REPRESENTATION TYPE OF THE BLOCKS OF CATEGORY O_{S}

BRIAN D. BOE AND DANIEL K. NAKANO

Abstract. In this paper the authors investigate the representation type of the blocks of the relative (parabolic) category O_{S} for complex semisimple Lie algebras. A complete classification of the blocks corresponding to regular weights is given. The main results of the paper provide a classification of the blocks in the “mixed” case when the simple roots corresponding to the singular set and S do not meet.

1. Introduction

1.1. The study of indecomposable modules is a central theme in the representation theory of finite-dimensional algebras. One can place a finite-dimensional algebra in one of three classes depending on the indecomposable modules the algebra admits. A finite-dimensional algebra has finite representation type if the algebra has finitely many indecomposable modules up to isomorphism. If the algebra does not have finite representation type then the algebra is said to be of infinite representation type. Algebras of infinite representation type are either tame or wild. Tame algebras are the ones where there is some reasonable chance to determine all the indecomposable modules.

Cline, Parshall and Scott [CPS] introduced the concept of a quasi-hereditary algebra in connection with their study of highest weight categories in 1988. In this paper they proved that these highest weight categories are equivalent to the module category for some quasi-hereditary algebra. Quasi-herditary algebras are well-behaved in the sense that the projectives admit filtrations by certain standard modules. With such information one can sometimes deduce the structure of the projective modules directly. This knowledge allows one to express the algebra in terms of a quiver and relations (i.e. the basic algebra) and to determine the representation type of the algebra. The classical Schur algebras which arise in the study of the general linear groups and symmetric groups are important examples of quasi-hereditary algebras. The complete classification of the representation type of these algebras was given by Doty, Erdmann, Martin and the second author [Erd2, DN, DEMN]. The quantum Schur algebras were recently classified in [EN]. The ordinary Bernstein-Gelfand-Gelfand Category O for finite-dimensional complex semisimple Lie algebras is another well-known example of a highest weight category. Category O has a block decomposition and each block O_{μ} is equivalent to the module category for some finite-dimensional quasi-hereditary algebra. Futorny, Pollack and the second author completely classified the representation type for the blocks of Category O [FNP]. Brüstle, König, and Mazorchuk provided another proof of this result using covariant algebras [BKM]. The complete statement of these Category O results will be given in Section 1.2.

Date: August 11, 2003.

Research of the second author partially supported by NSF grant DMS-0102225.

In our paper we investigate the blocks of the relative category O_{S} for all finite-dimensional complex simple Lie algebras, where S is a subset of the simple roots. The blocks for the categories O_{S} in general are much more complicated than the blocks for ordinary Category O for several reasons. First, for a given central character the corresponding block of O_{S} may not contain any simple modules (see Corollary 4.2). Second, blocks for O_{S} need not be indecomposable (see Sections 8.2.1–8.2.3). Although there was some progress made on studying these categories in the late 1980s, there are still many open questions related to these types of issues. Finally, whereas the classifications given for ordinary Category O did not use the full force of the Kazhdan-Lustzig conjecture, this paper will rely heavily on the entire theory. In fact there are several instances where computer calculations of Kazhdan-Lusztig polynomials were necessary to verify the results.

We will look at a natural family of blocks for category O_{S} which we will call the “mixed case”. These blocks will be denoted by O(𝔤, S, J) where J is a subset of simple roots. This situation is particularly natural because the blocks are always non-empty when S ∩ J = ∅. A remarkable fact, described in Section 2, provides conditions which insure that the Ext^{1}- quivers of O(𝔤, S, J) and O(𝔤, J, S) naturally coincide. This result will always holds when the longest element of the Weyl group acts as −1. In classifying these blocks for category O_{S}, we provide concrete realizations of the blocks of finite and tame representation type. The process of classifying these blocks has led us to establish some new tools involving both translation functors and the Jantzen-Vogan U_{α}-algorithm studied (and generalized) earlier by Irving. Furthermore, we also extend results involving categorical equivalences of Enright and Shelton [ES] to a wider class of algebras.

The paper is organized as follows. In the remainder of Section 1, we present the main results of the paper. Section 2 is devoted to developing the machinery that will be used throughout the paper. In Section 3, we classify the representation type of the regular blocks of category O_{S}. With these results in addition to work of Enright and Shelton [ES] we determine (in Section 4) the representation type of blocks of O_{S} representing a Hermitian symmetric pair with a stabilizer of type A_{1}. The majority of the paper (Sections 5–9) is devoted to providing a complete classification of the representation type of the blocks O_{S}^{μ} of category O_{S} such that Φ_{μ} = Φ_{J} with J ∩ S = ∅. We call this the “mixed case” because these results simultaneously generalize both the results on the blocks of ordinary Category O and the regular blocks of category O_{S}.

We acknowledge Vyacheslav Futorny who suggested that we look at this problem. We thank Jon Carlson, Bill Graham, Ron Irving, Catharina Stroppel and Toshiyuki Tanisaki for several useful discussions pertaining to the details of the paper. The second author would also like to express his gratitude to the Maths Institute at Oxford University for their hospitality during the writing of this paper.

1.2. Notation. Let 𝔤 be a simple Lie algebra over the field of complex numbers ℂ. Let 𝔥 ⊂ 𝔤 be a maximal toral subalgebra and Φ be the root system of 𝔤 relative to 𝔥. Let Δ be a set of simple roots for Φ and denote by Φ^{+} (respectively Φ^{−}) the set of positive (respectively negative) roots with respect to Δ. For a given root system of rank n, the simple roots will be denoted by α_{1}, α_{2}, …, α_{n}. We use the same standard ordering of roots as given in Bourbaki [Bou]. In particular, for type B_{n}, α_{n} denotes the unique short simple root and for type C_{n}, α_{n} denotes the unique long simple root. Let 𝔤_{α} be the root subspace of 𝔤 corresponding to the root α ∈ Φ. We have the following Cartan decomposition

𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+},

where 𝔫^{+} = ∑_{α∈Φ^{+}} 𝔤_{α} and 𝔫^{−} = ∑_{α∈Φ^{+}} 𝔤_{−α}. Set 𝔟^{±} = 𝔥 ⊕ 𝔫^{±}.

Let ≤ be the natural partial order on 𝔥^{∗} defined by μ < λ if and only if λ − μ is a sum of positive roots. Let W be the Weyl group generated by the reflections s_{α} corresponding to the roots α, with longest element w_{0}, and let ρ be the half-sum of all positive roots. For i = 1, 2, …, n, let s_{i} := s_{α_{i}} be the reflection given by the simple root α_{i} ∈ Δ. The Weyl group W acts on 𝔥^{∗} via the dot action: w · μ = w(μ + ρ) − ρ. The standard inner product on 𝔥^{∗} will be denoted by (−, −). If α ∈ Φ, then let α̌ = 2α/(α, α) be the coroot of α.

Write ℕ (resp. ℤ_{+}) for the set of positive (resp. nonnegative) integers. Let X be the integral weight lattice and X^{+} be the set of quasi-dominant integral weights with respect to Φ defined by X^{+} = {μ ∈ X : (μ + ρ, α̌) ∈ ℤ_{+}, α ∈ Φ^{+}}. For μ ∈ X, let

Φ_{μ} = {α ∈ Φ : (μ + ρ, α̌) = 0}.

1.3. Let S be a subset of Δ and O_{S}^{μ} be the full subcategory of O_{S} containing the simple modules of highest weight ν with ν ∈ W · μ. The formal definition of the category O_{S} can be found in Section 2.1. The block O_{S}^{μ} is completely determined by specifying the triple (Φ, Φ_{S}, Φ_{μ}). We note that if Φ_{μ} ≠ ∅ then Φ_{μ} is a subroot system of Φ and μ is called singular. If Φ_{μ} = ∅ then μ is called regular. When μ is regular we may write O_{S}^{reg} for O_{S}^{μ}; by the Translation Principle these blocks are all Morita equivalent. The following theorem provides a summary of the representation type of the blocks of ordinary Category O (i.e. when S = ∅) [FNP, BKM]. Theorem. Let 𝔤 be a finite-dimensional complex simple Lie algebra with μ ∈ X and S = ∅.

(a) O_{S}^{μ} is semisimple if and only if (Φ, Φ_{S}, Φ_{μ}) = (Φ, ∅, Φ);

(b) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{μ}) = (A_{1}, ∅, ∅) or (A_{2}, ∅, A_{1});

(c) O_{S}^{μ} has tame representation type if and only if (Φ, Φ_{S}, Φ_{μ}) = (A_{3}, ∅, A_{2}) or (B_{2}, ∅, A_{1});

(d) In all other cases (where S = ∅), the block O_{S}^{μ} has wild representation type.

1.4. Statement of the main results: regular case. Note that the Weyl groups of types B_{n} and C_{n} are isomorphic, along with their Bruhat orders. It follows that corresponding blocks in B_{n} and C_{n} have the same representation type, so we combine these types here and in the sequel and label them as BC_{n}. The following theorem classifies the representation type of the blocks of category O_{S} for regular weights. Theorem. Let 𝔤 be a finite-dimensional complex simple Lie algebra with S ⊆ Δ and J = ∅.

(a) O_{S}^{reg} is semisimple if and only if (Φ, Φ_{S}, Φ_{J}) = (Φ, Φ, ∅);

(b) O_{S}^{reg} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{n}, A_{n−1}, ∅), (BC_{n}, BC_{n−1}, ∅), or (G_{2}, A_{1}, ∅);

(c) In all other cases (where J = ∅), the block O_{S}^{reg} has wild representation type.

1.5. Statement of the main results: Hermitian symmetric semiregular case. Using an equivalence of categories of Enright-Shelton, we will immediately deduce the following result about semiregular blocks (i.e. |J| = 1) in Hermitian symmetric categories.

Corollary. Let 𝔤 be a finite-dimensional complex simple Lie algebra, S ⊆ Δ, and suppose (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair. Let μ ∈ X be such that Φ_{μ} is of type A_{1}. (We assume Φ_{μ} is generated by a simple root, possibly in S.)

(a) O_{S}^{μ} is semisimple if and only if (Φ, Φ_{S}, Φ_{μ}) = (A_{n}, A_{n−1}, A_{1}) or (B_{n}, B_{n−1}, A_{1});

(b) O_{S}^{μ} has finite representation type if and only if

(Φ, Φ_{S}, Φ_{μ}) = \begin{cases}(A_{n}, A_{1} × A_{n−2}, A_{1}) & n ≥ 2 \\ (C_{n}, A_{n−1}, A_{1}) & n = 3, 4 \\ (D_{n}, D_{n−1}, A_{1}) & n ≥ 4 \\ (D_{n}, A_{n−1}, A_{1}) & n = 4, 5 \\ (E_{6}, D_{5}, A_{1}); \end{cases}

(c) In all other Hermitian symmetric semiregular cases, the block O_{S}^{μ} has wild representation type.

1.6. Statement of the main results: mixed case. We classify the blocks of O_{S}^{μ}, where Φ_{μ} = Φ_{J} with J ⊂ Δ, in the case where S ∩ J = ∅. When S = ∅ we recover the results for ordinary Category O. On the other hand when J = ∅ we recover the results for the regular blocks of category O_{S}. Theorem (A). Let 𝔤 be a finite-dimensional complex simple Lie algebra, S ⊆ Δ, and μ ∈ X with singular simple roots J ⊂ Δ. Let O_{S}^{μ} be a block of category O_{S} where S ∩ J = ∅ and S ∪ J ≠ Δ.

(a) If |Δ| − |S ∪ J| ≥ 2 then O_{S}^{μ} has wild representation type.

(b) Suppose that |Δ| − |S ∪ J| = 1. Then

(i) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{n}, A_{n−1}, ∅), (BC_{n}, BC_{n−1}, ∅), (G_{2}, A_{1}, ∅), (A_{1}, ∅, ∅), or (A_{2}, ∅, A_{1});

(ii) O_{S}^{μ} has tame representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{3}, ∅, A_{2}) or

(B_{2}, ∅, A_{1}); (iii) In all other cases when |Δ| − |S ∪ J| = 1, the block O_{S}^{μ} has wild representation

type.

Theorem (B). Let 𝔤 be a finite-dimensional complex simple Lie algebra, S ⊆ Δ, and μ ∈ X with singular simple roots J ⊂ Δ. Let O_{S}^{μ} be a block of category O_{S} where S ∩ J = ∅ and S ∪ J = Δ.

(a) If Φ is of type A_{n} then (i) O_{S}^{μ} is semisimple if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{n}, A_{n−r}, A_{r}) for r = 0, 1, …, n; (ii) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{n}, A_{1} × A_{r}, A_{n−r−1}) for r = 1, 2, …, n − 2 or (A_{n}, A_{n−r−1}, A_{1} × A_{r}) for r = 1, 2; (iii) O_{S}^{μ} has tame representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (A_{n}, A_{n−4}, A_{1} ×

A_{3}); (iv) In all other cases O_{S}^{μ} has wild representation type.

(b/c) If Φ is of type BC_{n} then

(i) O_{S}^{μ} is semisimple if and only if (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{1}, BC_{n−1}), (BC_{n}, BC_{n−1},

A_{1}), (BC_{n}, BC_{n}, ∅), or (BC_{n}, ∅, BC_{n});

(ii) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, BC_{n−2}, A_{2}),

(BC_{n}, A_{r}, BC_{n−r}) for r = 2, 3, (BC_{3}, A_{2}, A_{1}) or (BC_{4}, A_{3}, A_{1}); (iii) O_{S}^{μ} has tame representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, BC_{n−3}, A_{3}); (iv) In all other cases O_{S}^{μ} has wild representation type.

(d) If Φ is of type D_{n} then (i) O_{S}^{μ} is semisimple if and only if S = Δ or J = Δ; (ii) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{r}, D_{n−r})

for r = 1, 2, or (D_{n}, D_{n−1}, A_{1}); (iii) O_{S}^{μ} has tame representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (D_{n}, D_{n−2}, A_{2})

or (D_{5}, A_{1}, A_{4}); (iv) In all other cases O_{S}^{μ} has wild representation type.

(e) If Φ is of type E_{6}, E_{7}, or E_{8} then (i) O_{S}^{μ} is semisimple if and only if S = Δ or J = Δ; (ii) O_{S}^{μ} has finite representation type if and only if (Φ, Φ_{S}, Φ_{J}) = (E_{6}, D_{5}, A_{1}); (iii) In all other cases O_{S}^{μ} has wild representation type.

(f) If Φ is of type F_{4} then (i) O_{S}^{μ} is semisimple if and only if S = Δ or J = Δ; (ii) In all other cases O_{S}^{μ} has wild representation type.

(g) If Φ is of type G_{2} then O_{S}^{μ} is always semisimple.

2. Preliminaries

2.1. Category O_{S}. Let 𝔤 be a simple Lie algebra of rank n as in Section 1.2. The universal enveloping algebra of 𝔤 will be denoted by U(𝔤). Fix a subset S ⊂ Δ; we will often regard S as a subset of {1, …, n}, via our fixed ordering of the simple roots Δ = {α_{1}, …, α_{n}}. Then S determines, in the usual way (see [RC] for instance), a standard parabolic subalgebra 𝔭_{S} = 𝔪_{S} ⊕ 𝔲_{S}^{+} ⊇ 𝔟^{+} with Levi factor 𝔪_{S} and nilradical 𝔲_{S}^{+}. When appropriate the subscript S on 𝔭_{S}, 𝔪_{S}, and 𝔲_{S}^{+} will be suppressed. The root system Φ_{S} of 𝔪_{S} is Φ ∩ ∑_{α∈S} ℤα with simple roots S, and positive system Φ_{S}^{+} = Φ^{+} ∩ Φ_{S}. Let W_{S} be the Weyl group of Φ_{S} (viewed as a subgroup of W), and w_{S} the longest element of W_{S}.

Consider the category O_{S} (defined by Rocha-Caridi in [RC]) of 𝔤-modules V satisfying the following conditions.

(2.1.1) V is U(𝔤)-finitely generated.

(2.1.2) Viewed as a U(𝔪_{S})-module, V is a direct sum of finite-dimensional U(𝔪_{S})-modules.

(2.1.3) V is locally 𝔲_{S}^{+}-finite; i.e., dim_{ℂ} U(𝔲_{S}^{+})v < ∞ for all v ∈ V.

The key objects in this category are the generalized Verma modules. Let F (μ) be a finite-dimensional simple 𝔪_{S}-module with highest weight μ ∈ 𝔥^{∗}. (A necessary and sufficient condition for finite-dimensionality is that μ ∈ X_{S}^{+} = { μ ∈ 𝔥^{∗} : (μ, α̌) ∈ ℤ_{+} for all α ∈ Φ_{S}^{+} }.) Extend F (μ) to a 𝔭_{S}-module by letting 𝔲_{S}^{+} act by zero. The induced module

V (μ) = U(𝔤) ⊗_{U(𝔭)} F (μ)

is a generalized Verma module (GVM). It is a highest weight module for 𝔤 with highest weight μ, and thus a quotient of the (ordinary) Verma module M(μ) with highest weight μ. The GVM V (μ) is an object in category O_{S} which has finite length and a unique maximal submodule. The unique simple quotient will be denoted by L(μ) (which is also the unique simple quotient of M(μ)); see [Lep1] for the basic properties of GVMs. Moreover, all simple modules in O_{S} are obtained in this way.

In [RC] it was shown that the category O_{S} has enough projectives; i.e., each module has a projective cover. This implies that there is one-to-one correspondence between simple modules and projective indecomposable modules. For μ ∈ X_{S}^{+}, let P (μ) be the projective cover of L(μ) in O_{S}. Each projective module has a filtration with subquotients of the form V (ν) with ν ∈ X_{S}^{+}. Let [P (μ) : V (ν)] be the multiplicity of V (ν) as a subquotient of P (μ) in such a filtration, and [V (ν) : L(μ)] be the multiplicity of L(μ) in a composition series of V (ν). We have the following reciprocity law [RC, Theorem 6.1].

(2.1.4) Let μ, ν ∈ X_{S}^{+}. Then [P (μ) : V (ν)] = [V (ν) : L(μ)].

2.2. Blocks of O_{S}. Let Z(U(𝔤)) be the center of U(𝔤). A module V has a central character χ ∈ Z(U(𝔤))^{∗} if zv = χ(z)v for all z ∈ Z(U(𝔤)) and all v ∈ V. Let O_{S}^{χ} be the full subcategory of O_{S} consisting of those modules V on which z − χ(z) acts locally nilpotently for all z ∈ Z(U(𝔤)). The category O_{S} decomposes as

O_{S} = ⨁_{χ∈Z(U(𝔤))^{∗}} O_{S}^{χ}.

Under this decomposition every module in O_{S} decomposes into a direct sum of modules with each summand belonging to one of the subcategories O_{S}^{χ}. We refer to the subcategories O_{S}^{χ} as blocks of category O_{S} (although technically this is an abuse of language, because O_{S}^{χ} may further decompose into a direct sum of full subcategories of indecomposable components with nontrivial extensions between modules in different components).

Each Verma module M(μ) (or any subquotient thereof) admits a central character denoted by χ_{μ}. For every χ ∈ Z(U(𝔤))^{∗}, there exists μ ∈ 𝔥^{∗} such that χ = χ_{μ}. If χ = χ_{μ} we often write O_{S}^{μ} = O(𝔤, S, μ) = O(𝔤, 𝔭_{S}, μ) instead of O_{S}^{χ}. The following linkage principle of Harish-Chandra holds for central characters (see [Hum]):

(2.2.1) χ_{μ} = χ_{ν} if and only if ν ∈ W · μ.

Hence, we may parametrize the simple modules in the block O_{S}^{μ} as L(w_{S}w · μ) for those w ∈ W with w_{S}w · μ ∈ X_{S}^{+}. By replacing μ + ρ with a W -translate if necessary, we may (and do) assume henceforth that μ + ρ is antidominant (meaning that (μ + ρ, α̌) ∉ ℕ for all α ∈ Φ^{+}). We assume for the remainder of the paper that all weights are integral. If μ + ρ is also regular, then the set of all w such that w_{S}w · μ ∈ X_{S}^{+} is

SW = { w ∈ W : l(s_{i}w) = l(w) + 1 for all i ∈ S } = { w ∈ W : w^{−1}(Φ_{S}^{+}) ⊂ Φ^{+} },

the set of minimal length right coset representatives for W_{S} in W. By the Jantzen-Zuckerman translation principle, the categories O(𝔤, S, μ) for μ + ρ regular antidominant integral are all Morita equivalent; as mentioned previously, we usually write O_{S}^{reg} for any such category.

Suppose μ + ρ is singular. If α ∈ Φ^{+} and (μ + ρ, α) = 0, write α = ∑_{i=1}^{n} a_{i}α_{i} with a_{i} ∈ ℤ_{+}. Since antidominance (and integrality) of μ + ρ implies that (μ + ρ, α_{i}) ≤ 0 for all i, it follows that (μ + ρ, α_{i}) = 0 for all i such that a_{i} > 0. Hence, setting

J = { α ∈ Δ : (μ + ρ, α) = 0 },

we see that the set of roots on which μ + ρ is singular is the root system Φ_{J} = Φ ∩ ∑_{α∈J} ℤα with simple roots J and positive system Φ_{J}^{+} = Φ^{+} ∩ Φ_{J}. As with S, we will routinely view J as a subset of {1, … n} via our fixed indexing of Δ. Set

(2.2.2)  ^{S}W^{J} = { w ∈ ^{S}W : w < ws_{j} ∈ ^{S}W for all j ∈ J }.

Proposition. Let μ + ρ be a singular antidominant integral weight, and let J as above be the set of simple roots on which μ + ρ is singular. Then the inequivalent irreducible modules in O_{S}^{μ} are the L(w_{S}w · μ) for w ∈ ^{S}W^{J}.

Proof. We first claim that w_{S}w · μ ∈ X_{S}^{+} if and only if wW_{J} ⊂ ^{S}W. Then, since W_{J} is the stabilizer of μ, it remains only to pick one representative from each such left coset wW_{J}. We show that ^{S}W^{J} is the set of minimal length elements of these cosets.

Now w_{S}w · μ ∈ X_{S}^{+} if and only if (w_{S}w(μ + ρ), α̌) > 0 for all α ∈ Φ_{S}^{+}. Since (w_{S}w(μ + ρ), α̌) = (μ+ρ, w^{−1}w_{S}α̌), w_{S}(Φ_{S}^{+}) = −Φ_{S}^{+}, and μ+ρ is antidominant, a necessary condition is that w^{−1}(Φ_{S}^{+}) ⊂ Φ^{+}; i.e., w ∈ ^{S}W. Since (w_{S}wz)·μ = w_{S}w·μ for all z ∈ W_{J}, we conclude that wW_{J} ⊂ ^{S}W. Conversely, assume that wW_{J} ⊂ ^{S}W. Then (μ + ρ, w^{−1}w_{S}α̌) ≥ 0 for all α ∈ Φ_{S}^{+}; equivalently, (μ + ρ, w^{−1}α̌) ≤ 0 for all α ∈ Φ_{S}^{+}. Suppose (μ + ρ, w^{−1}α̌) = 0 for some such α. Then w^{−1}α ∈ Φ_{J}^{+}. But then w_{J}^{−1}w^{−1}α ∈ −Φ_{J}^{+} ⊂ −Φ^{+} (where w_{J} is the longest element of W_{J}), implying that ww_{J} ∉ ^{S}W. This contradiction completes the proof of the first claim.

Now suppose that wW_{J} ⊂ ^{S}W and w is the minimal length element of this coset. Then certainly w < ws_{j} for all j ∈ J, so w ∈ ^{S}W^{J}. Conversely, assume that w ∈ ^{S}W^{J}. First, note that for j ∈ J, wα_{j} > 0 (because w < ws_{j}). Moreover, if wα_{j} = β ∈ Φ_{S}^{+}, then s_{j}w^{−1}β = −α_{j} < 0, contradicting ws_{j} ∈ ^{S}W. Hence, since a sum of roots in Φ^{+} ∖ Φ_{S}^{+} cannot be in Φ_{S}^{+},

(2.2.3)  wΦ_{J}^{+} ⊆ Φ^{+} ∖ Φ_{S}^{+}

We now claim that if z ∈ W_{J} and j ∈ J satisfy wz ∈ ^{S}W and z < zs_{j}, then wz < wzs_{j} and wzs_{j} ∈ ^{S}W. This will prove that wW_{J} ⊂ ^{S}W and w is the minimal length element in wW_{J}, by induction on l(z). It remains to prove the claim. Since z < zs_{j}, zα_{j} > 0, and since z ∈ W_{J}, zα_{j} ∈ Φ_{J}; thus zα_{j} ∈ Φ_{J}^{+}. It follows from (2.2.3) that wzα_{j} ∈ Φ^{+} ∖ Φ_{S}^{+}. In particular, wzα_{j} > 0, hence wz < wzs_{j}. Finally, given any β ∈ Φ_{S}^{+}, we have z^{−1}w^{−1}β ∈ Φ^{+} (since wz ∈ ^{S}W), and z^{−1}w^{−1}β ≠ α_{j} (since wzα_{j} ∉ Φ_{S}^{+}). Thus s_{j}z^{−1}w^{−1}β ∈ Φ^{+}, and wzs_{j} ∈ ^{S}W. This completes the proof of the claim, and hence of the lemma. _{□}

Corollary. ^{S}W^{J} = ^{S}W ∩ w_{S}W^{J}

Proof. This is stated in [Irv4]. Its proof follows quickly from (2.2.3) and the characterization W^{J} = { w ∈ W : wΦ_{J}^{+} ⊂ Φ^{+} }. For if w ∈ ^{S}W^{J}, then (2.2.3) implies w_{S}wΦ_{J}^{+} ⊂ Φ^{+} ∖ Φ_{S}^{+}, so w_{S}w ∈ W^{J}, and, since w_{S}^{−1} = w_{S}, w ∈ w_{S}W^{J}.

Conversely, if w ∈ ^{S}W ∩ w_{S}W^{J}, then w_{S}wΦ_{J}^{+} ⊂ Φ^{+} implies wΦ_{J}^{+} ⊂ (Φ^{+} ∖ Φ_{S}^{+}) ∪ (−Φ_{S}^{+}). But w ∈ ^{S}W implies w^{−1}Φ_{S}^{+} ⊂ Φ^{+} so wΦ^{+} ∩ (−Φ_{S}^{+}) = ∅. Hence wΦ_{J}^{+} ⊂ Φ^{+} ∖ Φ_{S}^{+}, and in particular w < ws_{j} for all j ∈ J, so w ∈ ^{S}W^{J}. _{□}

We will see another characterization of ^{S}W^{J} in Section 2.4. Notation. When an integral weight μ with μ + ρ antidominant has been fixed, we write V (𝔤, 𝔭, w) = V (w) := V (w_{S}w·μ), and similarly for L(𝔤, 𝔭, w) = L(w) and P (𝔤, 𝔭, w) = P (w). This convention relates to the Bruhat order and length function on W as follows:

[V (w) : L(x)] > 0 ⟹ x ≤ w ⟹ l(x) ≤ l(w).

The GVM V (e) always equals L(e). This choice of parametrization (with μ + ρ antidominant rather than dominant, and translating by w_{S}) has been made for compatibility with the equivalence of categories in Section 2.9—provided the simple roots of 𝔩 and 𝔤 are numbered compatibly, then a GVM for 𝔩 parametrized by w corresponds to the GVM for 𝔤 parametrized by the same w.

In summary, under our assumption that μ+ρ is antidominant integral, the non-isomorphic simple modules in the category O(𝔤, S, μ) = O(𝔤, S, J) are the L(w), w ∈ ^{S}W^{J}, and all projective indecomposable modules in this block are of the form P (w), w ∈ ^{S}W^{J}. Moreover, the category O(𝔤, S, μ) is a highest weight category (in the sense of [Don, CPS]) with respect to generalized Verma modules V (w), w ∈ ^{S}W^{J} [RC, Sections 5–6].

2.3. Extensions. We present here some facts about extensions in category O_{S} and its subcategories; uncited material in this subsection appears in [Irv3, Section 1.6]. Each generalized Verma module has a finite radical filtration, which equals its weight filtration [BBD, Cas]. If V is a finite length module, put rad^{0}V = V, and for i ≥ 0, rad^{i+1}V = rad(rad^{i}V) and rad_{i}V = rad^{i}V/rad^{i+1}V, the ith layer of the radical filtration of V. Since category O_{S} is a highest weight category, every extension between two irreducibles arises from an extension between them in layers 0 and 1 of the radical filtration of some generalized Verma module. As a consequence, one has Ext_{O_{S}}^{1}(L_{1}, L_{2}) ≃ Ext_{O}^{1}(L_{1}, L_{2}) when L_{1} and L_{2} are simple modules in O_{S}. Moreover, the category O_{S} has a contravariant duality which fixes simple modules. This implies that Ext_{O_{S}}^{1}(L_{1}, L_{2}) ≃ Ext_{O_{S}}^{1}(L_{2}, L_{1}).

Suppose λ is an antidominant integral regular weight, and μ is an antidominant integral weight, with J its set of singular simple roots. Given x, w ∈ ^{S}W^{J}, one has

(2.3.1)  [rad_{i}V (w_{S}w · μ) : L(w_{S}x · μ)] = [rad_{i}V (w_{S}w · λ) : L(w_{S}x · λ)].

This latter multiplicity is given by the appropriate coefficient of an inverse “relative” Kazhdan-Lusztig (KL) polynomial (associated to the relative category O_{S}) [CC]. In particular, let μ_{S}(−, −) be the relative Kazhdan-Lusztig μ-function: for x < w, both in ^{S}W, μ_{S}(x, w) is defined to be the coefficient of q^{(l(w)−l(x)−1)/2} in the relative KL polynomial P_{x,w}^{S}(q). Then [rad_{1}V (w_{S}w · λ) : L(w_{S}x · λ)] = μ_{S}(x, w). Therefore, when x, w ∈ ^{S}W^{J} with x < w, dim Ext_{O_{S}}^{1}(L(w_{S}w · μ), L(w_{S}x · μ)) = μ_{S}(x, w).

2.4. Parabolic-Singular Duality. Given 𝔤 and fixed subsets S and J of Δ, we shall call the two categories O(𝔤, S, J) and O(𝔤, J, S) complementary. In this subsection we will show that, under favorable circumstances (which occur “most of the time”), the Ext^{1}- quivers of complementary categories are canonically isomorphic. This follows from results of Beilinson-Ginzburg-Soergel [BGS] in the case S = ∅, and of E. Backelin [Bac] in the general case. However, these authors use techniques from the theory of perverse sheaves, so we will give more elementary proofs here. Following them, we shall call this situation parabolic-singular duality, or (in brief) S–J–duality.

2.4.1. Parameters for complementary categories. Lemma. ^{S}W^{J} = { w ∈ W: wΦ_{J}^{+} ⊂ Φ^{+} ∖ Φ_{S}^{+} and w^{−1}Φ_{S}^{+} ⊂ Φ^{+} ∖ Φ_{J}^{+} }.

Proof. Let w ∈ ^{S}W^{J}. We showed in (2.2.3) that wΦ_{J}^{+} ⊂ Φ^{+} ∖ Φ_{S}^{+}. When combined with w^{−1}Φ_{S}^{+} ⊂ Φ^{+} (since w ∈ ^{S}W) this also implies w^{−1}Φ_{S}^{+} ⊂ Φ^{+} ∖ Φ_{J}^{+}.

Conversely assume w belongs to the set on the right in the statement of the lemma. Since w^{−1}Φ_{S}^{+} ⊂ Φ^{+} we have w ∈ ^{S}W. Now let j ∈ J. Since w^{−1}Φ_{S}^{+} ⊂ Φ^{+} ∖ {α_{j}} we have s_{j}w^{−1}Φ_{S}^{+} ⊂ Φ^{+} ∖ {α_{j}}, and thus ws_{j} ∈ ^{S}W. Finally, wΦ_{J}^{+} ⊂ Φ^{+} implies w(α_{j}) > 0 so w < ws_{j}. By (2.2.2), w ∈ ^{S}W^{J}. _{□}

Corollary. ^{S}W^{J} = (^{J}W^{S})^{−1}.

The corollary implies that there is a one-to-one correspondence (induced by the map w ↦ w^{−1}) between the simple modules of the complementary categories O(𝔤, S, J) and O(𝔤, J, S).

2.4.2. Duality on ^{S}W^{J}. Recall that w_{0} is the longest element of the Weyl group W, and that −w_{0}(Δ) = Δ. We write w_{S} (resp. w_{J}) for the unique longest element of the parabolic subgroup W_{S} (resp. W_{J}). Each of these elements of W is its own inverse. Note also that w_{S}(Φ_{S}^{+}) = −Φ_{S}^{+} and hence (by length considerations) w_{S}(Φ^{+} ∖ Φ_{S}^{+}) = Φ^{+} ∖ Φ_{S}^{+}; similarly for w_{J}. Lemma. Let S be a subset of Δ. Then −w_{0}(S) = S if and only if w_{S}w_{0} = w_{0}w_{S}.

Proof. Assume that −w_{0}(S) = S. For α ∈ S we have w_{0}s_{α}w_{0}^{−1} = s_{w_{0}(α)} = s_{−w_{0}(α)} ∈ W_{S}. Thus w_{0}w_{S}w_{0}^{−1} ∈ W_{S}, and cannot have shorter length that w_{S}, otherwise conjugating by w_{0} again would produce an expression for w_{S} having fewer than l(w_{S}) simple factors. Hence w_{0}w_{S}w_{0}^{−1} = w_{S}.

Conversely, assume that w_{S}w_{0} = w_{0}w_{S}, and let α ∈ S. Put w_{0}(α) = −β where β ∈ Δ, and suppose β ∉ S. Since w_{S}(Φ^{+} ∖ Φ_{S}^{+}) = (Φ^{+} ∖ Φ_{S}^{+}), we have w_{S}w_{0}(α) = −w_{S}(β) ∈ Φ^{−}, whereas w_{0}w_{S}(α) ∈ w_{0}(Φ^{−}) = Φ^{+}. This is a contradiction. _{□}

Definition. If X and Y are subsets of Coxeter groups, a map f: X → Y is lengthcomplementary if there is a constant N such that l(f(x)) = N − l(x) for all x ∈ X. Proposition. (a) There is a length-complementary bijection ^{S}f^{J}: ^{S}W^{J} → ^{J}W^{−w_{0}S} given by ^{S}f^{J}(w) = w_{J}w^{−1}w_{S}w_{0}.

(b) There is a length-complementary bijection ^{S}g^{J}: ^{S}W^{J} → ^{−w_{0}J}W^{S} given by ^{S}g^{J}(w) = w_{0}w_{J}w^{−1}w_{S}.

Proof. Let w ∈ ^{S}W^{J}. Using Lemma 2.4.1 we have (^{S}f^{J}(w))^{−1}Φ_{J}^{+} = w_{0}w_{S}ww_{J}Φ_{J}^{+} = −w_{0}w_{S}wΦ_{J}^{+} ⊂ −w_{0}w_{S}(Φ^{+} ∖ Φ_{S}^{+}) = −w_{0}(Φ^{+} ∖ Φ_{S}^{+}) = Φ^{+} ∖ Φ_{−w_{0}S}^{+}. Also (^{S}f^{J}(w))Φ_{−w_{0}S}^{+} = w_{J}w^{−1}w_{S}w_{0}Φ_{−w_{0}S}^{+} = −w_{J}w^{−1}w_{S}Φ_{S}^{+} = w_{J}w^{−1}Φ_{S}^{+} ⊂ w_{J}(Φ^{+} ∖ Φ_{J}^{+}) = Φ^{+} ∖ Φ_{J}^{+}. Thus Sf^{J}(w) ∈ ^{J}W^{−w_{0}S}. Similarly ^{S}g^{J}(w) ∈ ^{−w_{0}J}W^{S}. The composition ^{J}g^{−w_{0}S} ∘^{S}f^{J}: ^{S}W^{J} → SW^{J} sends w to w_{0}w_{−w_{0}S}(w_{J}w^{−1}w_{S}w_{0})^{−1}w_{J} = w_{0}(w_{0}w_{S}w_{0})(w_{0}w_{S}ww_{J})w_{J} = w. Similarly Sf^{J} ∘^{J}g^{−w_{0}S} is the identity on ^{J}W^{−w_{0}S}. This proves that ^{S}f^{J} (similarly ^{S}g^{J}) is a bijection.

Recall that w is the minimal-length element in its coset wW_{J} and that wW_{J} ⊂ ^{S}W. Thus l(ww_{J}) = l(w) + l(w_{J}) and l(w_{S}ww_{J}) = l(w_{S}) + l(w) + l(w_{J}). Thus l(w_{J}w^{−1}w_{S}w_{0}) = l(w_{0}) − l(w_{J}w^{−1}w_{S}) = l(w_{0}) − l(w_{S}ww_{J}) = l(w_{0}) − l(w_{S}) − l(w_{J}) − l(w), proving that ^{S}f^{J} (and similarly ^{S}g^{J}) is length-complementary, with constant N = l(w_{0}) − l(w_{S}) − l(w_{J}). _{□}

Corollary. (a) Assume that −w_{0}(S) = S. Then w ↦ w^{♭} := w_{0} w_{S} w w_{J} is a lengthcomplementary involution of ^{S}W^{J}.

(b) Assume that −w_{0}(J) = J. Then w ↦ w^{#} := w_{S} w w_{J} w_{0} is a length-complementary involution of ^{S}W^{J}.

Proof. (a) Assume that −w_{0}(S) = S. Then ^{S}f^{J}: ^{S}W^{J} → ^{J}W^{S} is a length-complementary bijection, and (by Corollary 2.4.1) inversion is a length-preserving bijection from ^{J}W^{S} to SW^{J}. Thus the composition, which sends w to w^{♭}, is a length-complementary bijection of SW^{J} onto itself. Lemma 2.4.2 implies that it is actually an involution.

Part (b) follows similarly. _{□}

Remark. When −w_{0}(J) = J and −w_{0}(S) = S, w^{#} and w^{♭} need not be equal. Indeed, when the hypothesis is satisfied, w^{#} = w^{♭} ⟺ w w_{0} = w_{0} w ⟺ w_{0}ww_{0} = w. Take 𝔤 to be of type A_{2}, J = S = ∅ (so that ^{S}W^{J} = W), and w = s_{1}; then w_{0}s_{1}w_{0} = s_{2}.

2.4.3. Ext^{1}-quivers of complementary categories. Theorem. (a) ^{S}f^{J} induces an isomorphism between the Ext^{1}-quivers of O(𝔤, S, J) and O(𝔤, J, −w_{0}(S)).

(b) ^{S}g^{J} induces an isomorphism between the Ext^{1}-quivers of O(𝔤, S, J) and O(𝔤, −w_{0}(J), S).

(c) In particular, if −w_{0}(S) = S or −w_{0}(J) = J then the Ext^{1}-quivers of O(𝔤, S, J) and O(𝔤, J, S) are naturally isomorphic.

Proof. (a) Set O_{1} = O(𝔤, S, J) and O_{2} = O(𝔤, J, −w_{0}(S)). Write μ(−, −) for the ordinary Category O Kazhdan-Lusztig μ-function. Let x, w ∈ ^{S}W^{J}. Then

dim Ext_{O_{1}}^{1}(L(x), L(w)) = μ_{S}(x, w)
= μ(w_{S} x, w_{S} w)
= μ(x^{−1} w_{S}, w^{−1} w_{S})
= μ(w_{J} w_{J} w^{−1} w_{S} w_{0}, w_{J} w_{J} x^{−1} w_{S} w_{0})
= μ(w_{J} ^{S}f^{J}(w), w_{J} ^{S}f^{J}(x))
= μ_{J}((^{S}f^{J}(w),^{S}f^{J}(x))
= dim Ext_{O_{2}}^{1}(L(^{S}f^{J}(w)), L(^{S}f^{J}(x)),

where the second and second-last equalities follow from [Deo2, Proposition 3.4], the third from [Dye, p. 356], and the fourth from (a “right-hand” version of) [KL, Corollary 3.2].

(b) The proof for ^{S}g^{J} is almost identical, replacing the right side of the fourth equation with μ(w_{0} w_{J} w_{0} w_{0} w_{J} w^{−1} w_{S}, w_{0} w_{J} w_{0} w_{0} w_{J} x^{−1} w_{S} ), and using the fact that w_{0} w_{J} w_{0} = w_{−w_{0}(J)} in the fifth equation. More conceptually, (b) follows from (a) by applying the automorphism −w_{0} to (Δ, J, −w_{0}(S)) to obtain (Δ, −w_{0}(J), S).

Part (c) is an immediate consequence of (a) and (b). _{□}

The conclusion of (c) can be false if −w_{0} preserves neither S nor J. For example, the quiver for (A_{4}, {1, 2}, {3}) consists of a “domino” (two squares sharing a common edge) with an extra vertex attached to one of its corners, whereas the quiver for (A_{4}, {3}, {1, 2}) consists of two squares sharing a common vertex. (All edges are actually bi-directional, as is true in the Ext^{1}-quiver of any category O_{S}^{μ}.) Nor do the conditions S ∪ J = Δ, S ∩ J = ∅ ensure S–J–duality; for instance, the quivers for (A_{5}, {1, 3, 4}, {2, 5}) and its complement are not isomorphic.

On the other hand, the conclusion of (c) is sometimes true even when no natural duality exists on ^{S}W^{J}. The quivers of (A_{6}, {1, 2, 3, 5, 6}, {4}) and its complement are isomorphic graphs on 10 vertices, but there is no isomorphism which is length-complementary on the corresponding Weyl group parameters.

2.4.4. The previous theorem has some interesting consequences. Corollary (A). For any 𝔤 and S, O(𝔤, S, ∅) and O(𝔤, ∅, S) have (naturally) isomorphic Ext^{1}-quivers.

However, the categories O(𝔤, S, ∅) and O(𝔤, ∅, S) need not be Morita equivalent. A striking example is provided by (G_{2}, A_{1}, ∅), which has finite representation type, whereas (G_{2}, ∅, A_{1}) has wild representation type. In the former category, the GVMs other than the irreducible one are uniserial of radical length two, whereas in the latter, the GVMs are multiplicity-free uniserial of lengths 1, 2, …, 6. But the Ext^{1}-quivers separate into two copies of the quiver of type A_{6}. Corollary (B). O_{S}^{reg} is always a single indecomposable block.

Proof. A category is indecomposable if and only if its Ext^{1}-quiver is connected. This is the case for O(𝔤, ∅, S), because W^{S} is connected by Bruhat covering relations [Deo1, Corollary 3.8], and these always correspond to non-zero Ext^{1} groups between the associated simple modules. Now use the previous corollary. _{□}

2.5. Translation functors. For each λ ∈ w_{0} · X^{+}, let pr_{λ} be the exact functor taking a module in O to its component in O^{χ_{λ}}. Define the translation functor as follows. For λ, μ ∈ w_{0} · X^{+} let ν ∈ W (μ − λ) ∩ X_{Δ}^{+} where X_{Δ}^{+} is the set of dominant weights and set

(2.5.1)  T_{λ}^{μ}M = pr_{μ}((pr_{λ}M) ⊗ L(ν)).

For a fixed set S ⊆ Δ, let Tr : O → O and Res : O → O_{S} be defined as in [Irv3, pp. 55– 56]. By definition Tr(M) is the smallest submodule of M such that Res(M) := M/Tr(M) is in O_{S}. Observe that Res is right exact, but not exact (as stated in [Irv3, Proposition 5.1.2]). To see why the functor Res is right exact, let σ : M → N be an epimorphism.

Set τ : M → N → Res(N) be the composition of σ with the canonical epimorphism N → Res(N). Then M/ker τ ≅ Res(N) is in O_{S}. Therefore, by definition of Tr(M), we have an epimorphism Res(M) → Res(N) induced by σ. Let L^{j}(−) := L^{j}Res(−) denote the higher left derived functors of Res(−).

We claim that Res is acyclic on all (ordinary) Verma modules (i.e. L^{j}(M(w · λ)) = 0 for all w ∈ W and j > 0). For δ maximal with respect to the ordering on weights in a block of O, M(δ) = P (δ) is projective, and so L^{j}(M(δ)) = L^{j}(P (δ)) = 0 for j > 0. Now let δ be arbitrary and assume inductively that for all γ > δ, we have L^{j}(M(γ)) = 0 for j > 0. We have a short exact sequence

0 → N → P (δ) → M(δ) → 0

with N having a Verma module filtration with factors M(γ) with γ > δ. It follows by using the long exact sequence in cohomology and the induction hypothesis that L^{j}(M(δ)) = 0 for all j > 0, and the functor Res preserves exact sequences of modules having Verma module filtrations. The following proposition extends Irving’s result [Irv3, Lemma 5.2.1] with some corrections and modifications to the proof. Proposition. Let λ, μ ∈ w_{0} · X^{+} with Φ_{λ} = Φ_{J} (resp. Φ_{μ} = Φ_{J′}), and w ∈ ^{S}W^{J}. Then T_{λ}^{μ}(V (w · λ)) has a filtration with subquotients of the form V (ww_{1} · μ) where w_{1} ∈ W^{λ} := Stab_{W} λ and ww_{1} ∈ ^{S}W^{J′}. All such GVMs occur with multiplicity one in the filtration.

Proof. One can verify using the proof in [Jan2, II 7.13] that T_{λ}^{μ}(M(w · λ)) has a filtration such that the factors are of the form M(ww_{1} · μ) with w_{1} ∈ W^{λ}. Each M(ww_{1}) occurs once as a factor in the filtration. Since Res is acyclic on Verma modules and takes Verma modules to generalized Verma modules, Res(T_{λ}^{μ}(M(w · λ))) has a filtration such that the factors are of the form V (ww_{1} · μ) with w_{1} ∈ W^{λ} with each such factor occuring once.

We need to prove that Res(T_{λ}^{μ}(M(w · λ))) ≅ T_{λ}^{μ}(Res(M(w · λ))) ≅ T_{λ}^{μ}(V (w · λ)). The last isomorphism follows by [Irv3, Proposition 5.1.2(ii)]. We will prove in general that Res(T_{λ}^{μ}(M)) ≅ T_{λ}^{μ}(Res(M)) for M ∈ O^{λ}.

First, we claim that if Tr(N) = N (N ∈ O^{λ}) then Tr(T_{λ}^{μ}(N)) = T_{λ}^{μ}(N). Suppose that T_{λ}^{μ}(N) ≠ 0 and L(σ) ⊂ rad_{0}T_{λ}^{μ}(N). It follows by adjointness that

0 ≠ Hom_{O_{S}}(T_{λ}^{μ}(N), L(σ)) = Hom_{O_{S}}(N, T_{μ}^{λ}(L(σ))).

If L(σ) is in O_{S} then T_{μ}^{λ}(L(σ)) is in O_{S} and then some composition factor in rad_{0}N is in O_{S}. This contradicts the fact that Tr(N) = N. Hence, L(σ) is not in O_{S} and T_{λ}^{μ}(N) = Tr(T_{λ}^{μ}(N)).

By definition we have a short exact sequence

(2.5.2)  0 → Tr(M) → M → Res(M) → 0.

Since T_{λ}^{μ} is exact and Res is right exact we get the following long exact sequence, by applying Res ∘ T_{λ}^{μ} to (2.5.2):

(2.5.3) ⋯ → L^{1}(T_{λ}^{μ}(Res(M))) → Res(T_{λ}^{μ}(Tr(M))) → Res(T_{λ}^{μ}(M)) → T_{λ}^{μ}(Res(M)) → 0.

In the last term we used the fact that T_{λ}^{μ} preserves O_{S} and hence Res(T_{λ}^{μ}(Res(M))) = T_{λ}^{μ}(Res(M)). It now suffices to show that Res(T_{λ}^{μ}(Tr(M))) = 0. But, let N = Tr(M) then Tr(N) = N and T_{λ}^{μ}(N) = Tr(T_{λ}^{μ}(N)). Consequently, Res(T_{λ}^{μ}(Tr(M)) = Res(Tr(T_{λ}^{μ}(N))) = 0. _{□}

2.6. U_{α}-algorithm. We will sometimes need to use the so-called U_{α}-algorithm to compute radical filtrations of GVMs. In this section, if M is a module in category O_{S}, write M_{i} := rad_{i}M.

2.6.1. Regular generalized Verma modules: right action. Fix an antidominant regular integral weight λ and a simple reflection s = s_{α}. In this subsection, when w ∈ ^{S}W, we write V (w) for V (w_{S}w · λ), and similarly L(w) for its simple quotient. There is an exact covariant functor θ_{s} on O_{S} called wall-crossing. It is the composition of the translation functors onto and off the α-wall; see [Vog, GJ, Jan1, CC, Deo2, Irv3]. If w ∈ ^{S}W then θ_{s}L(w) = 0 unless w < ws ∈ ^{S}W, in which case θ_{s}L(w) has a radical filtration consisting of three layers: (θ_{s}L(w))_{0} = (θ_{s}L(w))_{2} = L(w), and (θ_{s}L(w))_{1} is denoted U_{s}L(w) (or U_{α}L(w)). (The fact that rad(θ_{s}L(w))/soc(θ_{s}L(w)) is semisimple is known as Vogan’s Conjecture, and is equivalent to the Kazhdan-Lusztig Conjecture, which is a theorem in every context in which we shall need it.) Moreover,

U_{s}L(w) = L(ws) ⊕ ⨁ μ_{S}(x, w)L(x),

where the sum is over x ∈ ^{S}W with either x > xs or xs ∉ ^{S}W, and μ_{S}(x, w) is the Kazhdan-Lusztig μ-function of Section 2.3. The term L(ws) is called the special composition factor of U_{s}L(w). In addition, if w ∈ ^{S}W with w < ws ∈ ^{S}W, then θ_{s}V (w) is a non-split extension of V (ws) by V (w). These facts allow one inductively to compute the composition factors of each V (w), starting with V (e) = L(e), a process known as the U_{α}-algorithm.

Moreover, there is a “filtered” version of the algorithm, which computes not just composition factors but radical filtrations of GVMs; this is due to R. Irving [Irv3]. (For a graded version, see [BGS, Str].) Given a module M with a filtration {M^{i}}, define σM to be the same module with filtration (σM)^{i} = M^{i−1}. More generally, σ^{j}M denotes M with filtration (σ^{j}M)^{i} = M^{i−j}. Let w < ws be elements of ^{S}W. There is a short exact sequence of filtered modules

0 → σV (ws) → θ_{s}V (w) → V (w) → 0.

The GVMs are filtered by their radical filtrations, and the filtration of θ_{s}V (w) is constructed as follows. For each i ≥ 0 and each composition factor L(y) of V (w)_{i} for which θ_{s}L(y) ≠ 0 (equivalently, y < ys ∈ ^{S}W), (θ_{s}L(y))_{j} occurs in (θ_{s}V (w))_{i+j} (j = 0, 1, 2). (See the picture in [Irv3, Proposition 6.3.1]. Note that Irving consistently filters θ_{s}(−) beginning at layer −1 instead of 0, so his index-shift functors are applied to quotients rather than submodules.)

In the sequel, a “U_{α}-algorithm” will refer to a process for constructing, from the radical filtration of a GVM V (w), an “amalgamation” of the radical filtrations of V (w) and V (ws) (w < ws). The ith “layer” of the amalgamation will be V (w)_{i} ⊕ V (ws)_{i−1}, i ≥ 0. Only certain composition factors L(y) of V (w) are involved in the construction of the amalgamation. Such an L(y) in V (w)_{i} always contributes a copy of L(y) to each of layers i and i + 2 of the amalgamation, and a direct sum of certain other simple modules to layer i + 1. Deleting the (known) radical filtration of V (w) from the amalgamation leaves the radical filtration of V (ws).

2.6.2. Singular generalized Verma modules: right action. Let λ be as above, and let μ be an antidominant singular integral weight, so that the GVMs and simple modules in O_{S}^{μ} are parametrized by ^{S}W^{J}. Write V (w) (resp. \overline{V} (w)) to denote GVMs in O_{S}^{λ} (resp. O_{S}^{μ}). We abuse notation and write L(w) for the simple quotient of either V (w) or \overline{V} (w); the meaning should always be clear from context. The structure of the posets ^{S}W^{J} can be quite complicated; in particular, the lengths of the elements which occur need not be consecutive integers. Nonetheless it sometimes happens that w < ws are both in ^{S}W^{J} (for a simple reflection s = s_{α}). In this situation, a natural question to ask is whether there is some kind of U_{α}-algorithm for obtaining the radical filtration of \overline{V} (ws) from that of \overline{V} (w). It turns out that the answer is yes, provided that α is orthogonal to Φ_{J}. One uses Irving’s filtered U_{α}-algorithm for ^{S}W, but ignoring composition factors whose parameters are not in ^{S}W^{J}. The proof is based on Irving’s simple description of the transition from the radical filtrations of regular to singular GVMs, (2.3.1).

Let w ∈ ^{S}W^{J} with w < ws ∈ ^{S}W. Set A(w, s) = L(ws) if ws ∈ ^{S}W^{J}, otherwise A(w, s) = 0. Let B(w, s) be the direct sum of all composition factors L(x) of \overline{V} (w)_{1} (counted with multiplicities) satisfying x > xs or xs ∉ ^{S}W. Define

(2.6.1)  U_{s}L(w) = A(w, s) ⊕ B(w, s).

It can happen that U_{s}L(w) = 0. Proposition. Suppose w < ws are both in ^{S}W^{J}, where s = s_{α} is a simple reflection with α orthogonal to Φ_{J}. Then the radical filtration of \overline{V} (ws) is obtained from that of \overline{V} (w) by a U_{α}-algorithm. The contributing factors of \overline{V} (w) are those L(y) with y < ys ∈ ^{S}W. Such an L(y) ⊂ \overline{V} (w)_{i} contributes U_{s}L(y) (defined by (2.6.1)) to layer i + 1 of the amalgamation.

Proof. Recall that \overline{V} (w)_{i} is obtained from V (w)_{i} by “erasing” those composition factors whose parameters don’t lie in ^{S}W^{J}. The proposition is saying that we get the same result by ignoring all such composition factors at each stage of the U_{α}-algorithm. Suppose this fails. Then V (w) has a composition factor L(y) with y ∈ ^{S}W \^{S}W^{J}, where U_{s}L(y) has a composition factor L(x) with x ∈ ^{S}W^{J}. We must have y < ys ∈ ^{S}W in order for U_{s}L(y) to be defined. Also, since y ∉ ^{S}W^{J}, there exists j ∈ J with y > ys_{j} ∈ ^{S}W (since ^{S}W is closed under right-multiplication by simple reflections which decrease length).

First, x cannot be ys, because ys(α_{j}) = y(α_{j}) < 0 (the equality by our orthogonality hypothesis), giving ys > yss_{j}, and thus ys ∉ ^{S}W^{J}. Thus L(x) must be a non-special composition factor of U_{s}L(y). But an elementary property of Kazhdan-Lusztig polynomials says P^{S}x, y = P_{xs_{j},y}^{S} when y > ys_{j}. This in turn implies μ_{S}(x, y) = 0 (since x < xs_{j}) unless x = ys_{j}. But our orthogonality hypothesis and the fact that ys_{j} < y < ys imply ys_{j} < ys_{j}s = yss_{j} < ys, and since ys ∈ ^{S}W, the closure property implies yss_{j} ∈ ^{S}W. Then L(ys_{j}) does not appear in U_{s}L(y), so x ≠ ys_{j}. Thus no such x can exist. _{□}

Remark. It is possible to have w < ws_{α} both in ^{S}W^{J} where α is not orthogonal to Φ_{J}. A simple example occurs in (B_{3}, A_{1} ×A_{1}, A_{1}) in which w = s_{2}s_{3} < w′ = ws_{1} are both in ^{S}W^{J}; of course, α_{1} is not orthogonal to α_{2} ∈ Φ_{J}. In this case, \overline{V} (w) is simple, so the singular U_{α}-algorithm would predict that \overline{V} (w′) would have exactly two composition factors, L(w′) and L(w). But in fact, it has an additional composition factor, L(e), in layer 1. Thus the singular U_{α}-algorithm cannot be applied in this non-orthogonal situation.

2.6.3. Singular Verma modules: left action. Occasionally we shall have two parameters in ^{S}W^{J} which are linked by a simple reflection on the left but not on the right, and we shall need a version of the U_{α}-algorithm to relate the radical filtrations of the associated singular GVMs. This relies on a (left-side) version of the U_{α}-algorithm for singular (ordinary) Verma modules, again due to Irving [Irv5]. When μ is singular, the necessary translation functors are not defined on O^{μ}, and one must work in a category of Harish-Chandra modules. We briefly review some of the most pertinent facts, referring to [Irv5] and [Jan1] for details. In this subsection, we write M(w) for the (ordinary) Verma module M(w · μ) and L(w) for its simple quotient, w ∈ W^{J}. If w ∈ ^{S}W^{J} we write V (w) for the GVM V (w_{S}w · μ), but note that its simple quotient is L(w_{S}w).

Let 𝔨 be a diagonal copy of 𝔤 in 𝔤 × 𝔤, and let HC be the category of Harish-Chandra modules for (𝔤×𝔤, 𝔨). Fix an antidominant integral weight μ and a dominant regular integral weight ψ, and set ν = (μ, ψ). A theorem of Bernstein-Gelfand [BG] gives an equivalence of categories between O^{μ} and a certain full subcategory HC_{#}^{ν} of HC. For w ∈ W^{J}, the Verma module M(w) corresponds to a principal series module M(w), the irreducible L(w) to an irreducible Harish-Chandra module L(w), and its projective cover P (w) to the projective cover P(w) of L(w). Let HC^{ν} be the block of HC whose simple modules lie in HC_{#}^{ν}.

For each simple reflection s, there is defined on HC^{ν}, via translation functors on ψ, a (left-side) wall-crossing functor _{s}θ. (It does not preserve HC_{#}^{ν} when μ is singular.) The action extends to the filtered version of HC^{ν}. If we normalize so that all filtrations begin in degree 0, then the action of _{s}θ is described as follows. Let w ∈ W^{J}. If w < sw ∈ W^{J} then sθM(w) ≃ _{s}θM(sw), and there is a nonsplit exact sequence of filtered modules

0 → σM(sw) → _{s}θM(w) → M(w) → 0.

When sw ∉ W^{J} there is a nonsplit exact sequence

0 → σ^{2}M(w) → _{s}θM(w) → M(w) → 0.

If sw < w then _{s}θL(w) = 0. Otherwise _{s}θL(w) has radical length three, with layers 0 and 2 equal to L(w); its layer 1 is given by

(2.6.2)  _{s}UL(w) = \begin{cases}L(sw) ⊕ ⨁ μ(x, w)L(x) & if sw ∈ W^{J} \\ ⨁ μ(x, w)L(x) & if sw ∉ W^{J}, \end{cases}

where in each case the sum is over all x ∈ W^{J} with x > sx. The filtration of _{s}θM(w) is constructed from the radical filtration of M(w) by the usual process: the contributing factors of M(w) are those L(y) with y < sy, and such a factor in M(w)_{i} contributes _{s}UL(y) to layer i + 1 of _{s}θM(w). Via the equivalence of categories one immediately deduces a U_{α}- algorithm for O^{μ}. For w ∈ W^{J} with w < sw define _{s}UL(w) as in (2.6.2) (changing each L to L).

Theorem (Irving). Suppose w < sw are both in W^{J}, where s = s_{α} is a simple reflection. Then the radical filtration of M(sw) is obtained from that of M(w) by a U_{α}-algorithm. The contributing factors of M(w) are those L(y) with y < sy. Such an L(y) ⊂ M(w)_{i} contributes _{s}UL(y) (defined as in (2.6.2)) to layer i + 1 of the amalgamation.

2.6.4. Singular generalized Verma modules: left action. We can now pass to a left-side U_{α}-algorithm for singular GVMs by mimicking the (corrected) method of [Irv3, Sections 5.1–7.1]. We view O_{S}^{μ} as a subcategory of HC_{#}^{ν} ⊂ HC^{ν}. Let HC_{S}^{ν} be the full subcategory of HC^{ν} whose simple modules are { L(w_{S}w) : w ∈ ^{S}W^{J} }. For w ∈ ^{S}W^{J} let V(w) ∈ HC_{S}^{ν} correspond under the category equivalence to V (w) ∈ O_{S}^{μ}. Let P be the direct sum in HC^{ν} of the modules { P(w) : w ∈ W^{J} ∖ w_{S}^{S}W^{J} }. For M ∈ HC^{ν} let Tr(M) be the sum in M of all homomorphic images of P in M. Then Tr(M) is the largest submodule of M all of whose top composition factors have parameters w ∉ w_{S}^{S}W^{J}. Equivalently, by a standard Nakayama Lemma-type argument, Tr(M) is the smallest submodule of M for which M/Tr(M) lies in HC_{S}^{ν}. Set Res(M) = M/Tr(M). As in [Irv3, Proposition 5.1.2], Res(M(w_{S}w)) = V(w) for w ∈ ^{S}W^{J}. The analogs of Lemma 5.2.1 and Corollary 5.2.2 of [Irv3] are true in HC_{S}^{ν}; the proofs are the same. Lemma. Let s = s_{α} be a simple reflection with α orthogonal to Φ_{S}. Then _{s}θ: HC_{S}^{ν} → HC_{S}^{ν}.

Proof. We will repeatedly use the fact that w_{S}s = sw_{S}. It suffices to show that if w ∈ ^{S}W^{J} then _{s}θL(w_{S}w) ∈ HC_{S}^{ν}. We may assume that w_{S}w < sw_{S}w; equivalently, w < sw.

First, suppose the special factor of _{s}UL(w_{S}w) exists; i.e., sw_{S}w ∈ W^{J}. We must show that sw ∈ ^{S}W^{J}. Let i ∈ S. Then (sw)^{−1}(α_{i}) = w^{−1}s(α_{i}) = w^{−1}(α_{i}) > 0 since w ∈ ^{S}W. So sw ∈ ^{S}W ∩ w_{S}W^{J} = ^{S}W^{J} (Corollary 2.2).

Second, let L(w_{S}x) be a non-special factor of_{s}UL(w_{S}w). Then w_{S}x ∈ W^{J}, w_{S}x > sw_{S}x, and μ(w_{S}x, w_{S}w) ≠ 0. We must show that x ∈ ^{S}W^{J}. By Corollary 2.2 again, it suffices to show x ∈ ^{S}W. Suppose not. Using the fact that y ∈ ^{S}W if and only if y is of minimal length in its coset W_{S}y, if and only if w_{S}y is of maximal length in the same coset, we conclude that there exists i ∈ S such that w_{S}x < s_{i}w_{S}x. On the other hand, w_{S}w > s_{i}w_{S}w, so an argument with KL polynomials (similar to that in the proof of Proposition 2.6.2) shows that μ(w_{S}x, w_{S}w) = 0 unless s_{i}w_{S}x = w_{S}w. But in this case, (w_{S}w)^{−1}(α) = (w_{S}x)^{−1}s_{i}(α) = (w_{S}x)^{−1}(α) < 0, contradicting w_{S}w < sw_{S}w. _{□}

Assume for the remainder of this section that s = s_{α} where α is orthogonal to Φ_{S}. Then all of Chapter 6 of [Irv3] (with the exception of those results involving duality) extends to this setting.

We can now follow the proof of [Irv3, Theorem 7.1.2]. Assume w < sw with w, sw ∈ ^{S}W^{J}. Our assumption on s in the previous paragraph implies that w_{S}w < sw_{S}w. By [Irv5, Corollary 4.3], we have an exact sequence of filtered modules

0 → σM(sw_{S}w) → _{s}θM(w_{S}w) → M(w_{S}w) → 0.

Apply Res, which commutes with _{s}θ, and use the fact that radical filtrations are preserved under homomorphic images, to obtain the exact sequence of filtered modules

0 → σV(sw) → _{s}θV(w) → V(w) → 0.

By the Bernstein-Gelfand equivalence of categories, we have proved a version of the U_{α}- algorithm for the singular category O_{S}^{μ} with s acting on the left.

To state the theorem, we revert to our standard notation: L(w) = L(w_{S}w · μ) when w ∈ ^{S}W^{J}.

Theorem. Suppose w < sw are both in ^{S}W^{J}, where s = s_{α} is a simple reflection with α orthogonal to Φ_{S}. Then the radical filtration of V (sw) is obtained from that of V (w) by a U_{α}-algorithm. The contributing factors of V (w) are those L(y) with y < sy. The contribution of such an L(y) ⊂ V (w)_{i} to layer i + 1 of the amalgamation is the sum (with multiplicities) of all composition factors L(x) of V (y)_{1} satisfying x > sx, together with L(sy) if sy ∈ ^{S}W^{J}. Remark. The main difference between the right and left versions of the singular relative U_{α}-algorithms (Proposition 2.6.2 and Theorem 2.6.4) is in which irreducibles L(y) have U_{α}L(y) defined (where U_{α} = U_{s} or _{s}U, as appropriate). For the right-side version, it is only those satisfying y < ys ∈ ^{S}W, whereas for the left-side version, it is all those satisfying y < sy. The definition of U_{α}L(y) is modified accordingly to include all those composition factors L(x) of V (y)_{1} for which U_{α}L(x) is not defined.

2.7. Diamonds. In this section, we show that O(𝔤, S, J) has wild representation type whenever the poset ^{S}W^{J} contains a “diamond,” as defined below. For x, w ∈ W, write x → w, and say w covers x, if x < w and l(w) = l(x) + 1. Definition. Four distinct elements w, x_{1}, x_{2}, y ∈ W such that y → x_{k} → w (k = 1, 2) are called a diamond. Proposition. Let 𝔤 be any simple Lie algebra. Suppose that ^{S}W^{J} contains a diamond. Then O(𝔤, S, J) has wild representation type.

Proof. Let λ and μ be as in Section 2.3. By the comments at the end of that section, the quiver contains a diamond formed by the corresponding four nodes. Two possibilities arise: either there is an extension between one of these four irreducibles and another irreducible in O_{S}^{μ}, or there is not. In the first case, the quiver contains a “kite” (Figure 5, page 28). According to [DR], O_{S}^{μ} is of wild representation type.

Assume we are in the second case. Let u < v be two elements of the diamond. The relative KL polynomial P_{u,v}^{S} is equal to the ordinary (Category O) KL polynomial P_{w_{S}u,w_{S}v} [Deo2, Proposition 3.4]. Since this polynomial has constant term 1 and degree at most (l(w_{S}v) − l(w_{S}u) − 1)/2 = (l(v) − l(u) − 1)/2 ≤ 1/2, the polynomial is identically 1. It is easy to invert the portion of the matrix of (signed) relative KL polynomials associated to the diamond and deduce that the GVMs in O_{S}^{μ} associated to the diamond parameters have the following structure (levels indicate radical filtration layers):

V (w) : \begin{matrix}L(w) \\ L(x_{1}) L(x_{2}) \\ L(y) \end{matrix}  V (x_{1}) : \begin{matrix}L(x_{1}) \\ L(y) \end{matrix}  V (x_{2}) : \begin{matrix}L(x_{2}) \\ L(y) \end{matrix}  V (y) : L(y)

(There cannot be any additional composition factors, because that would entail an extension between some fifth irreducible and one of the diamond irreducibles, contrary to our assumption.) The analysis of the nonsingular Category O in type A_{1} × A_{1} [FNP, Section 4.2] shows that O_{S}^{μ} is of wild representation type. _{□}

Example. We present a typical verification that a poset ^{S}W^{J} contains a diamond. The general strategy is as follows. We specify a subset D = {y, x_{1}, x_{2}, w} ⊂ W which we wish to show is a diamond in ^{S}W^{J}; i.e.,

(1) u ∈ D ⟹ u ∈ ^{S}W^{J}, and
(2) y → x_{k} → w for k = 1, 2, and x_{1} ≠ x_{2}.

To show (1), we must show, for each u ∈ D:

(1a) u ∈ ^{S}W,
(1b) us_{j} ∈ ^{S}W for all j ∈ J, and
(1c) u < us_{j} for all j ∈ J.

For (1a), we show u^{−1}(α_{i}) > 0 for all i ∈ S. Given this, to show s_{j}u^{−1}(α_{i}) > 0 for i ∈ S and j ∈ J, it suffices to observe that u^{−1}(α_{i}) ≠ α_{j} in (1a); this will prove (1b). For (1c), we show u(α_{j}) > 0 for all j ∈ J.

Consider the category O(A_{n}, A_{n−4}, A_{2}×A_{2}) where S = {3, …, n−2} and J = {1, 2, n−1, n} and n ≥ 5. We claim that D = { y = s_{2}s_{3} … s_{n−3}s_{n−1}s_{n−2} … s_{3}, x_{1} = s_{1}y, x_{2} = s_{n}y, w = s_{1}s_{n}y } is a diamond in ^{S}W^{J}. When n ≥ 6 we compute:

y^{−1}(α_{i}) = \begin{cases}α_{2} + α_{3}, & i = 3 \\ α_{i}, & 3 < i < n − 2 \\ α_{n−2} + α_{n−1}, & i = n − 2, \end{cases}

whereas when n = 5, y^{−1}(α_{3}) = α_{2} + α_{3} + α_{4}. For u = x_{1} or x_{2} or w we have u^{−1}(α_{i}) = y^{−1}(α_{i}) for i ∈ S (since s_{1}(α_{i}) = s_{n}(α_{i}) = α_{i}). This proves (1a), and also (1b), by the earlier observation.

Next, we have

y(α_{j}) = \begin{cases}α_{1} + α_{2}, & j = 1 \\ α_{3} + ⋯ + α_{n−1}, & j = 2 \\ α_{2} + ⋯ + α_{n−2}, & j = n − 1 \\ α_{n−1} + α_{n}, & j = n, \end{cases}

so y < ys_{j} for all j ∈ J. Then

x_{1}(α_{j}) = \begin{cases}α_{2}, & j = 1 \\ α_{3} + ⋯ + α_{n−1}, & j = 2 \\ α_{1} + ⋯ + α_{n−2}, & j = n − 1 \\ α_{n−1} + α_{n}, & j = n, \end{cases}

so x_{1} < x_{1}s_{j} for all j ∈ J; similarly for x_{2}, by symmetry. Using w = s_{n}x_{1},

w(α_{j}) = \begin{cases}α_{2}, & j = 1 \\ α_{3} + ⋯ + α_{n}, & j = 2 \\ α_{1} + ⋯ + α_{n−2}, & j = n − 1 \\ α_{n−1}, & j = n, \end{cases}

so w < ws_{j} for all j ∈ J. This proves (1c).

In this example, because the expression for y involves neither s_{1} nor s_{n}, it is clear that l(x_{k}) = l(y) + 1 and l(w) = l(x_{k}) + 1, k = 1, 2. Clearly x_{1} ≠ x_{2}. This shows (2), and thus completes the verification that D is a diamond in ^{S}W^{J}.

2.8. D_{5}-quiver. The following result allows us to show that one has wild representation type if one can identify a certain subposet (shaped like the D_{5} Dynkin diagram) within the poset of ^{S}W^{J}.

Proposition. Let 𝔤 be any simple Lie algebra. Suppose ^{S}W^{J} contains a subset {w, x, y, z, v} in the pattern of Figure 1 (edges indicate Bruhat order covering relations, and higher elements are longer). Then O(𝔤, S, J) has wild representation type.

zy v x w Figure 1

Proof. Assuming that the poset ^{S}W^{J} has the indicated structure, one can conclude that the generalized Verma modules have the following quotients:

V (w) : \begin{matrix}L(w) \\ L(x) \end{matrix}  V (x) : \begin{matrix}L(x) \\ L(y) L(v) \end{matrix} V (y) : \begin{matrix}L(y) \\ L(z) \end{matrix}  V (v) : L(v)  V (z) : L(z).

Using reciprocity (2.1.4), the projective covers contain the following quotients:

P (w) : \begin{matrix}L(w) \\ L(x) \end{matrix}  P (x) : \begin{matrix}L(x) \\ L(y) L(v) L(w) \\ L(x) \end{matrix}  P (y) : \begin{matrix}L(y) \\ L(z) L(x) \\ L(y) L(v) \end{matrix}

P (v) : \begin{matrix}L(v) \\ L(x) \\ L(y) L(v) \end{matrix} P (z) : \begin{matrix}L(z) \\ L(y) \\ L(z). \end{matrix}

The Ext^{1}-quiver for B = O(𝔤, S, J) contains the subquiver given in Figure 2. Set A = End_{B}(P) where P is the direct sum of all projective indecomposable modules in B. Consider the idempotent given by e = 1_{y} +1_{z} +1_{v}. From the structure of the projectives, the algebra eAe has subquiver given in Figure 3. By [Erd1, I.10.8(iv)], this shows that eAe has wild representation type, thus A has wild representation type. _{□}

L(z)  L(y)  L(x)  L(w) L(v)

Figure 2. D_{5}-quiver

z y v Figure 3

2.9. Equivalences of categories. Given an integral weight δ, let π be the weight in the W -dot orbit of δ with π + ρ antidominant. Following Enright-Shelton [ES] define the truncated category O_{t}(𝔤, S, δ) to be the full subcategory of O(𝔤, S, π) consisting of modules M with the property:

if ν ∈ X_{S}^{+} and [M : L(ν)] > 0 then ν ≤ δ.

Let 𝔮 be a maximal parabolic subalgebra of 𝔤 which contains 𝔟 but not 𝔭, and denote by 𝔩 the Levi factor of 𝔮. Let ρ(𝔩) denote half the sum of the positive roots for 𝔩. Put 𝔯 = 𝔪 ∩ 𝔩, and write W_{𝔪}, W_{𝔩}, W_{𝔯} for the Weyl groups of 𝔪, 𝔩, 𝔯. Recall w_{S}, the longest element of W_{S} = W_{𝔪}. Let ^{𝔯}w_{𝔩} be the longest element of ^{𝔯}W_{𝔩}, the set of minimal length right coset representatives of W_{𝔯} in W_{𝔩}. Set ν_{1} = −ρ − ρ(𝔩), ν_{2} = −2ρ, and ν_{3} = w_{S} ^{𝔯}w_{𝔩} · ν_{2} (dot action relative to ρ). Proposition (A). [ES, Proposition 7.1] The categories O(𝔩, 𝔩 ∩ 𝔭, ν_{1}) and O_{t}(𝔤, 𝔭, ν_{3}) are equivalent, under an equivalence which maps V (𝔩, 𝔯, w) to V (𝔤, 𝔭, w) (w ∈ ^{𝔯}W_{𝔩}).

Notice that ν_{1} +ρ(𝔩) (resp. ν_{2} +ρ) is regular antidominant for 𝔩 (resp. 𝔤), and so ν_{3} (being in the W -dot orbit of ν_{2}) is regular for 𝔤. Thus the proposition says that O(𝔩, 𝔩 ∩ 𝔭, reg) is equivalent to a truncated subcategory of O(𝔤, 𝔭, reg). We remark that the proof given by Enright and Shelton of Proposition 2.9(A) can be extended to singular categories in the following situation. Let J be a subset of Δ such that S ∩ J = ∅. Set ν̃_{1} = −ρ − ρ(𝔩) + ρ_{J}, ν̃_{2} = −2ρ + ρ_{J}, and ν̃_{3} = w_{S} ^{𝔯}w_{𝔩} · ν̃_{2}. Proposition (B). The categories O(𝔩, 𝔩 ∩ 𝔭, ν̃_{1}) and O_{t}(𝔤, 𝔭, ν̃_{3}) are equivalent, under an equivalence which maps V (𝔩, 𝔯, w) to V (𝔤, 𝔭, w) (w ∈ ^{𝔯}W_{𝔩}^{J}).

2.10. Induction-Restriction. Let S ⊆ Δ and 𝔭 = 𝔪 ⊕ 𝔲^{+} be the corresponding parabolic subalgebra. In this section we will deal with the case when there is a parabolic subalgebra 𝔮 with 𝔭 ⊆ 𝔮. The parabolic subalgebra 𝔮 has a Levi decomposition

𝔮 = 𝔩 ⊕ 𝔲_{𝔮}^{+} = 𝔰 ⊕ 𝔱 ⊕ 𝔲_{𝔮}^{+},

where 𝔰 is a semisimple Lie algebra, 𝔱 is toral, [𝔰, 𝔱] = 0 and 𝔲_{𝔮}^{+} is nilpotent. Let 𝔥_{𝔰} ⊂ 𝔥 be a Cartan subalgebra of 𝔰. Clearly, 𝔥 = 𝔥_{𝔰} ⊕ 𝔱.

Let λ ∈ 𝔱^{∗} and M be in O(𝔩, 𝔩 ∩ 𝔭, ν) where ν ∈ 𝔥_{𝔰}^{∗}. By using the identification 𝔥 = 𝔥_{𝔰} ⊕ 𝔱, we have 𝔥^{∗} = 𝔥_{𝔰}^{∗} ⊕ 𝔱^{∗}, so one can consider ν + λ ∈ 𝔥^{∗}. Now we can view M as a 𝔮-module where 𝔲_{𝔮}^{+} acts trivially and 𝔱 acts by λ.

Next consider the induced module V (M, λ) = U(𝔤) ⊗_{U(𝔮)} M. One can easily verify by using the fact that 𝔭 ⊆ 𝔮 that V (M, λ) is a module in the category O_{S} of 𝔤-modules. In fact we know that simple modules in O(𝔩, 𝔩 ∩ 𝔭, ν) are sent to generalized Verma modules in O(𝔤, 𝔭, ν + λ). Therefore, for each λ ∈ 𝔱^{∗} we have a functor

F_{λ} : O(𝔩, 𝔩 ∩ 𝔭, ν) → O(𝔤, 𝔭, ν + λ)

such that F_{λ}(M) = V (M, λ).

Since 𝔮 is a parabolic subalgebra of 𝔤, 𝔤 = 𝔮 ⊕ 𝔲_{𝔮}^{−} where 𝔲_{𝔮}^{−} is a nilpotent subalgebra. Then as a U(𝔩)-module

F_{λ}(M) ≅ ⟨𝔲_{𝔮}^{−}U(𝔲_{𝔮}^{−}) ⊗ M⟩ ⊕ ⟨1 ⊗ M⟩

by the Poincare-Birkhoff-Witt theorem and the fact that [𝔩, 𝔲_{𝔮}^{−}] ⊆ 𝔲_{𝔮}^{−}.

We are now in a position to use the same arguments given in [FNP, Theorem 3.3] to deduce the following theorem. Theorem. Let 𝔮 = 𝔩 ⊕ 𝔲_{𝔮}^{+} = 𝔰 ⊕ 𝔱 ⊕ 𝔲_{𝔮}^{+} be a parabolic subalgebra of 𝔤 satisfying 𝔭 ⊂ 𝔮, and let ν ∈ 𝔥_{𝔰}^{∗}, λ ∈ 𝔱^{∗}. If O(𝔩, 𝔩∩𝔭, ν) is not semisimple (resp. not finite, not tame), then neither is O(𝔤, S, ν + λ).

2.11. Rank reduction. Finally, we can prove our key result which allows us to transfer information from smaller Levi subalgebras to the entire Lie algebra. Corollary. Let S ∩ J = ∅ where S, J ⊆ Δ. Let 𝔩 be any standard Levi subalgebra of 𝔤 and S′ ⊆ Δ be the corresponding set of simple roots associated with 𝔩. If O(𝔩, S ∩ S′, J ∩ S′) is not semisimple (resp. not finite, not tame), then neither is O(𝔤, S, J).

Proof. Let 𝔭 (resp. 𝔮) be the standard parabolic subalgebra corresponding to S (resp. S′). If 𝔭 ⊆ 𝔮 then we can apply Theorem 2.10. Therefore, we may assume that 𝔮 does not contain 𝔭. Then we can find a maximal parabolic subalgebra 𝔮′ with 𝔩′ as its Levi component such that 𝔮 ⊆ 𝔮′, also 𝔩 ⊆ 𝔩′ and 𝔮′ does not contain 𝔭. Therefore, by Theorem 2.10, if O(𝔩, S∩S′, J ∩S′) is not semisimple (resp. not finite, not tame), then neither is O(𝔩′, S ∩ Δ_{𝔩′}, J ∩ Δ_{𝔩′}). Now the result follows by Proposition 2.9(B). _{□}

3. Regular blocks of category O_{S}

3.1. Finite type. Let A be a quasi-hereditary algebra and suppose the simple modules are denoted by L_{0}, L_{1}, …, L_{t}. Suppose that the corresponding standard modules are given by V (L_{j}), j = 0, 1, …, t, with the following structures:

V (L_{0}) : L_{0} \begin{matrix}V (L_{j}) : \\ j = 1, …, t \end{matrix} \begin{matrix}L_{j} \\ L_{j−1} \end{matrix}

One can apply the reciprocity law to conclude that the projective covers of the simples have the following structures:

P (L_{0}) : \begin{matrix}L_{0} \\ L_{1} \\ L_{0} \end{matrix}  \begin{matrix}P (L_{j}) : \\ j = 1, …, t − 1 \end{matrix} \begin{matrix}L_{j} \\ L_{j−1} L_{j+1} \\ L_{j} \end{matrix}  P (L_{t}) : \begin{matrix}L_{t} \\ L_{t−1} \end{matrix}

The quiver for the algebra A is given by Figure 4 with relations α_{1}β_{1} = 0 = β_{t}α_{t}, α_{j}α_{j+1} = 0 = β_{j+1}β_{j}, and β_{j}α_{j} = α_{j+1}β_{j+1} for j = 1, 2, …, t − 1. Here we are identifying the basic algebra with the opposite algebra of the endomophism ring of the direct sum of the projective indecomposable A-modules.

From the structures of the projectives, one can deduce that the algebra A is a threenilpotent algebra but the only indecomposable modules of radical length three are projective. Consequently, A has the same representation type as A/J^{2} where J = Rad A. The quiver for A separates into two copies of the quiver of type A_{t+1}. Therefore, by Gabriel’s theorem [Gab], A has finite representation type. The structures of the indecomposable modules are given in [LN, Section 4.2].

L_{0}  L_{1}  L_{2}  L_{t−1}  L_{t}^{α_{1} β_{1} α_{2} β_{2} α_{t} β_{t}} Figure 4

The blocks of O_{S}^{reg} for (A_{n}, A_{n−1}, ∅), (B_{n}, B_{n−1}, ∅), (C_{n}, C_{n−1}, ∅), and (G_{2}, A_{1}, ∅) have quivers of the form given in Figure 4. We remark there are two choices for S that give rise to (A_{n}, A_{n−1}, ∅) and (G_{2}, A_{1}, ∅). However, both choices give rise to Morita equivalent algebras. We are using S = {α_{2}, α_{3}, …, α_{n}} for A_{n} and S = {α_{2}} for G_{2}. The correspondence between simple modules for these blocks and those in Figure 4 is given by the following tables. Set w_{j} = s_{1}s_{2} ⋯ s_{j} and z_{j} = s_{n−1}s_{n−2} ⋯ s_{n−j}.

\begin{matrix}Vertex & (A_{n}, A_{n−1}, ∅) & (BC_{n}, BC_{n−1}, ∅) \\ L_{0} & L(e) & L(e) \\ L_{1} & L(w_{1}) & L(w_{1}) \\ L_{2} & L(w_{2}) & L(w_{2}) \\ ⋮ & ⋮ & ⋮ \\ L_{n−1} & L(w_{n−1}) & L(w_{n−1}) \\ L_{n} & L(w_{n}) & L(w_{n}) \\ L_{n+1} & – & L(w_{n}z_{1}) \\ ⋮ & ⋮ & ⋮ \\ L_{2n−2} & – & L(w_{n}z_{n−2}) \\ L_{2n−1} & – & L(w_{n}z_{n−1}) \end{matrix}

\begin{matrix}Vertex & (G_{2}, A_{1}, ∅) \\ L_{1} & L(e) \\ L_{2} & L(s_{1}) \\ L_{3} & L(s_{1}s_{2}) \\ L_{4} & L(s_{1}s_{2}s_{1}) \\ L_{5} & L(s_{1}s_{2}s_{1}s_{2}) \\ L_{6} & L(s_{1}s_{2}s_{1}s_{2}s_{1}) \end{matrix}

We can summarize our findings of this section in the following proposition. Proposition. Let 𝔤 be a finite-dimensional complex simple Lie algebra, S ⊆ Δ. If

(Φ, Φ_{S}, Φ_{μ}) = \begin{cases}(A_{n}, A_{n−1}, ∅) \\ (BC_{n}, BC_{n−1}, ∅) \\ (G_{2}, A_{1}, ∅) \end{cases}

then the block O_{S}^{reg} has finite representation type.

3.2. Wild type. In this section we analyze certain “base case” blocks of category O_{S}^{reg} having a diamond in the poset ^{S}W. Such blocks were shown to have wild representation type in Section 2.7. For category O_{S}^{reg} with (Φ, Φ_{S}, ∅) = (A_{3}, A_{1} ×A_{1}, ∅), (B_{3}, A_{1} ×A_{1}, ∅), (C_{3}, A_{1} × A_{1}, ∅), (B_{3}, A_{2}, ∅), or (C_{3}, A_{2}, ∅) we give the elements of a diamond in ^{S}W in the following table. The proofs follow the paradigm given in Example 2.7.

Element of ^{S}W (A_{3}, A_{1} × A_{1}, ∅) (BC_{3}, A_{1} × A_{1}, ∅) (BC_{3}, A_{2}, ∅)
y  s_{2}  s_{2}  s_{3}s_{2}
x_{1}  s_{2}s_{1}  s_{2}s_{1}  s_{3}s_{2}s_{1}
x_{2}  s_{2}s_{3}  s_{2}s_{3}  s_{3}s_{2}s_{3}
w  s_{2}s_{1}s_{3}  s_{2}s_{1}s_{3}  s_{3}s_{2}s_{1}s_{3}

For the blocks of O_{S} arising from Lie algebras of type D, we consider the cases (Φ, Φ_{S}, ∅) = (D_{4}, A_{3}, ∅) and (D_{n}, D_{n−1}, ∅). The table below indicates which elements of ^{S}W give rise to a diamond.

Element of ^{S}W (D_{4}, A_{3}, ∅) (D_{n}, D_{n−1}, ∅)
y  s_{4}s_{2}  s_{1} … s_{n−2}
x_{1}  s_{4}s_{2}s_{3}  s_{1} … s_{n−2}s_{n}
x_{2}  s_{4}s_{2}s_{1}  s_{1} … s_{n−2}s_{n−1}
w  s_{4}s_{2}s_{3}s_{1}  s_{1} … s_{n−2}s_{n−1}s_{n}

Finally, we consider the blocks of O_{S}^{reg} corresponding to (F_{4}, B_{3}, ∅) and (F_{4}, C_{3}, ∅). In these cases one can also identify a diamond in ^{S}W. The table below identifies the elements.

Element of ^{S}W (F_{4}, B_{3}, ∅) (F_{4}, C_{3}, ∅)
y  s_{4}s_{3}s_{2}  s_{1}s_{2}s_{3}
x_{1}  s_{4}s_{3}s_{2}s_{1}  s_{1}s_{2}s_{3}s_{4}
x_{2}  s_{4}s_{3}s_{2}s_{3}  s_{1}s_{2}s_{3}s_{2}
w  s_{4}s_{3}s_{2}s_{1}s_{3} s_{1}s_{2}s_{3}s_{4}s_{2}

We can summarize our findings of this section in the following proposition. Proposition. Let 𝔤 be a finite-dimensional complex simple Lie algebra, S ⊆ Δ. If

(Φ, Φ_{S}, Φ_{μ}) = \begin{cases}(A_{3}, A_{1} × A_{1}, ∅) \\ (BC_{3}, A_{1} × A_{1}, ∅) \\ (BC_{3}, A_{2}, ∅) \\ (D_{4}, A_{3}, ∅) \\ (D_{n}, D_{n−1}, ∅), & n ≥ 4 \\ (F_{4}, BC_{3}, ∅) \end{cases}

then the block O_{S}^{reg} has wild representation type.

3.3. Proof of Main Theorem (Regular Blocks). We can now prove the main result given by Theorem 1.4.

Proof. (a) If S = Δ then O_{S}^{reg} has only one simple module. Any quasi-hereditary algebra with only one simple module must be semisimple. On the other hand, suppose that S ⊊ Δ and let I = {α} with α ∈ Δ − S. According to Theorem 1.3, O(𝔩_{I}, 𝔩_{I} ∩ 𝔭_{S}, reg) is not semisimple, thus O(𝔤, 𝔭_{S}, reg) is not semisimple by Corollary 2.11.

(b) Suppose that |Δ| − |S| ≥ 2. Let I = {α_{1}, α_{2}} where α_{i} ∉ S for i = 1, 2. By using Theorem 1.3, we have O(𝔩_{I}, 𝔩_{I} ∩ 𝔭_{S}, reg) has wild representation type, thus O(𝔤, 𝔭_{S}, reg) has wild representation type by Corollary 2.11.

(c) Assume that Δ − S = {α}. First suppose α corresponds to a node in the interior of the Dynkin diagram. Let β_{1}, β_{2} be simple roots adjacent to α and set I = {β_{1}, β_{2}, α}. Then O(𝔩_{I}, 𝔩_{I} ∩ 𝔭_{S}, reg) corresponds with (Φ_{I}, A_{1} × A_{1}, ∅). This is wild by Proposition 3.2, thus O(𝔤, 𝔭_{S}, reg) is wild.

We can now assume that α corresponds to a node at the end of the Dynkin diagram. Our analysis takes us systematically through all the irreducible root systems. The cases having finite representation type have already been treated in Proposition 3.1. For the others, we will use Corollary 2.11 to reduce to one of the wild “base cases” in Proposition 3.2. The notation S′ will refer to the set of simple roots for the smaller Levi algebras.

Type A_{n}: In this case we are done because O_{S}^{reg} has finite type for (A_{n}, A_{n−1}, ∅).

Type BC_{n}: The category O_{S′}^{reg} is wild for (BC_{3}, A_{2}, ∅). Therefore, O_{S}^{reg} is wild for (BC_{n}, A_{n−1}, ∅). On the other hand, O_{S}^{reg} has finite type for (BC_{n}, BC_{n−1}, ∅).

Type D_{n}: The category O_{S′}^{reg} is wild for (D_{4}, A_{3}, ∅). Therefore, O_{S}^{reg} is wild for (D_{n}, A_{n−1}, ∅). Moreover, O_{S}^{reg} has wild representation type for (D_{n}, D_{n−1}, ∅).

Type E_{6}: The category O_{S′}^{reg} is wild for (D_{5}, D_{4}, ∅) and (D_{5}, A_{4}, ∅). Consequently, the category O_{S}^{reg} is wild for (E_{6}, D_{5}, ∅) and (E_{6}, A_{5}, ∅).

Type E_{7}: From the D_{n} and E_{6} cases, the category O_{S′}^{reg} has wild representation type for (D_{6}, D_{5}, ∅), (E_{6}, D_{5}, ∅) and (E_{6}, A_{5}, ∅). Consequently, the category O_{S}^{reg} is wild for (E_{7}, E_{6}, ∅), (E_{7}, D_{6}, ∅) and (E_{7}, A_{6}, ∅).

Type E_{8}: In order to prove that the category O_{S}^{reg} is wild for (E_{8}, E_{7}, ∅), (E_{8}, D_{7}, ∅) and (E_{8}, A_{7}, ∅) one can just repeat the previous argument in the E_{7} case by adding one to every subscript.

Type F_{4}: The category O_{S}^{reg} is wild for (F_{4}, BC_{3}, ∅).

Type G_{2}: The category O_{S}^{reg} has finite representation type for (G_{2}, A_{1}, ∅). _{□}

4. Singular Hermitian Symmetric Categories

4.1. Proof of Corollary 1.5. In this subsection we characterize the representation type of “semiregular” (|J| = 1) categories O(Φ, S, J) where (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair. Our main tool is a beautiful equivalence of categories due to Enright-Shelton.

Proof. Case 1. Assume that (Φ, Φ_{S}) is of type (A_{n}, A_{n−1}), (B_{n}, B_{n−1}), or (D_{n}, D_{n−1}). Then the structure of the category O(Φ, Φ_{S}, ∅) is quite simple; the radical filtrations of the GVMs are well-known. These were originally discovered in [BJ, Section 4.5], and are listed more explicitly in [Irv2]. We apply (2.3.1) to deduce the radical filtrations of the GVMs in the semiregular category O(Φ, S, J). Write J = {j}.

In type A, |^{S}W^{J}| = 1 so O(Φ, S, J) contains only a single irreducible and is therefore semisimple.

In type B, there are two possibilities. If j < n, then |^{S}W^{J}| = 2, but these two parameters are not adjacent in the chain ^{S}W. Since the GVMs in (Φ, Φ_{S}, ∅) are uniserial of length at most two, with the two composition factors corresponding to adjacent parameters in ^{S}W, there are no extensions between the irreducibles corresponding to the two parameters in SW^{J} (recall from Section 2.3 that all extensions occur between layers 0 and 1 of the radical filtrations of GVMs). Thus O(Φ, S, J) is semisimple, consisting of two blocks. If j = n then |^{S}W^{J}| = 1 and O(Φ, S, J) is a single semisimple block.

In type D, |^{S}W^{J}| = 2 for all j. Inspection of the structure of the regular GVMs (cf. e.g. [Irv2, Section 7] and [Irv1, 6.2]) shows that there is an extension between the two semisimple irreducibles, and the category is of finite type.

Case 2. Assume that (Φ, Φ_{S}) is of type (A_{n}, A_{p−1}×A_{n−p}) with 1 < p < n, (C_{n}, A_{n−1}) with n ≥ 3, or (D_{n}, A_{n−1}) with n ≥ 4. Let (Φ′, Φ_{S′}) be of type (A_{n−2}, A_{p−2} × A_{n−p−1}), (C_{n−2}, A_{n−3}), or (D_{n−2}, A_{n−3}), respectively. In type C_{n}, assume furthermore that α_{j} is a short simple root. Then by [ES, Lemma 10.1 and Proposition 11.2], there is an equivalence of categories between O(Φ, S, J) and O(Φ′, S′, ∅). If α_{j} = α_{n} (the long simple root in type C_{n}), then by [ES, Corollary 12.14], the category O(Φ, S, J) splits into two blocks, each equivalent to O(Φ′, S′, ∅). In particular, the categories (A_{n}, A_{1} × A_{n−2}, A_{1}), (D_{4}, A_{3}, A_{1}), and (D_{5}, A_{4}, A_{1}) are equivalent to the finite-type categories (A_{n−2}, A_{n−3}, ∅), (A_{1} × A_{1}, A_{1}, ∅), and (A_{3}, A_{2}, ∅), respectively, while the categories (C_{3}, A_{2}, A_{1}), (C_{4}, A_{3}, A_{1}), are equivalent to direct sums of one or two copies of the finite-type categories (A_{1}, ∅, ∅), (B_{2}, B_{1}, ∅), respectively. These are the only finite-type possibilities in Case 2; the remainder are wild, by Theorem 1.4.

Case 3. Assume that (Φ, Φ_{S}) is of type (E_{6}, D_{5}) or (E_{7}, E_{6}). Using Bourbaki labeling, we may assume that S = {1, …, n−1}. The semiregular categories O(𝔤, S, {j}) for j = 1, …, n are all equivalent; see [BES]. The proof is the same as for [ES, Lemma 10.1], in which the key ingredient, a theorem of D. Vogan, only relies on the nilradical 𝔲_{S}^{+} being abelian; this is the case if and only if (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair. Thus it suffices to analyze O(𝔤, S, {n}). The radical filtrations of the regular category appear in [CIS]. For n = 6 the semiregular category contains six standard modules, with the structure described in Section 3.1. Therefore this category is of finite type. For n = 7, the poset ^{S}W^{J} contains a diamond, where y = s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}, x_{1} = ys_{2}, x_{2} = ys_{5}, w = ys_{2}s_{5}. Hence this category is of wild type. _{□}

4.2. Hermitian symmetric categories with non-A_{1} × ⋯ × A_{1} singularities. In this subsection we show that a singular Hermitian symmetric category O_{S}^{μ} having singularity type other than A_{1} × ⋯ × A_{1} is empty. Proposition. Assume that (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair, and let w ∈ ^{S}W. If α and β are nonorthogonal simple roots, then not both w < ws_{α} ∈ ^{S}W and w < ws_{β} ∈ ^{S}W can occur.

Proof. Suppose w < ws_{α}, w < ws_{β}, and ws_{α}, ws_{β} ∈ ^{S}W. Consider the positive roots α′ = w(α) and β′ = w(β). If α′ ∈ Φ_{S}^{+} then (ws_{α})^{−1}(α′) = −α < 0, which contradicts ws_{α} ∈ ^{S}W. Thus α′ ∈ Φ^{+} ∖ Φ_{S}^{+} = Φ(𝔲_{S}^{+}); similarly β′ ∈ Φ(𝔲_{S}^{+}). Since by hypothesis α + β is a root, so is w(α + β) = α′ + β′; hence α′ + β′ ∈ Φ(𝔲_{S}^{+}). Therefore [𝔤_{α′}, 𝔤_{β′}] = 𝔤_{α′+β′} ≠ 0. But this contradicts our Hermitian symmetric hypothesis, namely that 𝔲_{S}^{+} is abelian. _{□}

Corollary. Assume that (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair, and suppose J contains two simple roots adjacent in the Dynkin diagram of Φ. Then the category O(𝔤, S, J) is empty.

Proof. It is immediate from the proposition and the definition (2.2.2) of ^{S}W^{J} that the latter is empty. _{□} 5. Mixed case: Δ ≠ S ∪ J

We now introduce a convention which will be in effect throughout the remainder of the paper. We fix a regular antidominant integral weight λ, and an antidominant integral weight μ with J ⊂ Δ its set of singular simple roots. We also fix S ⊂ Δ with S ∩ J = ∅.

To help keep track of the various choices of S and J which arise in the mixed cases, the reader may find it useful to adopt the following convention: draw a Dynkin diagram and enclose the roots in S with one or more ovals (one for each connected component of S), and the roots in J with rectangles.

5.1. We first show that if |Δ| − |S ∪ J| ≥ 2 then O_{S}^{μ} must have wild representation type. This verifies the statement of Theorem 1.6(A)(a).

Let S and J be such that S ∩J = ∅ and |Δ|−|S ∪J| ≥ 2. Choose {β_{1}, β_{2}} ⊆ Δ−(S ∪J). Set T = {β_{1}, β_{2}} ∪ S. According to Theorem 1.4, O(𝔪_{T} , S, ∅) has wild representation type. Therefore, O_{S}^{μ} must have wild representation type by Theorem 2.10.

5.2. The remainder of Section 5 is devoted to proving Theorem 1.6(A)(b). Let 𝔤 be a simple complex Lie algebra with underlying root system Φ of type A_{3}, B_{3}, or C_{3}. We begin by analyzing these rank 3 cases where Φ_{S∪J} is of type A_{1} × A_{1}. According to Theorems 1.3 and 1.4, O_{S}^{μ} has wild representation type when S = ∅ (Φ_{J} = A_{1} × A_{1}) or when J = ∅ (Φ_{S} = A_{1} × A_{1}). So assume |S| = |J| = 1 with Φ_{S∪J} of type A_{1} × A_{1}. Then there exists a diamond in the poset ^{S}W^{J} (given in the table below), so these blocks are also of wild representation type by Proposition 2.7.

Element of ^{S}W^{J} (A_{3}, A_{1}, A_{1})  (BC_{3}, A_{1}, A_{1})  (BC_{3}, A_{1}, A_{1})
S = {α_{1}} J = {α_{3}} S = {α_{1}} J = {α_{3}} S = {α_{3}} J = {α_{1}}
y  s_{2}  s_{2}  s_{2}
x_{1}  s_{2}s_{1}  s_{2}s_{1}  s_{1}s_{2}
x_{2}  s_{3}s_{2}  s_{3}s_{2}  s_{2}s_{3}
w  s_{3}s_{2}s_{1}  s_{3}s_{2}s_{1}  s_{1}s_{2}s_{3}

5.3. The analysis of the rank 3 algebras in the preceding subsection can now be used to prove the following proposition. Proposition. Let O_{S}^{μ} be a block of O_{S} with S∩J = ∅ and |Δ|−|S∪J| = 1. If Δ−(S∪J) = {α} where α is a simple root in the interior of the Dynkin diagram then O_{S}^{μ} has wild representation type.

Proof. Let β_{1} and β_{2} be simple roots which are adjacent to α in the Dynkin diagram. Set T = {α, β_{1}, β_{2}}. From the results in Section 5.2, O(𝔪_{T} , {β_{1}, β_{2}}, J ∩ T) has wild representation type. Therefore, by Corollary 2.11, O_{S}^{μ} must have wild representation type.

□

5.4. More wild algebras. In this section we verify that certain algebras are wild for complex Lie algebras of ranks 3 and 4. First consider the block O_{S}^{μ} where (Φ, Φ_{S}, Φ_{J}) = (A_{3}, A_{1}, A_{1}) or (A_{4}, A_{1}, A_{2}). Here we must specify S and J to remove any ambiguity. In the first case S = {α_{3}} and J = {α_{2}}. For the second case S = {α_{4}} and J = {α_{2}, α_{3}}.

The Ext^{1}-quiver has a subquiver of the form given in Figure 5. By [DR] this shows that these blocks have wild representation type. The correspondence between vertices and simple modules is given in the table below.

Simple module (A_{3}, A_{1}, A_{1})  (A_{4}, A_{1}, A_{2})
S = {α_{3}} J = {α_{2}} S = {α_{4}} J = {α_{2}, α_{3}}
γ  L(s_{2}s_{1}s_{3})  L(s_{3}s_{2}s_{1}s_{4})
α  L(s_{1}s_{2}s_{1}s_{3})  L(s_{2}s_{3}s_{2}s_{1}s_{4})
β  L(s_{2}s_{1})  L(s_{3}s_{2}s_{1})
σ  L(s_{1})  L(s_{2}s_{1})
δ  L(e)  L(s_{1})

δ σ α β γ Figure 5

Now consider the block O_{S}^{μ} for the BC_{3}-algebras listed in the two tables below. These algebras are wild because the posets contain diamonds. Note these are the cases where the sets S and J are adjacent.

Element of ^{S}W^{J} (BC_{3}, A_{1}, A_{1})  (BC_{3}, A_{1}, A_{1})
S = {α_{1}} J = {α_{2}} S = {α_{2}} J = {α_{1}}
y  s_{2}s_{3}  s_{3}s_{2}
x_{1}  s_{2}s_{1}s_{3}  s_{3}s_{1}s_{2}
x_{2}  s_{3}s_{2}s_{3}  s_{3}s_{2}s_{3}
w  s_{3}s_{2}s_{1}s_{3}  s_{3}s_{1}s_{2}s_{3}

Element of ^{S}W^{J} (BC_{3}, A_{1}, A_{1})  (BC_{3}, A_{1}, A_{1})
S = {α_{3}} J = {α_{2}} S = {α_{2}} J = {α_{3}}
y  s_{2}s_{3}  s_{3}s_{2}
x_{1}  s_{1}s_{2}s_{3}  s_{3}s_{2}s_{1}
x_{2}  s_{2}s_{1}s_{3}  s_{3}s_{1}s_{2}
w  s_{1}s_{2}s_{1}s_{3}  s_{3}s_{1}s_{2}s_{1}

Finally consider the rank 4 cases listed in the following table. Again these blocks are wild because of the presence of a diamond in the quiver.

Element (BC_{4}, A_{1}, A_{2})  (BC_{4}, A_{2}, A_{1})  (F_{4}, B_{2}, A_{1})
S = {α_{4}} J = {α_{2}, α_{3}} S = {α_{2}, α_{3}} J = {α_{4}} S = {α_{2}, α_{3}} J = {α_{4}}
y  s_{3}s_{4}  s_{4}s_{3}  s_{1}s_{2}s_{3}
x_{1}  s_{1}s_{3}s_{4}  s_{4}s_{3}s_{1}  s_{1}s_{2}s_{4}s_{3}
x_{2}  s_{2}s_{3}s_{4}  s_{4}s_{3}s_{2}  s_{1}s_{2}s_{3}s_{2}
w  s_{1}s_{2}s_{3}s_{4}  s_{4}s_{3}s_{2}s_{1}  s_{1}s_{2}s_{4}s_{3}s_{2}

5.5. Two infinite families. We will now show that the blocks O_{S}^{μ} for two infinite families have wild representation type. Proposition. The block O_{S}^{μ} has wild representation type if (Φ, S, J) = (A_{n}, {α_{2}, …, α_{n−1}}, {α_{1}}), (A_{n}, {α_{2}, …, α_{n−1}}, {α_{n}}), or (B_{n}, {α_{2}, …, α_{n−1}}, {α_{n}}).

Proof. First assume we are in type A; by symmetry we may assume J = {α_{1}}. Consider the associated regular category (A_{n}, {α_{2}, …, α_{n−1}}, ∅). This pair (𝔤, 𝔭) corresponds in a natural way to the real rank one group SU(n, 1), for which the radical filtrations of the generalized Verma modules have been worked out in [Col, 6.2]. We adopt the notations (ij) and (ij)^{θ} of [Col, Figure 9.2] to label the parameters of ^{S}W. The poset is reproduced in Figure 6; edge labels denote simple reflections acting on the right. The parameters in ^{S}W^{J} are those with an edge labeled “1” emanating up.

Consider the parameter w = (1, n − 1). According to [Col], the radical filtration of V (w) is:

V (w) : \begin{matrix}L(1, n − 1) \\ L(0, n)^{θ} L(1, n − 1)^{θ} L(2, n − 2)^{θ} L(0, n − 2)^{θ} \\ L(0, n − 1)^{θ} L(1, n − 2)^{θ} \end{matrix}

We now apply the translation functor ψ_{1} to push the (regular) GVM V (w) to its counterpart \overline{V} (w) in the singular category (A_{n}, \{α_{2}, …, α_{n−1}\}, \{α_{1}\}). Abuse notation and write L(i, j) for ψ_{1}L(i, j), etc. Then \overline{V} (w) has radical filtration

\overline{V} (w) : \begin{matrix}L(1, n − 1) \\ L(0, n)^{θ} L(0, n − 2)^{θ} \\ L(0, n − 1)^{θ} \end{matrix}

This situation gives a “diamond” in the quiver (note that both parameters in layer 1 are related to the parameter in layer 2 by simple reflections, so these extensions exist). But there is also an extension between L(1, n − 2) and L(1, n − 1) for the same reason. Thus we have a subquiver of the form Figure 5, which implies that the singular category has wild representation type.

Now assume we are in the B_{n} case, and again consider the associated regular category (B_{n}, {α_{2}, …, α_{n−1}}, ∅). There are 3n elements forming a segment of ^{S}W, which we will denote simply as { i, i′, i′′ : 1 ≤ i ≤ n }, defined as follows:

i = s_{1}s_{2} … s_{i−1}
i′ = s_{n}s_{1}s_{2} … s_{i−1}
i′′ = s_{n}s_{n−1}s_{1} … s_{i−1},

01^{θ}

02^{θ} 11^{θ}

03^{θ}  12^{θ}  21^{θ}

(0, n − 1)^{θ} (n − 2, 1)^{θ}

(0, n)^{θ} (1, n − 1)^{θ} (n − 2, 2)^{θ} (n − 1, 1)^{θ}

0, n  1, n − 1  n − 2, 2 n − 1, 1

0, n − 1 n − 2, 1

03  12  21

02 11

01

n 1 n−1 1 n 2 n−2 1 n−1 2 n 3 3 n−2 2 1 3 n−2 n n−1 1 2 n−1 n 2 1 3 n−2 n n−1 3 n−2 n−2 1 n−1 2 n 3 n−1 1 n 2 n 1

Figure 6. Poset for (A_{n}, {α_{2}, …, α_{n−1}}, ∅)

where the element i = 1 denotes the identity. The poset is depicted in Figure 7, where edge labels denote simple reflections acting on the right. The parameters in ^{S}W^{J} are those with an edge labeled “n” emanating up.

Our immediate goal is to compute the radical filtration of V (n′). It is easy to use the filtered U_{α}-algorithm to deduce the following radical filtrations:

(5.5.1)

V (1) : L(1)  V (i) : \begin{matrix}L(i) \\ L(i − 1) \end{matrix}  V (1′) : \begin{matrix}L(1′) \\ L(1) \end{matrix}  V (j′) : \begin{matrix}L(j′) \\ L(j) L(j − 1)′ \\ L(j − 1) \end{matrix}

1 2  1′ 2′  1′′ 2′′ n − 2 n − 1  (n − 2)′ n  (n − 1)′ (n − 2)′′ n′  (n − 1)′′ n′′_{1 1 1 2 2 2 n−3 n−3 n−3 n−2 n−2 n−2 n−1 n−1 n−1 n n n n n−1 n n n−1 n n n−1 n n n n−2} Figure 7

where 2 ≤ i ≤ n and 2 ≤ j < n. Now apply θ_{n−1} to V (n − 1)′ to get

(5.5.2)  \begin{matrix}V (n − 1)′ \\ V (n′) \end{matrix} = \begin{matrix}θ_{n−1}L(n − 1)′ \\ θ_{n−1}L(n − 1) ⊕ θ_{n−1}L(n − 2)′ \end{matrix}

From Figure 7 and (5.5.1) we have

U_{n−1}L(n − 1)′ = L(n′)
U_{n−1}L(n − 1) = L(n) ⊕ L(n − 2)
U_{n−1}L(n − 2)′ = L(n − 2)′′ ⊕ L(n − 2).

Putting all of this information into (5.5.2) and “erasing” the radical filtration of V (n − 1)′ given in (5.5.1) produces the radical filtration of V (n′):

V (n′) : \begin{matrix}L(n′) \\ L(n) L(n − 1)′ L(n − 2)′′ L(n − 2) \\ L(n − 1) L(n − 2)′ \end{matrix}

Finally, apply ψ_{n} to obtain the radical filtration of the corresponding singular GVM:

\overline{V} (n′) : \begin{matrix}L(n′) \\ L(n) L(n − 2)′′ L(n − 2) \\ L(n − 1) \end{matrix}

In particular, the simple modules corresponding to n′, n, n − 2, n − 1 form a “diamond” in the quiver (though not in ^{S}W^{J}), and we have an additional extension between L(n′) and L(n′′). Thus we again have a subquiver of the form Figure 5, so this type B category has wild representation type. _{□}

5.6. We can now verify the statement in Theorem 1.6(A)(b). It suffices to show that all cases when both S ≠ ∅ and J ≠ ∅ have wild representation type. This will be accomplished by systematically looking at each type of root system. Proposition 5.3 reduces us to looking at cases when Δ−(S ∪J) = {β} where β is a simple root at the end of the Dynkin diagram. Assume that S ≠ ∅ and J ≠ ∅.

Type A_{n}: Without loss of generality we may assume that β = α_{1}. It follows from Proposition 5.5 and Corollary 2.11 that if α_{2} ∈ S then O_{S}^{μ} has wild representation type. Therefore, we can assume that α_{2} ∈ J. If α_{3}, α_{4} ∈ J then O_{S}^{μ} has wild representation type by Theorem 1.3 and Corollary 2.11. On the other hand, we know from Section 5.4 that O(A_{3}, {3}, {2}) and O(A_{4}, {4}, {2, 3}) have wild representation type; thus O_{S}^{μ} has wild representation type in the remaining cases.

Type BC_{n}: First let us look at the case when β = α_{1}. By using Proposition 5.5, we can assume that α_{2} ∈ J. Moreover, if α_{3}, α_{4} ∈ J then O_{S}^{μ} has wild representation type, by Theorem 1.3 and Corollary 2.11. Now the following all have wild representation type: O(BC_{4}, {4}, {2, 3}), O(A_{4}, {4}, {2, 3}), O(BC_{3}, {3}, {2}) and O(A_{3}, {3}, {2}). Consequently, O_{S}^{μ} has wild representation type.

Next assume β = α_{n}. One can now reduce down to the B_{3} Levi factor and use Theorem 1.3, Theorem 1.4, or Section 5.4, along with Corollary 2.11.

Type D_{n}: We can find a subset of simple roots T ⊊ Δ, spanning a subsystem of type A, such that β ∈ T, T ∩ S ≠ ∅ and T ∩ J ≠ ∅. The algebra O(𝔪_{T} , S ∩ T, μ) has wild representation type by our results for A_{n}. Therefore, O_{S}^{μ} has wild representation type.

Type E_{6}, E_{7}, E_{8}: Here O_{S}^{μ} has wild representation type by the same argument given in our analysis for type D_{n}.

Type F_{4}: Without loss of generality we may assume that β = α_{1}. There are six possibilities for S, J. Set T = {α_{1}, α_{2}, α_{3}}. In five of the possibilities, we can deduce from information about BC_{3} that O(𝔪_{T} , S ∩ T, μ) has wild representation type. The only remaining case is when (Φ, Φ_{S}, Φ_{J}) = (F_{4}, B_{2}, A_{1}). But, O_{S}^{μ} has wild representation type in this situation by Section 5.4.

Type G_{2}: Since the rank is two and |Δ − (S ∪ J)| = 1 it follows that either S = ∅ or J = ∅.

Our analysis above has reduced us to the cases when either S = ∅ or J = ∅. Theorem 1.6(A)(b) now follows by Theorems 1.3 and 1.4.

6. Mixed case: Δ = S ∪ J, Type A

Throughout this section, we assume that 𝔤 is of type A_{n}. We determine the representation type of O(𝔤, S, J) when S ∩ J = ∅, S ∪ J = Δ. The proof will proceed by a series of reductions, based on the total number of connected components of S and J in the Dynkin diagram of Φ.

6.1. One component. The blocks O(𝔤, ∅, Δ) and O(𝔤, Δ, ∅) are semisimple by Theorems 1.3 and 1.4.

6.2. Two components. By reversing the Dynkin diagram if necessary, we may assume that J = {α_{1}, …, α_{n−r}}, S = {α_{n−r+1}, …, α_{n}}. Writing weights for A_{n} as usual in coordinates in ℝ^{n+1}, but translating by a multiple of (1, …, 1) to make all coordinates nonnegative (for convenience), we may take μ + ρ = (0, …, 0, 1, 2, …, r). Then the only W - translate of μ+ρ which is regular dominant on all the roots in S is (0, …, 0, r, r−1, …, 1, 0). Since there is only one highest weight, we conclude that O(𝔤, S, J) is semisimple.

6.3. Four or more components. We will show that any category O(𝔤, S, J) in which the total number of components of S and J is four or more is wild. By Corollary 2.11, this will follow if we show it when the total number of components is exactly four; we assume this for the remainder of the subsection. By the same reasoning, it is enough to treat the cases where each of {α_{1}} and {α_{n}} is a component of S or J. By reversing the Dynkin diagram if necessary, we henceforth assume that {α_{1}} is a component of S and {α_{n}} is a component of J.

Assume 𝔤 is of type A_{n} with n ≥ 5, S = {α_{1}, α_{k+1}, …, α_{n−1}}, and J = {α_{2}, α_{3}, …, α_{k}, α_{n}} for some 2 ≤ k ≤ n − 2. Put

w = s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … s_{1}
x = s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … ŝ_{k} … s_{1}
y = \begin{cases}s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … ŝ_{k}ŝ_{k−1} … s_{1}, & k > 2 \\ s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … s_{4}s_{1}, & k = 2 \end{cases}

v = \begin{cases}s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … ŝ_{k+1}ŝ_{k} … s_{1}, & 2 < k < n − 2 \\ s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … s_{3}, & k = 2 \\ s_{2}s_{3} … s_{n−2}s_{n−1}s_{n−3} … s_{1}, & k = n − 2 \end{cases}

z = \begin{cases}s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … ŝ_{k+1}ŝ_{k}ŝ_{k−1} … s_{1}, & k > 2 \\ s_{2}s_{3} … s_{n−2}s_{n}s_{n−1} … s_{5}s_{1}, & k = 2, n > 5 \\ s_{2}s_{3}s_{4}s_{1}, & k = 2, n = 5, \end{cases}

where denotes an omitted factor. It is straightforward (along the lines of Example 2.7) to check that the set {w, x, y, z, v} satisfies the conditions of Proposition 2.8, and thus O(𝔤, S, J) is wild.

Finally, we are left with the case (A_{4}, {α_{1}, α_{3}}, {α_{2}, α_{4}}). Direct computation shows that this category contains exactly five irreducibles, parametrized by Weyl group elements w, x, y, z, v, where w, x, y, v are arranged as in Figure 1, but l(z) = l(y)−3 and z < y, z < v. Nonetheless, the corresponding GVM radical filtrations are as described in the proof of Proposition 2.8, and so the argument given there shows that this category is wild.

6.4. Three components.

6.4.1. J–S–J. Assume first that J has two components and S has one; then α_{1}, α_{n} ∈ J.

We claim that if each component of J contains at least two simple roots, then O(𝔤, S, J) is wild. The claim will follow, by Corollary 2.11, from the special case where each component of J contains exactly two roots. But in this situation, we have shown (in Example 2.7) that SW^{J} contains a diamond. Thus the category is wild, by Proposition 2.7.

So we are reduced to the situation where one component of J contains a single root; by symmetry, we may assume that this component is {α_{1}}. Let the other component of J have k simple roots, {α_{n−k+1}, …, α_{n}}. We will show that O(𝔤, S, J) is finite if k = 1 or 2, tame if k = 3, and wild if k = 4; it will follow from the last statement that the category is wild whenever k ≥ 4. Our plan is to show that for each of the indicated values of k, the category contains exactly k + 1 simple modules, and the corresponding GVMs are multiplicity-free and uniserial of lengths 1, 2, …, k + 1. From this the claimed representation types will follow, using previously known results.

An antidominant weight, regular on S and singular on J, is μ + ρ = (0, 0, 1, 2, …, m, …, m), with k + 1 occurrences of m = |S|. The remaining W -translates of μ + ρ which are regular antidominant on S are obtained from μ + ρ by interchanging the first zero with one of the last k m’s. This shows that there are k + 1 simple modules in the category, and in addition, that the parameters for k of them form a chain in the Bruhat order, linked by successive simple reflections (acting on the left).

We will compute the filtrations of the corresponding singular GVMs using the left-side singular relative U_{α}-algorithm, along with induction on k. We begin the induction by working in the associated regular category O(𝔤, S, ∅). For simplicity of notation, we express results in terms of the fixed m.

Suppose first that k = 1. Then the regular category is associated to the real rank one group SU(m + 2, 1). As discussed in Section 5.5, the GVM filtrations are given in [Col], in terms of^{S}W parameters in Figure 6. The two parameters which survive on the α_{1} and α_{m+2} walls are the ones which have edges labeled 1 and m + 2 = n emanating up, namely (01)^{θ} and (12). Write V (−) for a regular GVM in O(𝔤, S, ∅) and \overline{V} (−) for the corresponding singular GVM in O(𝔤, S, J); we abuse notation and write L(−) for a simple module in either the regular or singular category. Then V (01)^{θ} = L(01)^{θ} = \overline{V} (01)^{θ} is irreducible. According to [Col], V (12) has a multiplicity-free radical filtration, with L(12) as its top, L(01)^{θ} in the next layer, and other composition factors which do not survive on the wall. Thus we have the following radical filtrations:

(6.4.1)  \overline{V} (01)^{θ} : L(01)^{θ}  \overline{V} (12) : \begin{matrix}L(12) \\ L(01)^{θ} \end{matrix}

as claimed. By [FNP, Section 7.1], this category has finite representation type.

Next suppose k = 2. The singular category for k = 1, just discussed, embeds in this category as a truncated subcategory. There is one additional simple module, parametrized by s_{m+3}(12). The structures of \overline{V} (01)^{θ} and \overline{V} (12) are unchanged from (6.4.1) , so it remains to find the structure of \overline{V} (s_{m+3}(12)). We do this using the (singular) U_{α}-algorithm, Theorem 2.6.4, applying _{(m+3)}θ to \overline{V} (12). Note that every parameter x for the truncated subcategory has x < s_{m+3} x ∈ ^{S}W (because α_{m+3} is orthogonal to Φ_{S}). However, s_{m+3} x ∉ ^{S}W^{J} (except for the “new” parameter, with x = (12)). The algorithm gives

(m+3)θ \overline{V} (12) = \begin{matrix}\overline{V} (12) \\ \overline{V} (s_{m+3}(12)) \end{matrix} = \begin{aligned}L(12) \\ L(s_{m+3}(12)) & L(01)^{θ} \\ L(12) \\ & L(01)^{θ} \end{aligned}

whence

\overline{V} (s_{m+3}(12)) : \begin{matrix}L(s_{m+3}(12)) \\ L(12) \\ L(01)^{θ} \end{matrix}

as desired. By [FNP, Section 7.2], this category has finite representation type.

The case k = 3 proceeds similarly. The k = 2 category is a truncated subcategory, and there is one new parameter, s_{m+4}s_{m+3}(12). For x in the truncated subcategory, x < s_{m+4} x ∈ ^{S}W, and s_{m+4}x ∉ ^{S}W^{J} except when x = s_{m+3}(12). Applying _{(m+4)}θ to V (s_{m+3}(12)) and using Theorem 2.6.4 gives

\overline{V} (s_{m+4}s_{m+3}(12)) : \begin{matrix}L(s_{m+4}s_{m+3}(12)) \\ L(s_{m+3}(12)) \\ L(12) \\ L(01)^{θ} \end{matrix}

so by [FNP, Section 6], this category is tame.

Finally, the argument for k = 4 procceds in the same way, producing a uniserial GVM of length 5. By [FNP, Section 5.2], this category is wild.

6.4.2. S–J–S. Assume now that S has two components and J has one; then α_{1}, α_{n} ∈ S.

We claim that if each component of S contains at least two simple roots, then O(𝔤, S, J) is wild. The claim will follow, by Corollary 2.11, from the special case where each component of S contains exactly two roots. But in this situation, it is elementary to check (as in Example 2.7) that the following elements form a diamond in ^{S}W^{J}: x = s_{3}s_{4} … s_{n−2}s_{n−3} … s_{2}s_{n−1}, y_{1} = xs_{1}, y_{2} = xs_{n}, w = xs_{1}s_{n}. Thus the category is wild, by Proposition 2.7.

So we are reduced to the situation where one component of S contains a single root; by symmetry, we may assume that this component is {α_{1}}. Let the other component of S have k simple roots, {α_{n−k+1}, …, α_{n}}. We will show that O(𝔤, S, J) is finite.

An antidominant weight, regular on S and singular on J, is μ + ρ = (0, 1, 1, …, 1, 2, …, k + 1). The remaining W -translates of μ + ρ which are regular antidominant on S are (1, i + 1, 1, …, 1, 0, 1, …, { \atop i + 1}, …, k + 1) for 1 ≤ i ≤ k. This shows that there are k + 1 simple modules in the category. The corresponding elements of ^{S}W^{J} are e and w_{i} = s_{m} … s_{2} … s_{m}s_{1}s_{m+1} … s_{m+i}, 1 ≤ i ≤ k, where m = n − k.

We begin with k = 1 by analyzing the structure of the singular (ordinary) Verma modules in O(𝔤, ∅, J), and then project to the quotient category O(𝔤, S, J).

The poset W^{J} is the inverse of the poset ^{S}W for SU(n, 1) which appeared in Sections 5.5 and 6.4.1. Thus we may use the same labels (ij) and (ij)^{θ} for the parameters; the Bruhat order is also the same as in Figure 6, except that the simple reflections act on the left rather than on the right. We recall the notion of a standard composition factor. If x ≤ w in W and M(w) is an ordinary Category O Verma module, then [rad_{l(w)−l(x)}M(w) : L(x)] = 1, and this occurrence of L(x) (which arises from the inclusion of M(x) in M(w)) is called a “standard composition factor” of M(w). Any other occurrences of L(x) as a composition factor of M(w) must lie in rad_{i}M(w) for i < l(w) − l(x), and are called “nonstandard.” Analogous statements hold for singular Verma modules \overline{M}(w) in O(𝔤, ∅, J) when x ≤ w in W^{J}. Proposition (A). Let \overline{M}(w) be a (singular) Verma module in O(A_{n}, ∅, \{2, …, n − 1\}). Then all composition factors of \overline{M}(w) are standard, unless w = (ij) with i + j ≤ n − 2. In the latter case, set r = n − i − j − 1; then the non-standard composition factors of \overline{M}(w) are as follows, arranged according to their radical layers, beginning with rad_{1} at the top:

L(i + 1, j + 1)^{θ}
L(i + 2, j + 1)^{θ} L(i + 1, j + 2)^{θ}
⋮
L(i + r, j + 1)^{θ} L(i + r − 1, j + 2)^{θ} … L(i + 2, j + r − 1)^{θ} L(i + 1, j + r)^{θ}.

Proof. This is an elementary (if somewhat tedious) application of Irving’s singular U_{α}- algorithm, Theorem 2.6.3. _{□}

Now let w ∈ ^{S}W^{J}. The GVM \overline{V} (w) is a quotient of \overline{M}(w_{S}w), where w′ := w_{S}w ∈ W^{J} satisfies w′ > w′s_{i} for i ∈ S = {1, n}. Inspection shows that there are exactly two such parameters w′ ∈ W^{J}, namely (12)^{θ} and (01), corresponding (respectively) to w = e and w = w_{1}, in the notation introduced earlier in this subsection. Of course \overline{V} (e) = L(e) is irreducible. To compute \overline{V} (w_{1}), we apply the functor Res (recall Section 2.5) to \overline{M}(01). By [Lep2], the kernel of this projection is the sum in \overline{M}(01) of its submodules \overline{M}(s_{1}(01)) and \overline{M}(s_{n}(01)). By Proposition 6.4.2(A), we obtain

(6.4.2)  \overline{V} (w_{1}) : \begin{matrix}L(w_{1}) \\ L(e). \end{matrix}

In particular, by [FNP, Section 7.1], this category has finite representation type.

Now assume k > 1 and note that the k = 1 category embeds as a truncated subcategory; in particular, the structure of \overline{V} (w_{1}) is (6.4.2). The parameters w_{1}, … w_{k} ∈ ^{S}W^{J} form a simple chain, with w_{i+1} = w_{i}s_{m+i+1} for i < k. Lemma. Let notation be as in this subsection, with k > 1. Then w_{i} < w_{i}s_{m+j} ∉ ^{S}W for 1 ≤ i, j ≤ k, j ≠ i, i + 1.

Proof. Use the criterion: w ∈ ^{S}W ⟺ w^{−1}Φ_{S}^{+} ⊂ Φ^{+}. It is straightforward to check that if i+1 < j ≤ k then (w_{i}s_{m+j})^{−1}α_{m+j} = −α_{m+j}, while if 1 ≤ j < i then (w_{i}s_{m+j})^{−1}α_{m+j+1} = −α_{m+j+1}. Finally, the well-known fact that if w ∈ ^{S}W, s is a simple reflection, and w > ws, then ws ∈ ^{S}W (or a direct check with roots) verifies the inequalities. _{□}

Proposition (B). Let notation be as in this subsection, with k > 1. For 1 ≤ i < k the radical filtration of \overline{V} (w_{i+1}) is given by

\overline{V} (w_{i+1}) : \begin{matrix}L(w_{i+1}) \\ L(w_{i}) \end{matrix}

Proof. By induction on i. Apply the singular (relative) U_{α}-algorithm to \overline{V} (w_{i}), beginning with (6.4.2). Note that α_{m+i+1} is orthogonal to Φ_{J} since J = {α_{2}, …, α_{m}}, so Proposition 2.6.2 applies. Combined with the lemma, it gives

θ_{m+i+1}\overline{V} (w_{i}) = \begin{matrix}\overline{V} (w_{i}) \\ \overline{V} (w_{i+1}) \end{matrix} = \begin{matrix}L(w_{i}) \\ L(w_{i+1}) L(w_{i−1}) \\ L(w_{i}) \end{matrix}

when i > 1, and similarly when i = 1 replacing w_{i−1} by e. Hence the structure of \overline{V} (w_{i+1}) is as claimed. _{□}

As in Section 3.1, this category has finite representation type. This completes the verification of Theorem 1.6(B)(a).

7. Mixed case: Δ = S ∪ J, Type D

We now assume that Φ is of type D_{n}, n ≥ 4. We determine the representation type of O(𝔤, S, J) when S ∩ J = ∅, S ∪ J = Δ. The BC case will be investigated in the following section where several of the arguments in this section will be repeated. We will proceed as in type A, by analyzing the cases based on the total number of connected components of S and J in the Dynkin diagram of Φ.

7.1. One component. The block is semisimple in this case. This can be seen by using the results for ordinary Category O or the regular blocks of category O_{S}.

7.2. Two components. Let us first consider the case when (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{n−1}, A_{1}). This is a Hermitian symmetric semiregular case, so by Corollary 1.5, the block has finite representation type for n = 4, 5, and is wild for n ≥ 6.

Next suppose that (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{1}, A_{n−1}). For n = 6, this block is wild because the poset contains a diamond, thus for n ≥ 6 the block is wild by Corollary 2.11. One can directly verify for n = 4, this block has finite representation type (there are two simple modules and the corresponding GVM structure is as in Section 3.1) and for n = 5 this has tame representation type (there are four simples and the corresponding GVMs are multiplicity-free uniserial of lengths 1, 2, 3, and 4).

In the following subsections we analyze the remaining two-component cases, (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{r}, D_{n−r}) and (D_{n}, D_{n−r}, A_{r}) with 1 ≤ r ≤ n − 3.

7.2.1. r = 1. The category O(D_{n}, D_{n−1}, A_{1}) has two simple modules and is a semiregular Hermitian symmetric category. This category has finite representation type by Corollary 1.5. From Theorem 2.4.3 and the result above, it follows that O(D_{n}, A_{1}, D_{n−1}) must also have finite representation type.

7.2.2. r = 2. We will consider the blocks (D_{n}, D_{n−2}, A_{2}) and (D_{n}, A_{2}, D_{n−2}). It is easy to see (by examining weights in coordinates) that there are four simple modules in each of these blocks. We will label them A, B, C, D. The correspondence between weights and these labels is given in the table below. Set b = s_{2}s_{3} … s_{n}s_{n−2} … s_{3}, c = s_{1}b.

Simple module (D_{n}, D_{n−2}, A_{2}) (D_{n}, A_{2}, D_{n−2})
A  L(e)  L(e)
B  L(b)  L(b^{−1})
C  L(c)  L(c^{−1})
D  L(bc)  L(c^{−1}b^{−1})

The standard embeddings of the categories (D_{n−1}, D_{n−2}, A_{1}) and (D_{n−1}, A_{1}, D_{n−2}) show that we have the following structure for the GVMs of A and B.

\overline{V} (A) : A \overline{V} (B) : \begin{matrix}B \\ A \end{matrix}

The weights for the simple modules B and C differ by a simple reflection (on the right or the left) so the appropriate U_{α}-algorithm (Proposition 2.6.2 or Theorem 2.6.4) shows that the GVM \overline{V} (C) has two possible structures: Case 1, (D_{n}, A_{2}, D_{n−2}):

\overline{V} (C) : \begin{matrix}C \\ B \end{matrix}

Case 2, (D_{n}, D_{n−2}, A_{2}):

\overline{V} (C) : \begin{matrix}C \\ B \\ A \end{matrix}

We need to determine the structure of \overline{V} (D) in both cases. We will do this by carefully examining the structures of the projective indecomposable modules. Recall that from standard facts in homological algebra one has for μ_{1}, μ_{2} ∈ X

(7.2.1)  dim Ext_{O}^{1}(L(μ_{1}), L(μ_{2})) = [rad_{1}P (μ_{1}) : L(μ_{2})]

Moreover, since O_{S} is a highest weight category, if μ_{1} ≤ μ_{2} in X_{S}^{+} then

(7.2.2)  dim Ext_{O}^{1}(L(μ_{1}), L(μ_{2})) = [rad_{1}V (μ_{2}) : L(μ_{1})]

We shall also employ BGG reciprocity and make use of the following: if L is a simple module appearing in the socle of a GVM then the projective cover of L, P (L), is self-dual [Irv2, Addendum].

Let O_{1} (resp. O_{2}) be the block corresponding to (D_{n}, A_{2}, D_{n−2}) (resp. (D_{n}, D_{n−2}, A_{2})). From the structures of the GVMs above, one can conclude that Ext_{O_{j}}^{1}(A, B) ≅ ℂ for j = 1, 2. Note that the length-complementary map ^{S}f^{J} of Theorem 2.4.3 must interchange A with D and B with C. Also, for our particular S and J, we have −w_{0}S = S and −w_{0}J = J regardless of the parity of n. We can conclude that Ext_{O_{j}}^{1}(D, C) ≅ ℂ for j = 1, 2. Also, since l(c) is even, it follows by parity (using the table near the beginning of this section) that Ext_{O_{j}}^{1}(A, C) = Ext_{O_{j}}^{1}(B, D) = 0.

Case 1, (D_{n}, A_{2}, D_{n−2}): Suppose that [\overline{V} (D) : A] ≠ 0. Then D must extend A since \overline{V} (D) appears in P (A), \overline{V} (C) does not appear, and Ext_{O}^{1}(B, D) = 0. It follows that there is a factor of D in the second radical layer of P (A) and P (A) has a quotient of the form given in Figure 8. Moreover, P (A) is self-dual since A is the socle of \overline{V} (A). By self-duality, there is a submodule of P (A) with top factor D and bottom factor A. Since D corresponds to the GVM with the largest weight, P (A) has a filtration with GVM subquotients such that \overline{V} (D) is a submodule of P (A). Moreover, [\overline{V} (D) : D] = 1 implies that \overline{V} (D) must have top factor D and bottom factor A. This contradicts the self-duality of P (A). Hence, [\overline{V} (D) : A] = 0.

A B D A Figure 8

Suppose that [\overline{V} (D) : B] ≠ 0. We know by BGG reciprocity that P (B) has a quotient of the form given in Figure 9. Now \overline{V} (D) must appear in P (B). For P (B) to be self-dual

B C A B Figure 9

(since B ⊂ soc \overline{V} (C)), [\overline{V} (D) : A] ≠ 0 because Ext^{1}(B, D) = 0 yet A appears in the second radical layer of P (B). This is a contradiction, so [\overline{V} (D) : B] = 0.

From our discussion above we have [\overline{V} (D) : C] ≠ 0 because in fact Ext_{O}^{1}(C, D) ≅ ℂ. Moreover, A and B are not composition factors of \overline{V} (D), thus we have

\overline{V} (D) : \begin{matrix}D \\ C \end{matrix}

Hence, we can conclude that (D_{n}, A_{2}, D_{n−2}) has finite representation type.

Case 2, (D_{n}, D_{n−2}, A_{2}): Suppose that Ext_{O}^{1}(D, A) ≠ 0. Then the same argument given at the beginning of Case 1 leads to a contradiction. Thus Ext_{O}^{1}(D, A) = 0.

If [\overline{V} (D) : A] ≠ 0 then P (A) has a quotient of the form given in Figure 10; we use the fact that D does not extend A or B, but does extend C. Since P (A) is self-dual one can use the argument given in Case 1 to see that \overline{V} (D) has the following structure.

(7.2.3)  \overline{V} (D) : \begin{matrix}D \\ C \\ B \\ A \end{matrix}

A B D A C B A Figure 10

In this situation, we can conclude that the block corresponding to (D_{n}, D_{n−2}, A_{2}) is Morita equivalent to the block for (A_{3}, ∅, A_{2}). This has tame representation type by Theorem 1.3.

Now suppose that [\overline{V} (D) : A] = 0. Suppose that C is in the socle of \overline{V} (D). Recall that D extends only C. The self-duality of P (C) shows that \overline{V} (D) has two composition factors D and C. This shows that P (C) cannot be self-dual after all. Thus, C is not in the socle of \overline{V} (D).

We know \overline{V} (D) is reducible since Ext_{O}^{1}(C, D) ≠ 0. The only remaining possibility is that B must be in the socle or \overline{V} (D), so P (B) is self-dual. The module P (B) has a quotient of the form described in Figure 11.

B D A C B Figure 11

Therefore, \overline{V} (D) must be

\overline{V} (D) : \begin{matrix}D \\ C \\ B \end{matrix}

In this instance one can easily verify that the basic algebra is a homomorphic image of the basic algebra corresponding to the block for (A_{3}, ∅, A_{2}) given in [FNP, §6.1]. This implies that this algebra cannot be wild. A similar argument to the one given in [FNP, §6.1] shows that the algebra is not finite. Thus we again conclude that O(D_{n}, D_{n−2}, A_{2}) is tame. Remark. Examples strongly suggest that \overline{V} (D) always has the structure (7.2.3) in the (D_{n}, D_{n−2}, A_{2}) case. However, we have been unable to prove in general that [\overline{V} (D) : A] ≠ 0.

7.2.3. r ≥ 3. We assume throughout this subsection that r ≤ n − 3. We will first show that the algebra (D_{n}, A_{r}, D_{n−r}) is wild for r = 3; thus by Corollary 2.11, for r ≥ 3, this algebra has wild representation type.

The poset ^{S}W^{J} consists of the 8 elements listed in the table below, with Bruhat order depicted in Figure 12, where the indicated simple reflections act on the right; the dashed lines represent length > 1 Bruhat order relations.

Parameter reduced expression
w_{1}  e
w_{2}  s_{4} … s_{n}s_{n−2} … s_{3}
w_{3}  w_{2}s_{2}
w_{4}  w_{3}w_{2}
w_{5}  w_{3}s_{1}
w_{6}  w_{4}s_{1}
w_{7}  w_{6}s_{2}
w_{8}  w_{7}w_{2}

The elements w_{1}, …, w_{4} parametrize the truncated subcategory corresponding to (D_{n−1}, A_{2}, D_{n−3}). By the previous subsection, this latter category has finite representation type, with GVMs uniserial of length two (except \overline{V} (w_{1}), which is irreducible). Apply the singular relative U_{α}-algorithm with s = s_{1} to obtain the radical filtration of \overline{V} (w_{6}) from that of \overline{V} (w_{4}); note that Proposition 2.6.2 applies because α_{1} is orthogonal to Φ_{J}. From the table,

w_{2} < w_{2}s_{1} = s_{1}w_{2} ∉ ^{S}W. Beginning with \overline{V} (w_{4}) = \begin{matrix}L(w_{4}) \\ L(w_{3}) \end{matrix} we get

θ_{1}\overline{V} (w_{4}) = \begin{matrix}\overline{V} (w_{4}) \\ \overline{V} (w_{6}) \end{matrix} = \begin{aligned}L(w_{4}) \\ L(w_{6}) & L(w_{3}) \\ L(w_{4}) & L(w_{5}) L(w_{2}) \\ & L(w_{3}) \end{aligned}

from which we deduce the radical filtration

\overline{V} (w_{6}) : \begin{matrix}L(w_{6}) \\ L(w_{5}) L(w_{4}) L(w_{2}) \\ L(w_{3}) \end{matrix}

In particular, Ext^{1}(L(w_{5}), L(w_{6})) ≠ 0. Since we also have Ext^{1}(L(w_{i}), L(w_{j})) ≠ 0 for (i, j) = (4, 6), (3, 4), (3, 5), and (2, 3), the elements w_{2}, …, w_{6} determine a subquiver of the form shown in Figure 5. Hence this category is wild. Furthermore, by Theorem 2.4.3, the Ext^{1}-quivers for (D_{n}, A_{3}, D_{n−3}) and (D_{n}, D_{n−3}, A_{3}) are the same. Therefore, from our analysis above (for (D_{n}, A_{3}, D_{n−3})) we can conclude that (D_{n}, D_{n−3}, A_{3}) is also wild. It follows from Corollary 2.11 that (D_{n}, D_{n−r}, A_{r}) is wild for r ≥ 3.

7.3. Three components.

7.3.1. J–S–J. Let us consider the block where (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{r}, A_{1} × D_{n−r−1}) with n − r − 1 ≥ 3. For n − r − 1 ≥ 5, this block is wild because (A_{m}, A_{r}, A_{1} × A_{m−r−1}) is wild for m − r − 1 ≥ 4 and by applying Corollary 2.11. Also, for r ≥ 3, this block is wild by our two component analysis for type D (Section 7.2.3). Therefore, we are reduced

w_{1} w_{2} w_{3} w_{4} w_{5} w_{6} w_{7} w_{8}_{2 1 1 2}

Figure 12. ^{S}W^{J} for (D_{n}, A_{3}, D_{n−3}) and (D_{n}, D_{n−3}, A_{3})

to looking at the cases when r = 1, 2 and n − r − 1 = 3, 4. One can verify that in these cases the blocks have wild representation type because they have diamonds in their posets SW^{J} (Proposition 2.7). Therefore, by Corollary 2.11, (D_{n}, A_{r}, A_{s} × D_{n−s−r}) has wild representation type for r ≥ 1, s ≥ 1, n − s − r ≥ 3.

Next we consider the cases (D_{n}, A_{n−2}, A_{1} × A_{1}) (with J = {α_{1}, α_{n}}). Note that for n ≥ 7 these algebras have wild representation type by our two component analysis for type D (Section 7.2). The blocks for n = 5, 6 have diamonds in their posets and thus have wild representation type. Consequently, by Corollary 2.11, (D_{n}, A_{r}, A_{n−r−1} × A_{1}) (with J = {α_{1}, …, α_{n−r−1}, α_{n}}) also has wild representation type as long as r ≥ 3.

Finally, we must consider the case r = 2. The block (D_{5}, A_{2}, A_{2} × A_{1}) (where S = {α_{3}, α_{4}}) has wild representation type because there is a diamond in the poset. Thus, (D_{n}, A_{2}, A_{n−3} × A_{1}) (where S = {α_{n−2}, α_{n−1}}) has wild representation type for n ≥ 5. The case n = 4 will be discussed in Section 7.3.3 below.

7.3.2. S–J–S. Consider the block where (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{1} × D_{n−r−1}, A_{r}) with n − r − 1 ≥ 3. For r ≥ 1, these algebras are wild because they have diamonds in their posets, as listed in the following table (notation follows Section 2.7) . Set u = s_{2} … s_{r} … s_{1}s_{r+1} … s_{n−2}.

Element of ^{S}W^{J} (D_{n}, A_{1} × D_{n−r−1}, A_{4})
y  u
x_{1}  us_{n−1}
x_{2}  us_{n}
w  us_{n−1}s_{n}

By Corollary 2.11, (D_{n}, A_{s} × D_{n−s−r}, A_{r}) has wild representation type for r ≥ 1, s ≥ 1, n − s − r ≥ 3.

Next we consider the cases (D_{n}, A_{1} × A_{1}, A_{n−2}) (where S = {α_{1}, α_{n}}). For n ≥ 7, these algebras have wild representation type by Corollary 2.11 and Section 7.2. For n = 5, 6, there are diamonds in the poset, so these algebras are wild. Hence, by Corollary 2.11, (D_{n}, A_{n−r−1} × A_{1}, A_{r}) (with S = {α_{1}, …, α_{n−r−1}, α_{n}}) has wild representation type as long as r ≥ 3.

Finally consider the case r = 2. The block (D_{5}, A_{2} × A_{1}, A_{2}) (where J = {α_{2}, α_{3}}) has wild representation type because there is diamond in the poset. Therefore, the block (D_{n}, A_{n−3} × A_{1}, A_{2}) (where J = {α_{n−2}, α_{n−1}}) has wild representation type for n ≥ 5. In the next subsection we will cover the case n = 4.

7.3.3. Other Cases. We still need to analyze the 3-component cases when (Φ, Φ_{S}, Φ_{J}) = (D_{n}, A_{r}, A_{1} ×A_{1}) (J = {α_{n−1}, α_{n}}) and (D_{n}, A_{1} ×A_{1}, A_{s}) (S = {α_{n−1}, α_{n}}). For r = 3 and s = 3, these blocks have wild representation type. This can be verified by looking at the explicit filtrations of the generalized Verma modules, which show that the quiver contains a subquiver as in Figure 5 (page 28). Consequently, for r, s ≥ 3 these blocks are wild. The r = 2 block is finite (its four GVMs have filtrations as in Section 3.1) and the s = 2 block is tame (its four GVMs are multiplicity-free and uniserial of radical lengths 1, 2, 3, 4).

7.4. Four and more components. The only blocks for type D which have three components and are not wild are (D_{4}, A_{1} × A_{1}, A_{2}) and (D_{4}, A_{2}, A_{1} × A_{1}). Also, every block for type A with four or more components is wild. Therefore, in order to show that all blocks for type D which have four or more components are wild, we simply need to consider (D_{5}, A_{1} × A_{1} × A_{1}, A_{2}), (D_{5}, A_{2}, A_{1} × A_{1} × A_{1}), (D_{4}, A_{1}, A_{1} × A_{1} × A_{1}), and (D_{4}, A_{1} × A_{1} × A_{1}, A_{1}). One can verify that these algebras are indeed wild. For the D_{5} cases the wildness follows because there is a diamond in the poset. In order to prove that the D_{4} cases are wild one can use the explicit description of the radical filtrations of the generalized Verma modules and show there is a subquiver of the type given in Figure 5.

This completes the verification of Theorem 1.6(B)(d).

8. Mixed case: Δ = S ∪ J, Types BC

We now assume that Φ is of type B_{n} or C_{n}. We determine the representation type of O(𝔤, S, J) when S ∩ J = ∅, S ∪ J = Δ. As in the case for types A and D, the proof will proceed by a series of reductions, based on the total number of connected components of S and J in the Dynkin diagram of Φ.

8.1. One component. The block is semisimple in this case. This can be seen by using the results for ordinary Category O or the regular blocks of category O_{S}.

8.2. Two components. We consider the case when (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, BC_{n−r}, A_{r}) or (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{r}, BC_{n−r}), 1 ≤ r < n.

8.2.1. r = 1. For r = 1 there are two simples in the category. The (BC_{n}, BC_{n−1}, A_{1}) case is Hermitian symmetric semiregular, which was analyzed in Corollary 1.5. In the (BC_{n}, A_{1}, BC_{n−1}) case, the two parameters for the simples, e and s_{2}s_{3} … s_{n}s_{n−1} … s_{1}, have the same parity so there cannot be an extension between them. Thus in both cases the category splits into two semisimple components. One can also deduce the second case by using the first case and Theorem 2.4.3.

8.2.2. r = 2. For r = 2, there are four simple modules in each of these categories. We will label them A, B, C, D. The correspondence between Weyl group elements and these labels is given in the table below. Set a = e, b = s_{3} … s_{n}s_{n−1} … s_{2}, c = bs_{1}, d = cb.

Simple module (BC_{n}, A_{2}, BC_{n−2}) (BC_{n}, BC_{n−2}, A_{2})
A  L(a)  L(a^{−1})
B  L(b)  L(b^{−1})
C  L(c)  L(c^{−1})
D  L(d)  L(d^{−1})

Let O_{1} (resp. O_{2}) be the block corresponding to (BC_{n}, A_{2}, BC_{n−2}) (resp. (BC_{n}, BC_{n−2}, A_{2})). Because of parity of length of elements, we have Ext_{O_{j}}^{1}(A, B) = 0, and Ext_{O_{j}}^{1}(C, D) = 0 for j = 1, 2. Thus \overline{V} (A) and \overline{V} (B) are both irreducible. In (BC_{n}, A_{2}, BC_{n−2}) (resp. (BC_{n}, BC_{n−2}, A_{2})) the parameters for B and C differ by the simple reflection s_{1} on the right (resp. left) with α_{1} orthogonal to Φ_{J} (resp. Φ_{S}), so we can apply the U_{α}-algorithm, Proposition 2.6.2 (resp. Theorem 2.6.4), to conclude that the GVMs for A, B, C have the following structure:

\overline{V} (A) : A  \overline{V} (B) : B  \overline{V} (C) : \begin{matrix}C \\ B \end{matrix}

We need to determine the structure of \overline{V} (D). First let us consider the case when (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{2}, BC_{n−2}). Our goal is to show that \overline{V} (A) is a submodule of \overline{V} (D). We will show this by translating to a new category O(𝔤, S, J′), where J′ = {2, …, n−1}, and then back again. This procedure extends a method used in [ES] to analyze the Hermitian symmetric semiregular category (C_{n}, A_{n−1}, A_{1}) . Let

ψ_{J}: O(𝔤, S, ∅) → O(𝔤, S, J) and ϕ_{J}: O(𝔤, S, J) → O(𝔤, S, ∅)

be the translation functors onto and off of the J-walls, respectively; define ψ_{J′} and ϕ_{J′} analogously. Set

T_{J}^{J′} = ψ_{J′}ϕ_{J}: O(𝔤, S, J) → O(𝔤, S, J′)
T_{J′}^{J} = ψ_{J}ϕ_{J′}: O(𝔤, S, J′) → O(𝔤, S, J).

Write \overline{V} (−) (resp. \overline{V}′(−), V (−)) for GVMs in O(𝔤, S, J) (resp. O(𝔤, S, J′), O(𝔤, S, ∅)). By Section 2.5, for w ∈ ^{S}W^{J}, [ϕ_{J} \overline{V} (w)] is a sum of terms [V (y)] with y ∈ wW_{J} ∩^{S}W. On the other hand, for y ∈ ^{S}W, [ψ_{J′}V (y)] is either 0 or [\overline{V}′(w′)], according to whether yW_{J′} ∩^{S}W^{J′} is empty or {w′} (the intersection contains at most one element because ^{S}W^{J′} consists of left W_{J′} coset representatives). This motivates defining

Y = { y ∈ ^{S}W: yW_{J} ∩^{S}W^{J} ≠ ∅ and yW_{J′} ∩^{S}W^{J′} ≠ ∅ }.

For y ∈ Y define w(y) and w′(y) by

{w(y)} = yW_{J} ∩^{S}W^{J}, {w′(y)} = yW_{J′} ∩^{S}W^{J′}.

Then for w ∈ ^{S}W^{J},

(8.2.1)  [T_{J}^{J′} \overline{V} (w)] = ∑_{y∈Y,w(y)=w}[\overline{V}′(w′(y))]

and symmetrically, for w′ ∈ ^{S}W^{J′},

(8.2.2)  [T_{J′}^{J}\overline{V} (w′)] = ∑_{y′∈Y,w′(y′)=w′}[\overline{V} (w(y′))].

The set ^{S}W decomposes as a disjoint union of two subsets ^{S}W_{0} and ^{S}W_{1}, where ^{S}W_{0} consists of those elements of ^{S}W having a reduced decomposition in which s_{1} does not occur, and ^{S}W_{1} the rest. Suppose y ∈ Y ∩^{S}W_{k} for k = 0 or 1. Since the minimal-length element of the coset yW_{J} can be obtained from y by applying a sequence of reflections corresponding to simple roots in J, with the length decreasing at each stage, and since α_{1} ∉ J, it is clear that w(y) ∈ ^{S}W_{k} (for the same k). Similarly, w′(y) ∈ ^{S}W_{k}. Consequently, if w ∈ ^{S}W_{k}^{J} := ^{S}W^{J} ∩^{S}W_{k} then T_{J′}^{J}T_{J}^{J′} \overline{V} (w) has a filtration by GVMs \overline{V} (x) with x ∈ ^{S}W_{k}^{J}.

Recall the parameters a, …, d ∈ ^{S}W^{J} from the table at the beginning of this subsection, and observe that a, b ∈ ^{S}W_{0} while c, d ∈ ^{S}W_{1}. Putting

u = s_{3} … s_{n}
u′ = s_{2} … s_{n−1}
a′ = au = bu′
c′ = cu = du′,

it is straightforward to check that a′, c′ ∈ ^{S}W^{J′} with a′ ∈ ^{S}W_{0} and c′ ∈ ^{S}W_{1}. It is also easy to see that b, d, a′, c′ ∈ Y. For example, b ∈ ^{S}W^{J} and bu′ = a′ ∈ ^{S}W^{J′} with u′ ∈ W_{J′}; the others are checked similarly. We have

(8.2.3)  \begin{aligned}w(a′) & = a, & w′(a′) & = a′ \\ w(b) & = b, & w′(b) & = a′ \\ w(c′) & = c, & w′(c′) & = c′ \\ w(d) & = d, & w′(d) & = c′. \end{aligned}

It follows that [\overline{V}′(a′)] is a summand in [T_{J}^{J′} \overline{V} (B)], corresponding to y = b in (8.2.1), and that [\overline{V} (A)] is a summand in [T_{J′}^{J}\overline{V}′(a′)], corresponding to y′ = a′ in (8.2.2). Therefore

(8.2.4)  [\overline{V} (A)] is a summand in [T_{J′}^{J}T_{J}^{J′} \overline{V} (B)].

Similarly,

(8.2.5)  [\overline{V} (D)] is a summand in [T_{J′}^{J}T_{J}^{J′} \overline{V} (C)].

Finally, we claim that O(𝔤, S, J) splits into a direct sum of two subcategories (the “even” and “odd” parts), O_{e}(𝔤, S, J) and O_{o}(𝔤, S, J), with the simples A and D in the former, B and C in the latter. From the structures of the generalized Verma modules, there are no extensions between A and either B or C. Because of parity there is no extension between D and C. Suppose there is an extension of B by D. Then P (B) would have GVM factors \overline{V} (B), \overline{V} (C), each with multiplicity one, and \overline{V} (D) with multiplicity possibly greater than or equal to one. But, this cannot happen because \overline{V} (B) = B, and hence P (B) is self-dual. Therefore, there are no extensions between B and D. Let Γ_{e} be the projection of O(𝔤, S, J) onto O_{e}(𝔤, S, J), an exact functor.

Since b and c are linked by a simple reflection, and \overline{V} (B) = B is irreducible, \overline{V} (B) ⊂ \overline{V} (C). Apply the exact functor T_{J′}^{J}T_{J}^{J′} to obtain T_{J′}^{J}T_{J}^{J′} \overline{V} (B) ⊂ T_{J′}^{J}T_{J}^{J′} \overline{V} (C). By previous remarks, T_{J′}^{J}T_{J}^{J′} \overline{V} (C) has a filtration with GVM subquotients in \{\overline{V} (C), \overline{V} (D)\}, and by (8.2.5), \overline{V} (D) appears at least once as such a subquotient; similarly T_{J′}^{J}T_{J}^{J′} \overline{V} (B) has a filtration with GVM subquotients in \{\overline{V} (A), \overline{V} (B)\} and \overline{V} (A) appears at least once. Apply Γ_{e} to conclude that \overline{V} (A) ⊂ \overline{V} (D).

Since A = \overline{V} (A) ⊂ \overline{V} (D), P (A) is self-dual and [\overline{V} (D) : A] ≠ 0. Since A, B and C do not extend A and P (A) is self-dual we have a submodule M of P (A) with two composition factors D and A, with rad_{0}M = D (same argument used in 7.2.2). The simple module A does not appear as a composition factor in \overline{V} (B) or \overline{V} (C), and [\overline{V} (A) : A] = 1. This implies that P (A) has a filtration with GVM factors \overline{V} (A) and \overline{V} (D) and shows that

\overline{V} (D) : \begin{matrix}D \\ A \end{matrix}

This proves that O(BC_{n}, A_{2}, BC_{n−2}) splits into two Morita-equivalent blocks, eqch of which has finite representation type.

From Theorem 2.4.3, we can handle the (BC_{n}, BC_{n−2}, A_{2}) case. Recall that it remains only to determine the structure of \overline{V} (D). We can immediate deduce that Ext_{O_{2}}^{1}(D, X) = 0 when X = B, C, D and

Ext_{O_{2}}^{1}(D, A) ≅ Ext_{O_{1}}^{1}(A, D) ≅ ℂ.

By the argument given in the preceding paragraph, \overline{V} (D) is uniserial of length two and has composition factors D and A. It follows that this category also splits into two equivalent blocks, each of which has finite representation type.

8.2.3. r = 3. We will consider the cases (BC_{n}, A_{3}, BC_{n−3}) and (BC_{n}, BC_{n−3}, A_{3}). The category in the first case will have finite representation type and the category in the second case will be tame. Observe that this situtation is similar to the type D blocks (see Section 7.2.2) (D_{n}, A_{2}, D_{n−2}) and (D_{n}, D_{n−2}, A_{2}).

The categories for (BC_{n}, A_{3}, BC_{n−3}) and (BC_{n}, BC_{n−3}, A_{3}) have 8 simple modules labelled A, B, …, H, with corresponding Weyl group parameters denoted w_{1}, … w_{8} in the table below. The Bruhat ordering is the same as in Figure 12 (page 42).

Simple module Parameter (BC_{n}, A_{3}, BC_{n−3}) (BC_{n}, BC_{n−3}, A_{3})
A  w_{1}  e  e
B  w_{2}  s_{4} … s_{n}s_{n−1} … s_{3} s_{3} … s_{n}s_{n−1} … s_{4}
C  w_{3}  w_{2}s_{2}  s_{2}w_{2}
D  w_{4}  w_{3}w_{2}  w_{2}w_{3}
E  w_{5}  w_{3}s_{1}  s_{1}w_{3}
F  w_{6}  w_{4}s_{1}  s_{1}w_{4}
G  w_{7}  w_{6}s_{2}  s_{2}w_{6}
H  w_{8}  w_{7}w_{2}  w_{2}w_{7}

From the embeddings of (BC_{n−1}, A_{2}, BC_{n−3}) (resp. (BC_{n−1}, BC_{n−3}, A_{2})) we know the structures of the GVMs for A, B, C and D (from Section 8.2.2).

\overline{V} (A) : A  \overline{V} (B) : B  \overline{V} (C) : \begin{matrix}C \\ B \end{matrix}  \overline{V} (D) : \begin{matrix}D \\ A \end{matrix}

We can determine the structure of the GVMs for E, F, and G by using the appropriate singular U_{α}-algorithm. The following Bruhat relations are needed for (BC_{n}, A_{3}, BC_{n−3}): w_{1} < w_{1}s_{1} ∉ ^{S}W; w_{1} < w_{1}s_{2} ∉ ^{S}W; w_{2} < w_{2}s_{1} ∉ ^{S}W; w_{4} < w_{4}s_{2} ∉ ^{S}W. The corresponding inequalities with the simple reflections acting on the left (and changing each “∉ ^{S}W ” to “∉ ^{S}W^{J}”) suffice to do the computations for (BC_{n}, BC_{n−3}, A_{3}). This yields two cases.

Case 1, (BC_{n}, A_{3}, BC_{n−3}):

\overline{V} (E) : \begin{matrix}E \\ C \end{matrix}  \overline{V} (F) : \begin{matrix}F \\ D \end{matrix}  \overline{V} (G) : \begin{matrix}G \\ F \end{matrix}

Case 2, (BC_{n}, BC_{n−3}, A_{3}):

\overline{V} (E) : \begin{matrix}E \\ C \\ B \end{matrix}  \overline{V} (F) : \begin{matrix}F \\ D \\ A \end{matrix}  \overline{V} (G) : \begin{matrix}G \\ F \\ D \\ A \end{matrix}

Let O_{1} (resp. O_{2}) be the block corresponding to (BC_{n}, A_{3}, BC_{n−3}) (resp. (BC_{n}, BC_{n−3}, A_{3})). From the structures of the GVMs given above and Theorem 2.4.3, we have

Ext_{O_{1}}^{1}(H, E) ≅ Ext_{O_{2}}^{1}(A, D) ≅ ℂ;
Ext_{O_{2}}^{1}(H, E) ≅ Ext_{O_{1}}^{1}(A, D) ≅ ℂ;
Ext_{O_{1}}^{1}(H, F) ≅ Ext_{O_{2}}^{1}(A, C) = 0;
Ext_{O_{2}}^{1}(H, F) ≅ Ext_{O_{1}}^{1}(A, C) = 0.

Finally, we need to describe the structure of \overline{V} (H).

Case 1, (BC_{n}, A_{3}, BC_{n−3}): By parity, Ext_{O_{1}}^{1}(H, X) = 0 for X = C, D, G. Moreover, from above Ext_{O_{1}}^{1}(H, F) = 0. If Ext_{O_{1}}^{1}(H, A) ≠ 0 then by the self-duality of P (A), \overline{V} (H) is uniserial of length 2 with composition factors H and A. Again this is not possible because of the reciprocity law and the self-duality of P (A). Hence, Ext_{O_{1}}^{1}(H, A) = 0. Also, Ext_{O_{1}}^{1}(H, B) = 0 by a similar argument. Since Ext_{O_{1}}^{1}(H, E) ≠ 0, this shows that H belongs to the indecomposable component containing B, C, and E.

Now P (B) and P (C) are self-dual projective modules. By the same type of arguments given in Section 7.2.2, we can conclude that [\overline{V} (H) : X] = 0 for X = B, C. From this and the fact that Ext_{O_{1}}^{1}(H, E) ≅ ℂ we see that

\overline{V} (H) : \begin{matrix}H \\ E \end{matrix}

This proves that O(BC_{n}, A_{3}, BC_{n−3}) splits into two Morita-equivalent blocks, each of finite representation type. Case 2, (BC_{n}, BC_{n−3}, A_{3}): We begin by investigating the question of extension of simples with H. By parity Ext_{O_{2}}^{1}(H, X) = 0 for X = C, D, G. Moreover, Ext_{O_{2}}^{1}(H, X) = 0 for X = A, B, otherwise this would contradict the self-duality of P (A) and P (B). From above we have Ext_{O_{2}}^{1}(H, F) = 0 and Ext_{O_{2}}^{1}(H, E) ≅ ℂ.

Finally, we will analyze the possible socles of \overline{V} (H). If E is in the socle of \overline{V} (H) then this would contradict the self-duality of P (E). If B or C is in the socle of \overline{V} (D) then by using the same argument given in Case 2 of Section 7.2.2, one sees that \overline{V} (H) is of the form (respectively)

(8.2.6)  \overline{V} (H) : \begin{matrix}H \\ E \\ C \\ B \end{matrix}  or  \begin{matrix}H \\ E \\ C \end{matrix}

This shows that O(BC_{n}, BC_{n−3}, A_{3}) splits into two blocks of tame representation type (see Section 7.2.2, Case 2).

Remark. Examples strongly suggest that in the (BC_{n}, BC_{n−3}, A_{3}) case, \overline{V} (D) always has the first structure depicted in (8.2.6) . This would imply that the two blocks are Morita equivalent. However, we have been unable to prove in general that [\overline{V} (H) : B] ≠ 0.

8.2.4. r ≥ 4. When r = 4, these blocks have wild representation type because the posets contain diamonds, as listed in the table below. It follows from Corollary 2.11 that the blocks are wild for r ≥ 4. Put u = s_{4} … s_{n−1}s_{2} … s_{n−2}s_{n} … s_{4}s_{n} … s_{5}.

Element of ^{S}W^{J} (BC_{n}, BC_{n−4}, A_{4}) (BC_{n}, A_{4}, BC_{n−4})
y  u  u^{−1}
x_{1}  s_{1}u  u^{−1}s_{1}
x_{2}  s_{3}u  u^{−1}s_{3}
w  s_{1}s_{3}u  u^{−1}s_{3}s_{1}

8.3. Three or more components. We shall show that all algebras arising with three or more components are wild. By Corollary 2.11 it suffices to show this for exactly three components.

8.3.1. J–S–J. We first consider the case when O(𝔤, S, J) corresponds to (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{s}, A_{1} × BC_{t}). From the two component analysis for type BC (Section 8.2.4) and Corollary 2.11, the block is wild for s ≥ 4. Therefore, we may assume that s = 1, 2, 3. Furthermore, from our type A three component analysis (Section 6.4.1) we may assume that t = 1, 2, 3, 4. There are 12 cases to check and one can verify that each of these algebras is wild. When 1 < s ≤ 3 and 1 < t ≤ 4 or when s = 3 and t = 1 the poset contains a diamond. In all the other cases except when s = t = 1 the poset has a subposet as in Figure 1, hence these blocks are wild by Proposition 2.8. For the case when s = t = 1 (i.e. (BC_{3}, A_{1}, A_{1} × A_{1})), there are five simple modules parametrized by (say) v, w, x, y, z. By using reciprocity (2.1.4) one can show that three of the projectives have quotients of the form:

P (z) : \begin{matrix}L(z) \\ L(w) \\ L(y) \end{matrix}  P (v) : \begin{matrix}L(v) \\ L(w) \\ L(y) L(v) L(x) \end{matrix}  P (y) : \begin{matrix}L(y) \\ L(w) \\ L(v) L(y) L(z) \end{matrix}

Set A = End_{B}(P) where P is the direct sum of all projective indecomposable modules in B := O(𝔤, S, J). Consider the idempotent given by e = 1_{y} + 1_{z} + 1_{v}: the algebra eAe has subquiver given in Figure 3. Therefore, eAe is wild, thus A is wild.

Now by Corollary 2.11, we have that O(𝔤, S, J) has wild representation type for (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{s}, A_{r} × BC_{t}).

8.3.2. S–J–S. Now consider the block O(𝔤, S, J) where (Φ, Φ_{S}, Φ_{J}) = (BC_{n}, A_{1} × BC_{t}, A_{s}). According to Corollary 2.11, we can use our previous results for the two component case for type BC (Section 8.2.4) to conclude that for s ≥ 4, this block has wild representation type. When s = 3 and t ≥ 1, or s = 2 and t ≥ 2, the poset contains a diamond, as listed in the following table. So in these cases the block is wild by Proposition 2.7. Set p = s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{6} … s_{n}s_{n−1} … s_{5}, q = s_{3}s_{2}s_{1}s_{4}s_{3}s_{5} … s_{n}s_{n−1} … s_{4}.

Element of ^{S}W^{J} (BC_{n}, A_{1} × BC_{t}, A_{3}) (BC_{n}, A_{1} × BC_{t}, A_{2}), t ≥ 2
y  s_{2}p  q
x_{1}  s_{3}s_{2}p  s_{2}q
x_{2}  s_{2}s_{3}p  qs_{5}
w  s_{2}s_{3}s_{2}p  s_{2}qs_{5}

For s = 1 and t ≥ 2, or s = 2 and t = 1, the poset contains a subposet as in Figure 1 (page 19), given in the table below, so the category is wild by Proposition 2.8. Set u = s_{2}s_{3} … s_{n}s_{n−1} … s_{4}.

Element of ^{S}W^{J} (BC_{n}, A_{1} × BC_{t}, A_{1}), t ≥ 2 (BC_{4}, A_{1} × A_{1}, A_{2})
w  us_{1}s_{2}s_{3}  s_{3}s_{2}s_{4}s_{3}s_{4}s_{2}s_{1}
x  us_{1}s_{3}  s_{3}s_{2}s_{4}s_{3}s_{4}s_{1}
y  us_{1}  s_{3}s_{2}s_{3}s_{4}s_{1}
v  us_{3}  s_{3}s_{2}s_{4}s_{3}s_{4}
z  us_{4}s_{1}  s_{2}s_{3}s_{4}s_{1}

Finally for s = t = 1, one can verify that the block (BC_{3}, A_{1} × A_{1}, A_{1}) is Morita equivalent to the block for (BC_{3}, A_{1}, A_{1} × A_{1}). The latter is wild from Section 8.3.1. Consequently, the block corresponding to (BC_{3}, A_{1} × A_{1}, A_{1}) is wild.

It now follows by Corollary 2.11 that the categories O(BC_{n}, A_{r} × BC_{t}, A_{s}) are wild.

This completes the verification of Theorem 1.6(B)(b/c).

9. Mixed case: Δ = S ∪ J, exceptional algebras

9.1. E_{6} base cases. Let Φ be of type E_{6}. The following blocks of O_{S}^{μ} have wild representation type, due to the presence of the indicated diamonds in their posets ^{S}W^{J}. Set u = s_{1}s_{3}s_{4}s_{6}s_{5}, v = s_{4}s_{2}s_{3}s_{4}s_{5}, z = s_{2}s_{4}s_{3}s_{5}s_{4}.

Element of ^{S}W^{J} (E_{6}, D_{4}, A_{1} × A_{1}) (E_{6}, A_{1} × A_{1}, D_{4}) (E_{6}, A_{2}, A_{4}) (E_{6}, A_{4}, A_{2})
y  us_{4}s_{3}  s_{3}s_{4}u^{−1}  v  v^{−1}
x_{1}  us_{2}s_{4}s_{3}  s_{3}s_{4}s_{2}u^{−1}  s_{1}v  v^{−1}s_{1}
x_{2}  us_{4}s_{2}s_{3}  s_{3}s_{2}s_{4}u^{−1}  vs_{6}  s_{6}v^{−1}
w  us_{2}s_{4}s_{2}s_{3}  s_{3}s_{2}s_{4}s_{2}u^{−1}  s_{1}vs_{6}  s_{6}v^{−1}s_{1}
Element of ^{S}W^{J} (E_{6}, A_{5}, A_{1}) (E_{6}, A_{1}, A_{5})
y  z  z^{−1}
x_{1}  zs_{1}  s_{1}z^{−1}
x_{2}  zs_{6}  s_{6}z^{−1}
w  zs_{1}s_{6}  s_{6}s_{1}z^{−1}

Let (Φ, Φ_{S}, Φ_{J}) = (E_{6}, A_{1}, D_{5}) with S = {α_{6}}. In this case there are six simple modules and six generalized Verma modules. Set w = s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}. The parameters for ^{S}W^{J} are given by e, w, s_{1}w, s_{3}s_{1}w, s_{4}s_{3}s_{1}w, and s_{2}s_{4}s_{3}s_{1}w. Our (computer) calculations show that the generalized Verma modules are uniserial, multiplicity-free and have consecutive radical lengths from 1 to 6. This shows that this block is Morita equivalent to the ordinary Category O block (A_{5}, ∅, A_{4}) which has wild representation type.

9.2. E_{7} and E_{8} base cases. Let Φ be of type E_{7}. The following blocks of O_{S}^{μ} have wild representation type, because their posets contain the indicated diamonds. Set t = s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{6}, u = s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}, v = s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3} and z = s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}.

Element of ^{S}W^{J} (E_{7}, D_{6}, A_{1}) (E_{7}, D_{5}, A_{2}) (E_{7}, D_{5}, A_{1} × A_{1}) (E_{7}, A_{1}, E_{6})
y  t  u  vs_{6}  z
x_{1}  ts_{5}  us_{1}  vs_{6}s_{5}  s_{2}z
x_{2}  ts_{7}  s_{7}u  vs_{7}s_{6}  s_{5}z
w  ts_{5}s_{7}  s_{7}us_{1}  vs_{7}s_{6}s_{5}  s_{2}s_{5}z

For Φ of type E_{8}, the following algebras are wild because their posets contain diamonds. Set p = s_{8}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{7}.

Element of ^{S}W^{J} (E_{8}, E_{7}, A_{1}) (E_{8}, A_{1}, E_{7})
y  p  p^{−1}
x_{1}  ps_{2}  s_{2}p^{−1}
x_{2}  ps_{5}  s_{5}p^{−1}
w  ps_{2}s_{5}  s_{5}s_{2}p^{−1}

9.3. F_{4} base cases. Let Φ be of type F_{4}. First consider the blocks O_{S}^{μ} where (Φ, Φ_{S}, Φ_{J}) = (F_{4}, B_{3}, A_{1}), (F_{4}, A_{1}, B_{3}). By analyzing the structures of the generalized Verma modules we can conclude that there is a subquiver of the Ext^{1}-quiver of the form given in Figure 5 (page 28). Therefore, these blocks are wild. We remark that this also shows that the blocks O_{S}^{μ} where (Φ, Φ_{S}, Φ_{J}) = (F_{4}, C_{3}, A_{1}), (F_{4}, A_{1}, C_{3}) have wild representation type. We list the parameters for the simple modules in the table below. Set p = s_{4}s_{3}s_{2}s_{3} and q = s_{1}s_{3}s_{2}s_{3}s_{4}.

Simple module (F_{4}, B_{3}, A_{1}) (F_{4}, A_{1}, B_{3})
γ  L(ps_{1}s_{2}s_{4}s_{3}) L(s_{3}s_{2}s_{4}q)
α  L(ps_{1}s_{2}s_{3}) L(s_{2}s_{3}s_{2}s_{4}q)
β  L(ps_{1}s_{2})  L(s_{3}s_{2}q)
σ  L(ps_{1})  L(s_{2}q)
δ  L(p)  L(q)

We next consider the blocks O_{S}^{μ} where (Φ, Φ_{S}, Φ_{J}) = (F_{4}, B_{2}, A_{1} ×A_{1}), (F_{4}, A_{1} ×A_{1}, B_{2}), and (F_{4}, A_{2}, A_{2}) (S = {α_{3}, α_{4}} and J = {α_{1}, α_{2}}). The algebras have wild representation type because of the presence of a diamond. The table below gives the specific elements. Set u = s_{1}s_{2}s_{3}s_{4}, v = s_{2}s_{3}.

Element of ^{S}W^{J} (F_{4}, B_{2}, A_{1} × A_{1}) (F_{4}, A_{1} × A_{1}, B_{2}) (F_{4}, A_{2}, A_{2})
y  us_{2}s_{3}s_{2}s_{3}  s_{3}s_{2}s_{3}s_{2}u^{−1}  v
x_{1}  us_{2}s_{1}s_{3}s_{2}s_{3}  s_{3}s_{2}s_{3}s_{1}s_{2}u^{−1}  s_{1}v
x_{2}  us_{3}s_{2}s_{1}s_{3}s_{2}  s_{2}s_{3}s_{1}s_{2}s_{3}u^{−1}  vs_{4}
w  us_{3}s_{2}s_{1}s_{3}s_{2}s_{3}  s_{3}s_{2}s_{3}s_{1}s_{2}s_{3}u^{−1} s_{1}vs_{4}

9.4. We can now verify the statements in Theorem 1.6(B)(e–g). We will proceed in a case-by-case manner.

Type E_{6}: From our analysis for D_{n}, O(D_{n}, Φ_{S′}, Φ_{J′}) has wild representation type if the total number of components of S′ and J′ is at least three. In particular this holds for D_{5} (where the converse of the statement is actually true, although we shall not need this). By analyzing the Dynkin diagram for E_{6}, we can conclude immediately that O_{S}^{μ} has wild representation type for (Φ, Φ_{S}, Φ_{J}) except possibly for the following cases: (E_{6}, E_{6}, ∅), (E_{6}, ∅, E_{6}), (E_{6}, D_{5}, A_{1}), (E_{6}, A_{1}, D_{5}), (E_{6}, A_{4}, A_{2}), (E_{6}, A_{2}, A_{4}), (E_{6}, A_{1} × A_{1}, D_{4}), (E_{6}, D_{4}, A_{1} × A_{1}) (E_{6}, A_{5}, A_{1}), and (E_{6}, A_{1}, A_{5}).

The first two cases are semisimple and the last seven cases are wild by Section 9.1. The remaining case (E_{6}, D_{5}, A_{1}) has finite representation type by Corollary 1.5(b).

Type E_{7}: We can rely on our analysis of E_{6}. Non-wild blocks can only possibly occur when the restriction of S and J to the E_{6} part of the Dynkin diagram is (E_{6}, E_{6}, ∅), (E_{6}, ∅, E_{6}), or (E_{6}, D_{5}, A_{1}). Consequently we are reduced to the cases when (Φ, Φ_{S}, Φ_{J}) is (E_{7}, E_{7}, ∅), (E_{7}, ∅, E_{7}), (E_{7}, E_{6}, A_{1}), (E_{7}, A_{1}, E_{6}), (E_{7}, D_{6}, A_{1}), (E_{7}, D_{5}, A_{1} × A_{1}), (E_{7}, D_{5}, A_{2}), or (E_{7}, D_{5} × A_{1}, A_{1}). The first two cases are semisimple. The third case is wild by Corollary 1.5(c). The next four cases are wild by Section 9.2. Finally, the last case is wild because the block for (D_{6}, D_{4} × A_{1}, A_{1}) is wild.

Type E_{8}: From our analysis of E_{7}, it suffices to only consider the possibilities when the restriction of S and J to E_{7} is (E_{7}, E_{7}, ∅) or (E_{7}, ∅, E_{7}). Therefore, the possible non-wild blocks are (E_{8}, E_{8}, ∅), (E_{8}, ∅, E_{8}), (E_{8}, E_{7}, A_{1}) and (E_{8}, A_{1}, E_{7}). Again the first two cases are semisimple and the two remaining cases are wild by Section 9.2.

Type F_{4}: We can rely on our analysis of B_{3} and C_{3}. By restricting S and J to these algebras the possible non-wild blocks are (F_{4}, F_{4}, ∅), (F_{4}, ∅, F_{4}), (F_{4}, BC_{3}, A_{1}), (F_{4}, A_{1}, BC_{3}), (F_{4}, B_{2}, A_{1} × A_{1}), (F_{4}, A_{1} × A_{1}, B_{2}), and (F_{4}, A_{2}, A_{2}). The first two cases are semisimple and the other five are wild by Section 9.3.

Type G_{2}: The cases one has to consider are all semisimple. These are (G_{2}, G_{2}, ∅), (G_{2}, ∅, G_{2}) and (G_{2}, A_{1}, A_{1}). In the last case the algebra O_{S}^{μ} has three semisimple blocks.

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54 BRIAN D. BOE AND DANIEL K. NAKANO

Department of Mathematics, University of Georgia, Athens, Georgia 30602

E-mail address: brian@math.uga.edu
E-mail address: nakano@math.uga.edu