite. It is cyclic by means of a shift of the Z-action and is depicted in Appendix A.1. There, given a representation M of a finite slice of Γ ∞ (2, 1), we denote by M (i) for i ∈ Z the shifted representation M , such that the support of M (i) is non-zero in the i-th row (numbered from bottom to top) of Γ ∞ (2, 1), but zero below. By Covering Theory [13], the algebra A ∞ (2, 1) is, thus, of finite representation type.
-If x = 2, then the algebra is tame: The covering algebra contains an Euclidean subquiver of type D 6 and hence A ∞ (2, 2) has infinite representation type. It is indeed tame by [20,Theorem 2.4]: The Galois covering is strongly simply connected locally bounded and the algebra does not contain a convex subcategory which is hypercritical (see a list in [23]) or pg-critical (see a list in [17]).
-If x = 1, as above, we compute the Auslander-Reiten quiver of Γ ∞ (3, 1), which is finite. It is also cyclic by means of a shift of the Z-action and is depicted in Appendix A.2. There, given a representation M of a finite slice of Γ ∞ (3, 1), we denote by M (i) for i ∈ Z the shifted representation M , such that the support of M (i) is non-zero in the i-th row (numbered from bottom to top) of Γ ∞ (3, 1), but zero below. By Covering Theory [13], the algebra A ∞ (3, 1) is, thus, of finite representation type.
• Let = 4. The covering quiver contains an euclidean (and therefore tame) subquiver of type E 7 , thus, we always have infinite representation type.
-If x = 1, then then algebra A ∞ (4, 1) is tame by [20,Theorem 2.4]: The Galois covering is strongly simply connected locally bounded and the algebra does not contain a convex subcategory which is hypercritical (see a list in [23]) or pg-critical (see a list in [17]).
-For x ≥ 2, the algebra A ∞ (4, x) is of wild representation type, since the covering quiver contains the wild subquiver (see [23]): • If ≥ 5, then the algebra A ∞ ( , x) has wild representation type. In this case, the covering quiver Γ ∞ ( , x) contains the wild subquiver Thus, A ∞ (5, x) is of wild representation type for all x.

The indecomposable representations
As for the usual enhanced nilpotent cone, we consider separately the two cases: indecompos- Classification of indecomposables of dimension vector (0, * ) In this case, we are basically studying indecomposable representations of A( , x). For i ∈ Z and N ∈ N\{0}, let U (i, N ) be the N -dimensional indecomposable module defined as follows: as a C-vector space, it has a basis v 0 , . . . , v N −1 , with v k being a basis vector of the vector space at vertex i + k ∈ Z of U (i, N ). The linear maps of the representation map v k to v k+1 if possible; and to 0, otherwise. We can draw a picture of the Q(4)-representation U (2, 10) as follows, which makes clear the structure of the indecomposables: Proof. The universal covering quiver Γ is an infinite quiver of type A. However, the corresponding relations are a bit tricky. They are that for all i ∈ Z (and note that the composition of x · maps is always equals 0, as one might expect). But we can essentially ignore this and note that every indecomposable representation U Γ (i, N ) of Γ is nilpotent (here i ∈ Z and N ≥ 1, and the representation is defined just as for the cyclic quiver). So it suffices to check which of these factors through A( , x) after applying the covering functor F : Thus, the indecomposable representations of A( , x) that are obtained from Γ( , x) via the covering functor are precisely those U (i, N ) such that (i, N ) ∈ U( , x). Since we know by [15] that the algebra A( , x) is representation-finite, Covering Theory [13] implies that these are, in fact, all isomorphism classes of indecomposables.

Classification of indecomposables of dimension vector (1, * )
The second case deals with indecomposable The classification is given by Frobenius circle diagrams.

There are canonical bijections between:
• The set of isomorphism classes of indecomposable nilpotent representations M of Q ∞ ( ) with (dim M ) ∞ = 1.
• The set C F ( ) of Frobenius circle diagrams.
• The set of all partitions.

These bijections restrict to bijections between: • The set of isomorphism classes of indecomposable representations
• The set {C ∈ C F ( ) | wt (C) ≤ x} of Frobenius circle diagrams of weight at most x.
• The set of all partitions {λ ∈ P | wt (λ) ≤ x} of weight at most x.
Proof. It is clear that statement (2)  Given a Frobenius circle diagram C, we denote by M C the corresponding canonical indecomposable nilpotent representation.

A combinatorial parametrization
Given a fixed dimension vector d ∞ = (1, d 0 of indecomposable representations, for some unique tuple (C , C) ∈ C 2,F (d).
Recall that the GL d∞ -orbits in Rep(A ∞ ( , x), d ∞ ) are the same as the GL d -orbits. We denote the GL d -orbit of the representation (2) by O C ,C . We deduce from Theorem 5.6 that: In applications to admissible D-modules We associate to Y ν := Y the multipartition ν, where ν (i) = N j1 ≥ N j2 ≥ . . . , where the j r run over all 1 ≤ j r ≤ k such that i jr = i. Thus, the question is simply to find all (λ; ν) ∈ P × P such that dim(M λ ⊕ Y ν ) = e ∞ + nδ. Under the identification of ZQ( ) with Z[Z ], we have dim U (i, N ) = σ i res (N ) and hence dim Y = sres (ν).
Therefore it suffices to show that d λ = dim M λ equals ε ∞ + res (λ). Recall that Γ ∞ is the covering quiver of Q ∞ ( ) and Γ the covering quiver of Q( ). As in the proof of Theorem 5.4, let F : rep C CΓ → rep C CQ( ) denote the covering functor. If M λ is the unique lift of M λ to Γ ∞ such that v (i) = 0 for all i 0 (see section 5.1), then where (a 1 > · · · > a r ; b 1 > · · · > b r ) is λ written in Frobenius form and as required.
In the proof of Proposition 5.8 we have shown that If ν = ∅ then (λ; ν) ∈ Q(n, ) if and only if res (λ) = nδ. The set of all such λ is precisely the set of partitions of n that have trivial -core. This set, in turn, is in bijection with the set P (n) of -multipartitions of n, the bijection given by taking the -quotient of λ i.e. if λ has trivial -core then it is uniquely defined by its -quotient.

Translating between the different parametrizations
The goal of this final section is to describe how to pass directly between the combinatorial parametrization of the GL d -orbits in the enhanced cyclic nilpotent cone given by Johnson [14], and our parametrization given in Proposition 5.8. In order to do this, we first recall the former.
A tuple (λ, ) is called an -coloured partition if λ = (λ 1 , ..., λ k ) ∈ P and = ( 1 , ..., k ) ∈ (Z/ Z) k . This coloured partition gives rise to a coloured Young diagram Y (λ, ) by defining the colour of the box (i, j) of Y (λ) to be χ(i, j) denotes the residue class of x modulo . Its signature is defined to be ξ(λ, ) = (ξ(λ, ) m ) 0≤m≤ −1 and In this language, the well-known classification of orbits in the cyclic nilpotent cone can be stated as: Given an -coloured partition (λ, ) of signature d, it is mapped to the A( )-representation and the basis can, thus, be depicted best by the coloured Young diagram Y (λ, ).
Note that this parametrization can be directly translated to our circle diagrams of Theorem 5.4: The circle diagram consists of (λ) circles of which the i-th starts in vertex i and is of length λ i . We denote this circle diagram by C(λ, ), it corresponds to the representation 1≤i≤ (λ) Let us call a tuple (λ, , ν) a marked coloured partition if (λ, ) is a coloured partition and ν : N → Z is a marking function, which satisfies ν i ≤ λ i for all i. We define µ := (µ i ) 1≤i≤ (λ) = (λ i − ν i ) 1≤i≤ (λ) . Note that we have switched the roles of µ and ν in comparison to [14] -this is consistent with our conventions in subsection 4.3. A marked coloured partition (λ, , ν) is called a striped -bipartition, if 3. ν j < ν i + and µ j < µ i + for each i < j.
In the case = 1, this yields the set of double partitions P 2 (n), where n is the dimension vector, since (µ, ν) is a bi-partition of n. The set of all striped -bipartitions is denoted P st ( ); the subset with fixed signature ξ is denoted P st ( , ξ). The classification of orbits in the enhanced cyclic nilpotent cone, as in [14], is then given by: We can write down Ξ explicitly . Let (λ, , ν) ∈ P st ( , d), then there is a coloured Jordan basis B := {v i,j } 1≤i≤l(λ),1≤j≤λi and a nilpotent A( )-representation N in normal-form adapted to the basis B as described above. Set v i,j = 0 if j ≤ 0. Then Ξ (λ, , ν) is defined to be the orbit of the nilpotent representation (v, N ) of the cyclic enhanced nilpotent cone, Pictorially, this means that the i-th circle is marked at position µ i (not ν i , as one might expect). Interpreting Ξ (λ, , ν) as an A ∞ ( )-representation, we obtain Ξ (λ, , ν) where n = −1 i=0 d i and the right hand part is given by a circle diagram and, technically speaking, a graded vector space with a cycle of maps. Furthermore, ι = (λ) i=1 e i,νi and e i,j is the standard embedding into the (i, j)-th basis vector v i,j of B.
This completes the proof of the theorem.
We end this section by giving an example.