Systems of Differential Operators and Generalized Verma Modules

In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and de...

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Дата:	2014
Автор:	Kubo, T.
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Опубліковано:	Інститут математики НАН України 2014
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Systems of Differential Operators and Generalized Verma Modules / T. Kubo // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ.

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spelling	irk-123456789-1468482019-02-12T01:24:17Z Systems of Differential Operators and Generalized Verma Modules Kubo, T. In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems. 2014 Article Systems of Differential Operators and Generalized Verma Modules / T. Kubo // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E46; 17B10; 22E47 DOI:10.3842/SIGMA.2014.08 http://dspace.nbuv.gov.ua/handle/123456789/146848 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.
format	Article
author	Kubo, T.
spellingShingle	Kubo, T. Systems of Differential Operators and Generalized Verma Modules Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kubo, T.
author_sort	Kubo, T.
title	Systems of Differential Operators and Generalized Verma Modules
title_short	Systems of Differential Operators and Generalized Verma Modules
title_full	Systems of Differential Operators and Generalized Verma Modules
title_fullStr	Systems of Differential Operators and Generalized Verma Modules
title_full_unstemmed	Systems of Differential Operators and Generalized Verma Modules
title_sort	systems of differential operators and generalized verma modules
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146848
citation_txt	Systems of Differential Operators and Generalized Verma Modules / T. Kubo // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kubot systemsofdifferentialoperatorsandgeneralizedvermamodules
first_indexed	2025-07-11T00:46:13Z
last_indexed	2025-07-11T00:46:13Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 008, 35 pages Systems of Differential Operators and Generalized Verma Modules Toshihisa KUBO Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan E-mail: toskubo@ms.u-tokyo.ac.jp URL: http://www.ms.u-tokyo.ac.jp/~toskubo/ Received April 08, 2013, in final form January 17, 2014; Published online January 24, 2014 http://dx.doi.org/10.3842/SIGMA.2014.008 Abstract. In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems. Key words: conformally invariant systems; quasi-invariant differential operators; intertwi- ning differential operators; real flag manifolds; generalized Verma modules; standard maps 2010 Mathematics Subject Classification: 22E46; 17B10; 22E47 1 Introduction Conformally invariant systems are systems of differential operators that are equivariant under an action of a Lie algebra. More precisely, let V → M be a vector bundle over a smooth manifold M and g0 a Lie algebra of first-order differential operators acting on smooth sections on V. A linearly independent list D1, . . . , Dm of differential operators on V is then said to be conformally invariant if, for each X ∈ g0, the condition [X,Di] ∈ n∑ j=1 C∞(M)Dj holds for all 1 ≤ i ≤ n, where [X,Di] = XDi−DiX. (For the precise definition see, for example, Section 2 of [2].) In this article we shall take M = N̄0Q0/Q0, an open dense submanifold of a certain class of flag manifolds G0/Q0, and take vector bundle V→M to be the restriction of a homogeneous line bundle Ls → G0/Q0 to N̄0Q0/Q0. Many examples of conformally invariant systems implicitly or explicitly exist in the literature, especially in the case of m = 1. The Laplacian ∆ on Rn and wave operator � on the Minkowski space R3,1 are, for instance, two outstanding examples. The Yamabe operator on Sn is another example in conformal geometry. (See, for instance, the series of works [26, 27], and [28] of Kobayashi–Ørsted.) In the case of m ≥ 2, an example may find in a work of Davidson–Enright– Stanke [8]. It would be also interesting to point out that such systems of operators are implicitly presented in the work [40] of Wallach and also related to a project of Dobrev on constructing intertwining differential operators of arbitrary order (see, for example, [10, 11], and [9]). The theory of conformally invariant systems consisting of one differential operator started with a work of Kostant [32]. He called such differential operators quasi-invariant. The notion of conformally invariant systems generalizes that of quasi-invariant differential operators, and it mailto:toskubo@ms.u-tokyo.ac.jp http://www.ms.u-tokyo.ac.jp/~toskubo/ http://dx.doi.org/10.3842/SIGMA.2014.008 2 T. Kubo agrees with the notion of conformal invariance introduced by Ehrenpreis [12]. As a generaliza- tion of quasi-invariant differential operators conformally invariant systems are also related with a work of Huang [14] for intertwining differential operators. As described above the theory of conformally invariant systems can be viewed as a geometric- analytic theory. Further, it is also closely related to the explicit construction of homomorphisms between generalized Verma modules. Homomorphisms between generalized Verma modules (equivalently, intertwining differential operators between degenerate principal series representa- tions) have received a lot of attentions from many points of view. See, for instance, [3, 4, 5, 15, 16, 17, 29, 30, 31, 36, 37, 38, 39], and the references therein. We wish to note that the list of the articles is far from complete. We hope that the reader will understand that it is impossible to attempt to give an exhaustive list, as there are numerous relevant contributions in the literature. In [32], Kostant showed that quasi-invariant differential operators explicitly yield homomor- phisms between appropriate generalized Verma modules. Conformally invariant systems also yield concrete homomorphisms between appropriate generalized Verma modules [2]. We shall describe our work for this direction more carefully later in this introduction. We now turn to a project on conformally invariant systems. A project for conformally invariant systems started with the work of [1] and [2]. Other progress on the project are reported in [18, 19, 20, 21, 22] and [33, 34, 35]. To see a recent development of the theory of conformally invariant systems the introduction of the work [22] of Kable is very helpful. Besides the recent development one can also find relationships between quasi-invariant differential operators and the classic work of Maxwell on harmonic polynomials in the introduction. Descriptions on Heisenberg Laplacian and Heisenberg ultrahyperbolic equation may also deserve one’s attention. (In the particular direction the introduction of [20] has more details.) The present work is also part of the project. The aim of this paper is to close the cases that were left open in [35] and [34]. To describe our work of this paper more precisely, we now briefly review the works in the papers. Let G be a complex, simple, connected, simply-connected Lie group with Lie algebra g. Give a Z-grading g = ⊕r j=−r g(j) on g so that q = g(0)⊕ ⊕ j>0 g(j) = l ⊕ n is a maximal parabolic subalgebra. Let Q = NG(q) = LN . For a real form g0 of g, define G0 to be an analytic subgroup of G with Lie algebra g0. Set Q0 = NG0(q). We consider a line bundle Ls → G0/Q0 for s ∈ C. As the homogeneous space G0/Q0 admits an open dense submanifold N̄0Q0/Q0, we restrict our bundle to this submanifold. By slight abuse of notation we refer to the restricted bundle as Ls. The systems that we shall construct act on smooth sections of the restricted bundle Ls → N̄0. Our systems of operators are constructed from L-irreducible constituents W of g(−r+k)⊗g(r) for 1 ≤ k ≤ 2r. We call the systems of operators Ωk systems. (We shall describe the construction more precisely in Section 2.2.) An Ωk system is a system of kth-order differential operators. There is no reason to expect that Ωk systems are conformally invariant on Ls for arbitrary s ∈ C; the conformal invariance of Ωk systems depends on the complex parameter s for the line bundle Ls. We then say that an Ωk system has special value sk if the system is conformally invariant on the line bundle Lsk . In [35], the parabolic subalgebra q = l ⊕ n was taken to be a maximal parabolic subalgebra of quasi-Heisenberg type, that is, a maximal parabolic subalgebra with nilpotent radical n with conditions that [n, [n, n]] = 0 and dim([n, n]) > 1. In this setting the Ωk systems for k ≥ 5 are zero. We then sought the special values for the Ω1 system and Ω2 systems. While the special value s1 for the Ω1 system was determined for each parabolic subalgebra q as s1 = 0, we left three cases open for Ω2 systems. To describe the open cases, let us now start explaining briefly the classification for the irreducible constituents W that contribute to Ω2 systems. In [35], we first observed that if irreducible constituents W contribute to Ω2 systems then their highest weights are of the form µ+ε, where µ is the highest weight for g(1) and ε is some weight for g(1). We called such irreducible constituents special and classified as Type 1a, Type 1b, Type 2, and Systems of Differential Operators and Generalized Verma Modules 3 Type 3 with respect to certain technical conditions for the highest weight µ + ε. (The precise conditions will be given in Definition 2.6.) Table 1 summarizes the types of special constituents. Here, for example, “Bn(i)” indicates the maximal standard parabolic subalgebra of g of type Bn, which is determined by the ith simple root αi. In the table V (µ+εγ), V (µ+εnγ), and V (µ+ε±nγ) denote the special constituents with highest weights µ + εγ , µ + εnγ , and µ + ε±nγ , respectively. (We shall precisely describe these highest weights in Section 2.4 and Section 3). As illustrated in Table 1, there are one, two, or three special constituents. A dash in the column for V (µ+εnγ) indicates that there is no special constituent V (µ+ εnγ) in the case. In [35], under the assumption that q is not of type Dn(n− 2), we found the special values s2 for Ω2 systems for the Type 1a and Type 2 constituents. The technique that we used allowed us to handle each case uniformly. However, since the technique relied on some technical conditions on the highest weights, we could not apply it to the systems coming from Type 1b and Type 3 constituents. Table 1. Types of special constituents. Parabolic subalgebra V (µ+ εγ) V (µ+ εnγ) Bn(i), 3 ≤ i ≤ n− 2 Type 1a Type 1a Bn(n− 1) Type 1a Type 1b Bn(n) Type 2 − Cn(i), 2 ≤ i ≤ n− 1 Type 3 Type 2 Dn(i), 3 ≤ i ≤ n− 3 Type 1a Type 1a E6(3) Type 1a Type 1a E6(5) Type 1a Type 1a E7(2) Type 1a − E7(6) Type 1a Type 1a E8(1) Type 1a − F4(4) Type 2 − Parabolic subalgebra V (µ+ εγ) V (µ+ ε+nγ) V (µ+ ε−nγ) Dn(n− 2) Type 1a Type 1a Type 1a If V (µ+ε) := V (µ+εγ), V (µ+εnγ), or V (µ+ε±nγ) then the missing cases may be summarized as follows: 1) the maximal parabolic subalgebra q is of type Dn(n− 2), 2) the special constituent V (µ+ ε) is of Type 1b, and 3) the special constituent V (µ+ ε) is of Type 3. These are the cases boxed in Table 1. Our goal in this article is to find the special values s2 of Ω2 systems in these three cases. In contrast to [35], in which each case was treated as uniformly as possible, we handle the three cases individually in this paper. For the case that q is of type Dn(n − 2), we first observe that each special constituent is of Type 1a. We then directly apply the technique used in [35]. For Type 1b and Type 3 cases, we use an explicit realization of a Lie algebra g. In this way certain computations can be carried out easily. The special values s2 for the type Dn(n− 2), Type 1b, and Type 3 cases are s2 = 1 (Corol- lary 3.2), s2 = 1 (Theorem 4.2), and s2 = n− i+1 (Theorem 5.8), respectively. Now, with these results together with ones in [35], if ∆ and ∆(g(1)) denote a (fixed) root system and the set of roots contributing to g(1), respectively, then we obtain the following beautiful consequence. 4 T. Kubo Consequence 1.1. Let q be a maximal parabolic subalgebra of quasi-Heisenberg type. The special value s2 of the Ω2 system associated to the special constituent V (µ+ ε) is s2 =  \|∆µ+ε(g(1))\| 2 − 1 if V (µ+ ε) is of Type 1, −1 if V (µ+ ε) is of Type 2, n− i+ 1 if V (µ+ ε) is of Type 3, where \|∆µ+ε(g(1))\| is the number of elements of ∆µ+ε(g(1)) := {α ∈ ∆(g(1)) \| (µ+ ε)−α ∈ ∆}. Here we combine Type 1b with Type 1a. This is because it turned out that the special value for Type 1b case can be given by the same formula as for Type 1a case (see Remark 4.3). If Ω2\|V (µ+ε)∗ denotes the Ω2 system coming from the special constituent V (µ + ε) then Table 2 exhibits the special values for all the Ω2 systems under consideration. Table 2. Line bundles with special values. Parabolic subalgebra Ω2\|V (µ+εγ)∗ Ω2\|V (µ+εnγ)∗ Bn(i), 3 ≤ i ≤ n− 2 L ( (n− i− 1 2)λi ) L(λi) Bn(n− 1) L ( 1 2λn−1 ) L(λn−1) Bn(n) L(−λn) − Cn(i), 2 ≤ i ≤ n− 1 L ( (n− i+ 1)λi ) L(−λi) Dn(i), 3 ≤ i ≤ n− 3 L ( (n− i− 1)λi ) L(λi) E6(3) L(λ3) L(2λ3) E6(5) L(λ5) L(2λ5) E7(2) L(2λ2) − E7(6) L(λ6) L(3λ6) E8(1) L(3λ1) − F4(4) L(−λ4) − Parabolic subalgebra Ω2\|V (µ+εγ)∗ Ω2\|V (µ+ε+nγ)∗ Ω2\|V (µ+ε−nγ)∗ Dn(n− 2) L(λn−2) L(λn−2) L(λn−2) Here λi denotes the fundamental weight for the simple root αi and L(sλi) := Ls, the restricted line bundle over N̄0. Now we turn to [34]. To describe the work of the paper, we first recall that Kostant showed in [32] that a quasi-invariant differential operator gives a homomorphism between suitable scalar generalized Verma modules with explicit image of a highest weight vector. As a full generalization of quasi-invariant differential operators, it is then shown in [2] that a conformally invariant system also explicitly yields a homomorphism between appropriate generalized Verma modules. Here the generalized Verma modules are not necessarily of scalar-type. A homomorphism between generalized Verma modules is called standard if it comes from a homomorphism between corresponding (full) Verma modules, and called non-standard other- wise [36]. While standard homomorphisms are well-understood [4, 36], the classification of non-standard homomorphisms is still an open problem. In [34], we classified the standardness of the homomorphisms ϕΩk between generalized Verma modules arising from the conformally invariant Ω1 and Ω2 systems. While the map ϕΩ1 was shown to be standard for each parabolic subalgebra q, the classification for the map ϕΩ2 was incomplete. This is because of the lack of the special values of the three cases mentioned above (the boxed cases in Table 1). Thus, in this paper, we also determine whether or not the maps ϕΩ2 coming from the conformally invariant Ω2 systems in the three cases are standard. Systems of Differential Operators and Generalized Verma Modules 5 The classification results are given in Theorem 3.3 (type Dn(n−2) case), Theorem 4.4 (Type 1b case), and Theorem 5.9 (Type 3 case). It turned out that in each case the map ϕΩ2 is non- standard. With the results from [34], Table 3 summarizes the classification of the standardness of the maps ϕΩ2 . Table 3. The classification of ϕΩ2 . Parabolic subalgebra Ω2\|V (µ+εγ)∗ Ω2\|V (µ+εnγ)∗ Bn(i), 3 ≤ i ≤ n− 2 standard non-standard Bn(n− 1) standard non-standard Bn(n) standard − Cn(i), 2 ≤ i ≤ n− 1 non-standard standard Dn(i), 3 ≤ i ≤ n− 3 non-standard non-standard E6(3) non-standard non-standard E6(5) non-standard non-standard E7(2) non-standard − E7(6) non-standard non-standard E8(1) non-standard − F4(4) standard − Parabolic subalgebra Ω2\|V (µ+εγ)∗ Ω2\|V (µ+ε+nγ)∗ Ω2\|V (µ+ε−nγ)∗ Dn(n− 2) non-standard non-standard non-standard Recall that, for each parabolic subalgebra q, the special value s1 for the Ω1 system is s1 = 0 and that the map ϕΩ1 is standard. Now, with the results and ones in Tables 1 and 3, we obtain the following interesting consequence. Consequence 1.2. Let q be a maximal parabolic subalgebra of quasi-Heisenberg type. The map ϕΩk for k = 1, 2 is non-standard if and only if the special value sk of the Ωk system is a positive integer. Here we wish to note that there is another interesting relationship between the special values of Ωk systems and the associated scalar generalized Verma modules. In [1], when the nilpotent radical n of parabolic subalgebra q = l⊕n is a Heisenberg algebra, it was shown that all the first reducibility points of the scalar generalized Verma modules associated with Ωk systems were accounted by the special values of the systems. In this case, as of quasi-Heisenberg case, the Ωk systems for k ≥ 5 are zero. To obtain the account the special values of Ω3 systems and Ω4 systems as well as these of the Ω1 system and Ω2 systems were used. In the quasi-Heisenberg case the results in [35] and in this paper for the Ω1 system and Ω2 systems do not give an account for the first reducibility points. The special values of Ω3 systems and Ω4 systems thus seem to require also in the quasi-Heisenberg case. We will report the spacial values of the systems elsewhere. Before closing this introduction let us make two remarks on this paper, although we under- stand that this introduction is already long enough. The first is on the technique to determine the special values of Ω2 systems. In [35], to determine the special values, we used some reduction techniques. Although the techniques significantly reduced the amount of computations, as sev- eral technical formulas on differential operators were used, the computations were still somewhat not straightforward. Now it is known that special values can be obtained by computations in generalized Verma modules (see, for instance, [20]). Then, in this paper, by combining the idea used in [35] with that for generalized Verma modules, we are successful to simplify computations further. The technique is given in Proposition 2.7. 6 T. Kubo The second remark is on the definition of special constituents. As stated in the introduction in [34], there were certain discrepancy on the terminology “special constituents” between [35] and [34]. In [35] (and in an earlier paragraph of this introduction) we defined the special constituents for Ω2 systems as ones whose highest weights satisfy certain conditions. On the other hand, in [34], we redefined such constituents for any Ωk systems as irreducible constituents that contribute to Ωk systems. The reason why we redefined special constituents as in [34] is that the later definition works not only for Ω2 systems but also for any Ωk systems, and also that the irreducible constituents with highest weights satisfying the technical conditions are highly expected to contribute to Ω2 systems. Indeed, in [35], the implication was verified except the three missing cases. In this paper we showed that the implication does hold also in the three cases. The results are described in Section 3 for the type Dn(n − 2) case, and stated in Propositions 4.1 and 5.6 for the Type 1b and Type 3 cases, respectively. Now the two notions of special constituents do agree for Ω2 systems. We now outline the rest of this paper. This paper consists of five sections (with this intro- duction) and one appendix. In Section 2 we review the works [35] and [34]. In particular we give the precise construction of Ωk systems. We also review about maximal parabolic subalgebras q of quasi-Heiseberg type in this section. In Section 3, when q is of type Dn(n − 2), we find the special values of the Ω2 systems and determine the standardness of the map ϕΩ2 . The special values are given in Corollary 3.2 together with Theorem 3.1. The standardness of ϕΩ2 is deter- mined in Theorem 3.3. Sections 4 and 5 are devoted to the Ω2 systems arising from Type 1b special constituent and Type 3 special constituent, respectively. The special values and the standardness of ϕΩ2 are determined in Theorems 4.2 and 4.4 for Type 1b case and Theorems 5.8 and 5.9 for Type 3 case, respectively. Finally, in Appendix A, we collect the miscellaneous data that will be helpful for the work of this paper. 2 Preliminaries The purpose of this section is to summarize the framework established in [35] and [34]. The notation and conventions remain in force in the rest of this paper. 2.1 A specialization of a vector bundle V→M We start with recalling from [35] the smooth manifold M and the vector bundle V→M to work with. This is nothing but the non-compact picture of a degenerate principal series representation. Let G be a complex, simple, connected, simply-connected Lie group with Lie algebra g. Fix a maximal connected solvable subgroup B, and write b = h ⊕ u for its Lie algebra with h the Cartan subalgebra and u the nilpotent subalgebra. Let q ⊃ b be a standard parabolic subalgebra of g. If Q = NG(q) then write Q = LN for the Levi decomposition of Q. Let g0 be a real form of g in which the complex parabolic subalgebra q has a real form q0. Let G0 be the analytic subgroup of G with Lie algebra g0. Define Q0 = NG0(q) ⊂ Q, and write Q0 = L0N0. We will work with G0/Q0 for a class of maximal parabolic subgroup Q0 whose complexified Lie algebra q is a maximal parabolic subalgebra of g of quasi-Heisenberg type (see Section 2.4). Let ∆ = ∆(g, h) denote the set of roots of g with respect to h. Write ∆+ for the positive system attached to b and denote by Π the set of simple roots. We write gα for the root space for α ∈ ∆. For each subset S ⊂ Π, let qS be the corresponding standard parabolic subalgebra. Write qS = lS⊕nS with Levi factor lS = h⊕ ⊕ α∈∆S gα and nilpotent radical nS = ⊕ α∈∆+\∆S gα, where ∆S = {α ∈ ∆ \|α ∈ span(Π\S)}. If q is a maximal parabolic subalgebra of g then there exists a unique simple root αq ∈ Π so that q = q{αq}. Let λq be the fundamental weight of αq. The weight λq is orthogonal to any roots α with gα ⊂ [l, l]. Hence it exponentiates to a character χq of L. For s ∈ C, we set χs := \|χq\|s, an analytic character for L0. As χq takes Systems of Differential Operators and Generalized Verma Modules 7 real values on L0, we have dχs = sλq. Let Cχs be the one-dimensional representation of L0 with character χs. The representation χs is extended to a representation of Q0 by making it trivial on N0. It then deduces a line bundle Ls on G0/Q0 with fiber Cχs . The group G0 acts on the space C∞χ (G0/Q0,Cχs) = {F ∈ C∞(G0,Cχs) \|F (gq) = χs(q−1)F (g) for all q ∈ Q0 and g ∈ G0} by left translation. The action of g0 on C∞χ (G0/Q0,Cχs) arising from this action is given by (Y •F )(g) = d dt F (exp(−tY )g) ∣∣∣∣ t=0 (2.1) for Y ∈ g0, where the dot • denotes the action of the Lie algebra as differential operators. This action is extended C-linearly to g and then naturally to the universal enveloping algebra U(g). Let N̄0 be the unipotent subgroup opposite to N0. The natural infinitesimal action of g on the image of the restriction map C∞χ (G0/Q0,Cχs) → C∞(N̄0,Cχs) induced by (2.1) gives an action of g on the whole space C∞(N̄0,Cχs) (see, for instance, Section 2 of [35]). The line bundle Ls → G0/Q0 restricted to N̄0 is the trivial bundle N̄0 × Cχs → N̄0. We shall work with the trivial bundle on N̄0. By slight abuse of notation, we refer to the trivial bundle over N̄0 as Ls. 2.2 The Ωk systems A systematic construction of systems of differential operators for the line bundle Ls → N̄0 was established in [35]. In this subsection we summarize the construction of the systems of differential operators. For a subspace W of g, we write ∆(W ) = {α ∈ ∆ \| gα ⊂W} and Π(W ) = ∆(W )∩Π. We keep the notation from the previous subsection, unless otherwise specified. Let g = ⊕r j=−r g(j) be a Z-grading on g with g(1) 6= 0. Take L to be the analytic subgroup of G with Lie algebra g(0). Observe that, as [g(0), g(j)] ⊂ g(j), each graded subspace g(j) is an L-module and so is g(−r + k) ⊗ g(r). Write R for the infinitesimal right translation of g0. As usual, we extend it C-linearly to g and then naturally to U(g). We build systems of differential operators in three steps. Step 1: First, for 1 ≤ k ≤ 2r, consider the L-equivariant polynomial map τk : g(1)→ g(−r + k)⊗ g(r), X 7→ ( ad(X)k ⊗ Id ) ω, where ω is the element in g(−r)⊗ g(r) defined by ω := ∑ γj∈∆(g(r)) X∗γj ⊗Xγj . Here Xγj are root vectors for γj , and X∗γj are the vectors dual to Xγj with respect to the Killing form κ, namely, X∗γj (Xγt) := κ(X∗γj , Xγt) = δj,t with δi,t the Kronecker delta. Step 2: Next, for an L-irreducible constituent W of g(−r + k)⊗ g(r), consider the associated L-intertwining operator τ̃k\|W ∗ ∈ HomL(W ∗,Pk(g)) defined by τ̃k\|W ∗(Y ∗)(X) := Y ∗(τk(X)), where W ∗ is the dual space for W with respect to the Killing form κ. Take an irreducible constituent W of g(−r + k)⊗ g(r) so that τ̃k\|W ∗ 6≡ 0. 8 T. Kubo Step 3: Last, to the space W ∗ dual to the space W taken in Step 2, apply the following algebraic procedure: W ∗ τ̃k\|W∗→ Pk(g(1)) ∼= Symk(g(−1)) σ ↪→ U(n̄) R→ D(Ls)n̄. (2.2) Here, Pk(g(1)) is the space of homogeneous polynomials on g(1) of degree k, the map σ : Symk(g(−1)) → U(n̄) is the symmetrization operator, and D(Ls)n̄ is the space of n̄-invariant differential operators for Ls. Let Ωk\|W ∗ : W ∗ → D(Ls)n̄ be the composition of the linear maps described in (2.2), namely, Ωk\|W ∗ = R◦σ◦ τ̃k\|W ∗ . For simplicity we write Ωk(Y ∗) = Ωk\|W ∗(Y ∗) for the differential operator arising from Y ∗ ∈W ∗. Now, given basis {Y ∗1 , . . . , Y ∗m} for W ∗, we have a system of differential operators Ωk ( Y ∗1 ) , . . . ,Ωk(Y ∗ m). We call the system of operators the Ωk\|W ∗ system. When the irreducible constituent W ∗ is not important, we simply refer to each Ωk\|W ∗ system as an Ωk system. We may want to note that Ωk systems are independent of a choice for a basis for W ∗ up to some natural equivalence (see Definition 3.5 of [35]). The Ωk systems are not conformally invariant for arbitrary complex parameter s ∈ C for the line bundle Ls. We say that an Ωk system has special value sk if the system is conformally invariant on the line bundle Lsk . 2.3 The Ωk systems and generalized Verma modules Conformally invariant systems yield non-zero U(g)-homomorphisms between appropriate gene- ralized Verma modules. Since the theory simplifies computations to find the special values of Ωk systems, in this subsection, we review how conformally invariant Ωk systems induce such homomorphisms. We start with a well-known fact on the duality between degenerate principal series and generalized Verma modules. A generalized Verma module Mq[E] := U(g) ⊗U(q) E is a U(g)- module that is induced from a finite dimensional simple q-module E. It is well-known that if E → G0/Q0 is the homogenous bundle with fiber E then there is a natural pairing between the space Γ(E) of smooth sections and generalized Verma module Mq[E ∗], where E∗ is the dual space for E (see, for example, [7] and [13]). Via this natural pairing, associated to the line bundle Ls is the generalized Verma module Mq[C−s], where C−s = C−sλq is the q-module derived from the Q0-representation (χ−s,C) with dχ = λq. Now, given irreducible constituent W of g(−r+k)⊗g(r), we define an L-intertwining operator ωk\|W ∗ : W ∗ → U(n̄) by ωk\|W ∗ = σ ◦ τ̃ \|W ∗ , so that Ωk\|W ∗ = R ◦ ωk\|W ∗ . If writing ωk(W ∗) = ωk\|W ∗(W ∗) then we obtain the following diagram: W ∗ ωk\|W∗ �� Mq[C−s] U(n̄) ·⊗C−s oo R // D(Ls)n̄ ωk(W ∗)⊗ C−s ωk(W ∗)�oo � // {Ωk(Y ∗ 1 ), . . . ,Ωk(Y ∗ m)}, (2.3) where {Y ∗1 , . . . , Y ∗m} is a given basis for W ∗. Here we indicate by the squiggly arrow that the right differentiation R is only applied for the basis elements ωk(Y ∗ 1 ), . . . , ωk(Y ∗ m) in ωk(W ∗). It can be seen from (2.3) that constructing the Ωk\|W ∗ system is equivalent to constructing the L-submodule ωk(W ∗)⊗ C−s. Proposition 2.1 below shows that there is a further relationship. Systems of Differential Operators and Generalized Verma Modules 9 Proposition 2.1 ([2, Theorem 19]). The Ωk\|W ∗ system is conformally invariant on the line bundle Lsk if and only if ωk ( W ∗ ) ⊗ C−sk ⊂Mq[C−sk ]n, where Mq[C−s]n = {v ∈Mq[C−s] \|X · v = 0 for all X ∈ n}. It follows from Proposition 2.1 that if the Ωk\|W ∗ system is conformally invariant on Lsk then the L-submodule ωk(W ∗) ⊗ C−sk induces a U(g)-module U(g) ⊗U(q) ( ωk(W ∗) ⊗ C−sk ) . The following proposition shows that this is indeed a generalized Verma module. Proposition 2.2 ([34, Proposition 3.4(1)]). If W ∗ has highest weight ν then ωk(W ∗)⊗ C−s is the simple L-submodule of Mq[C−s] with highest weight ν − sλq. Now it follows from Propositions 2.1 and 2.2 that a conformally invariant Ωk system induces a homomorphism from Mq[ωk(W ∗)⊗C−sk ] to Mq[C−sk ]. Indeed, if the Ωk\|W ∗ system is confor- mally invariant for Lsk then, by Proposition 2.1, there exists inclusion map ι ∈ HomL ( ωk(W ∗)⊗ C−sk ,Mq[C−sk ] ) . The inclusion map ι then induces a non-zero U(g)-homomorphism ϕΩk ∈ HomU(g),L ( Mq[ωk(W ∗)⊗ C−sk ],Mq[C−sk ] ) , that is given by Mq[ωk ( W ∗ ) ⊗ C−sk ] ϕΩk→ Mq[C−sk ], u⊗ ( ωk(Y ∗)⊗ 1) 7→ u · ι ( ωk(Y ∗)⊗ 1). (2.4) Observe that there is a quotient map from a (full) Verma module to a generalized Verma module. A homomorphism between generalized Verma modules is called standard if it comes from a homomorphism between the corresponding full Verma modules, and called non-standard otherwise [36]. In the rest of this subsection, to study the standardness of the map ϕΩk in (2.4), we give a simple criterion to determine whether or not the standard homomorphism ϕstd : Mq[ωk(W ∗) ⊗ C−sk ] → Mq[C−sk ] is zero. To do so, it is convenient to parametrize generalized Verma modules by their infinitesimal characters. Therefore we write Mq[C−skλq ] = Mq(−skλq + ρ), where ρ is half the sum of the positive roots. Similarly, if W ∗ has highest weight ν then, by Proposition 2.2, we write Mq[ωk ( W ∗ ) ⊗ C−sk ] = Mq(ν − skλq + ρ). Now, if v := ωk(Y ∗)⊗ 1 then (2.4) is expressed by Mq(ν − skλq + ρ) ϕΩk→ Mq(−skλq + ρ), u⊗ v 7→ u · ι(v). (2.5) To describe the criterion efficiently we recall a well-known definition for a link of two weights. Let 〈·, ·〉 be the inner product on h∗ induced from the Killing form κ. Write α∨ = 2α/〈α, α〉. Definition 2.3 (Bernstein–Gelfand–Gelfand). Let λ, δ ∈ h∗ and β1, . . . , βt ∈ ∆+. Set δ0 = δ and δi = sβi . . . sβ1δ for 1 ≤ i ≤ t. We say that the sequence (β1, . . . , βt) links δ to λ if (1) δt = λ and (2) 〈δi−1, β ∨ i 〉 ∈ Z≥0 for 1 ≤ i ≤ t. Let M(η) denote the (full) Verma module with highest weight η − ρ. As usual, if there is a non-zero U(g)-homomorphism from M(η) into M(ζ) then we write M(η) ⊂ M(ζ). If Π(l) denotes the set of simple roots α ∈ Π so that gα ⊂ l then the criterion is given as follows. Proposition 2.4 ([34, Proposition 4.6]). Let Mq(ν − skλq + ρ) and Mq(−skλq + ρ) be the generalized Verma modules in (2.5). Then the standard map from Mq(ν−skλq+ρ) to Mq(−skλq+ ρ) is zero if and only if there exists αν ∈ Π(l) so that −αν − skλq + ρ is linked to ν − skλq + ρ. 10 T. Kubo 2.4 The Ω2 systems associated to maximal parabolic subalgebras of quasi-Heisenberg type In Sections 3, 4, and 5, we study Ω2 systems associated to a certain class of maximal parabolic subalgebra q = l⊕ n, which we call quasi-Heisenberg type. More precisely we find their special values and determine the standardness of the maps ϕΩ2 . Then, in this subsection, we recall from [35] some observation on the Ω2 systems associated to such maximal parabolic subalgebras. First, we recall from [35] that a parabolic subalgebra q = l⊕n is called quasi-Heisenberg type if its nilpotent radical n satisfies the conditions that [n, [n, n]] = 0 and dim([n, n]) > 1. Let αq be a simple root, so that the maximal parabolic subalgebra q = q{αq} = l⊕n determined by αq is of quasi-Heisenberg type. Given Dynkin type T of g, if we write T (i) for the Lie algebra together with the choice of maximal parabolic subalgebra q = q{αi} determined by αi then the maximal parabolic subalgebras q = l⊕ n of quasi-Heisenberg type are classified as follows: Bn(i), 3 ≤ i ≤ n, Cn(i), 2 ≤ i ≤ n− 1, Dn(i), 3 ≤ i ≤ n− 2, and E6(3), E6(5), E7(2), E7(6), E8(1), F4(4). Here, the Bourbaki conventions [6] are used for the labels of the simple roots. Note that, in type An, any maximal parabolic subalgebra has abelian nilpotent radical, and also that, in type G2, the nilpotent radicals of two maximal parabolic subalgebras are a 3-step nilpotent or Heisenberg algebra. We next observe that a maximal parabolic subalgebra q of quasi-Heisenberg type induces a 2-grading on g. As q has two-step nilpotent radical, if λq is the fundamental weight for αq then, for all β ∈ ∆, the quotient 2〈λq, β〉/\|\|αq\|\|2 takes the values of 0, ±1, or ±2 (see for example Section 4.1 of [35]). Therefore, if Hq is the element in h so that β(Hq) = 2〈λq, β〉/\|\|αq\|\|2 for all β ∈ ∆, and if g(j) is the j-eigenspace of ad(Hq) on g then the adjoint action of Hq induces a 2-grading g = ⊕2 j=−2 g(j) on g with parabolic subalgebra q = g(0) ⊕ g(1) ⊕ g(2), where l = g(0) and n = g(1)⊕ g(2). The subalgebra n̄, the nilpotent radical opposite to n, is given by n̄ = g(−1) ⊕ g(−2). Here we have g(2) = z(n) and g(−2) = z(n̄), where z(n) (resp. z(n̄)) is the center of n (resp. n̄). Thus we denote the 2-grading on g by g = z(n̄)⊕ g(−1)⊕ l⊕ g(1)⊕ z(n) (2.6) with parabolic subalgebra q = l⊕ g(1)⊕ z(n). Now, for 1 ≤ k ≤ 4, the maps τk associated to the grading (2.6) are given by τk : g(1)→ g(−2 + k)⊗ z(n), X 7→ 1 k! ( ad(X)k ⊗ Id ) ω with ω = ∑ γj∈∆(z(n)) X∗γj ⊗Xγj . (2.7) In particular when k = 2, we have τ2 : g(1)→ l⊗ z(n), X 7→ 1 2 ( ad(X)2 ⊗ Id ) ω. (2.8) The Ω2 systems, the systems of differential operators that we study, are constructed from the τ2 map. Systems of Differential Operators and Generalized Verma Modules 11 By construction the Ω2 systems arise from irreducible constituents W of l⊗z(n) so that τ̃2\|W ∗ is not identically zero (see the procedures described in Subsection 2.2). Since such irreducible constituents play a role to determine the special values of the Ω2 systems, for the remainder of this section, we recall from [35] the observation on l and irreducible constituents W for which τ̃2\|W ∗ 6≡ 0. We start with the structure of l = z(l)⊕ [l, l], where z(l) is the center of l. First, as q = l⊕n is the maximal parabolic subalgebra determined by αq, we have z(l) = CHq. The semisimple part [l, l] is either simple or the direct sum of two or three simple ideals (see for example Appendix A of [35]). To characterize the simple ideals, let γ be the highest root for g. If g is not of type An then there is exactly one simple root not orthogonal to γ. It is well known that if αγ is the unique simple root then the nilpotent radical n′ of the parabolic subalgebra q′ = q{αγ} satisfies dim([n′, n′]) = 1. Recall from Subsection 2.2 that we say q = l⊕ n is of quasi-Heisenberg type if dim([n, n]) > 1. Hence, if q = q{αq} is a parabolic subalgebra of quasi-Heisenberg type then αγ is in Π(l) = Π\{αq}. In particular there is a unique simple ideal of [l, l] containing the root space gαγ for αγ . We denote by lγ the unique simple ideal containing gαγ . Similarly, when [l, l] consists of two (resp. three) simple ideals, we denote the other simple ideal(s) by lnγ (resp. l+nγ and l−nγ). The three simple factors occur only when q is of type Dn(n− 2). So, when q is not of type Dn(n− 2), the Levi subalgebra l may decompose into l = CHq ⊕ lγ ⊕ lnγ . (2.9) Similarly, when q is of type Dn(n− 2), one may write l = CHq ⊕ lγ ⊕ l+nγ ⊕ l−nγ . (2.10) Note that when [l, l] is a simple ideal, we have lnγ = {0} (l±nγ = {0}). It follows from the decompositions (2.9) and (2.10) that the tensor product l⊗ z(n) may be written as l⊗ z(n) =  ( CHq ⊗ z(n) ) ⊕ ( lγ ⊗ z(n) ) ⊕ ( lnγ ⊗ z(n) ) , q is not of type Dn(n− 2),( CHq ⊗ z(n) ) ⊕ ( lγ ⊗ z(n) ) ⊕ ( l+nγ ⊗ z(n) ) ⊕ ( l−nγ ⊗ z(n) ) , q is of type Dn(n− 2). To build an Ω2 system, it is necessary to choose an irreducible constituent W in l ⊗ z(n) so that the L-intertwining map τ̃2\|W ∗ ∈ HomL ( W ∗,P2(g(1)) ) is not identically zero. Now we give a necessary condition for irreducible constituents W so that τ̃2\|W ∗ 6≡ 0. To do so, for ν ∈ h∗ with 〈ν, α∨〉 ∈ Z≥0 for all Π(l), we denote by V (ν) the simple l-module with highest weight ν\|h∩[l,l]. Suppose that l⊗ z(n) has irreducible constituent V (ν). If the linear map τ̃2 ∣∣ V (ν)∗ : V (ν)∗ → P2(g(1)) is not identically zero then, via the isomorphism P2(g(1)) ∼= Sym2(g(1))∗, V (ν) should be an irreducible constituent of Sym2(g(1)) ⊂ g(1)⊗g(1). In particular if µ is the highest weight of g(1) then ν is of the form ν = µ+ ε for some ε ∈ ∆(g(1)). It was shown in Lemma 4.14 of [35] that the highest root γ is of this form. However, it follows from Proposition 6.5 of [35] that V (γ) does not occur in Sym2(g(1)). Based on this observation we give the following definition. Definition 2.5 ([35, Definition 6.7]). An irreducible constituent V (ν) of l⊗z(n) is called special1 if V (ν) satisfies the following two conditions: (C1) ν = µ+ ε for some ε ∈ ∆(g(1)). (C2) ν 6= γ. 1There is a certain discrepancy on the terminology “special constituents”. See the comments in the introduction on this matter. 12 T. Kubo It is shown in [35, Section 6] that, for q not of type Dn(n− 2), there are exactly one or two special constituents of l ⊗ z(n); one is an irreducible constituent of lγ ⊗ z(n) and the other is equal to lnγ ⊗ z(n). We denote by V (µ + εγ) and V (µ + εnγ) the special constituents so that V (µ + εγ) ⊂ lγ ⊗ z(n) and V (µ + εnγ) = lnγ ⊗ z(n). It will be shown in Section 3 that if q is of type Dn(n− 2) then there are three special constituents, namely, V (µ + εγ) ⊂ lγ ⊗ z(n) and V (µ+ ε±nγ) = l±nγ ⊗ z(n). To compute the special values of Ω2 systems efficiently, the special constituents V (µ+ ε) are classified as Type 1a, Type 1b, Type 2, or Type 3 as follows: Definition 2.6 ([35, Definition 6.20]). Let µ be the highest weight for g(1). We say that a special constituent V (µ+ ε) is of (1) Type 1a if µ+ ε is not a root with ε 6= µ and both µ and ε are long roots, (2) Type 1b if µ+ ε is not a root with ε 6= µ and either µ or ε is a short root, (3) Type 2 if µ+ ε = 2µ is not a root, or (4) Type 3 if µ+ ε is a root. Table 1 in the introduction shows the types of special constituents. A dash in the column for V (µ+ εnγ) indicates that lnγ = {0} for the case. (So there is no special constituent V (µ+ εnγ).) In Sections 4 and 5, we find the special values of the Ω2 systems coming from special con- stituents of Type 1b and Type 3, respectively. The following proposition will play a key role to determine the special values. Proposition 2.7. Let V (µ + ε)∗ be the dual module of an irreducible constituent V (µ + ε) of l ⊗ z(n) so that the operator Ω2\|V (µ+ε)∗ : V (µ + ε)∗ → D(Ls)n̄ is non-zero. If Xh is a highest weight vector for g(1) and if Y ∗l is a lowest weight vector for V (µ + ε)∗ then the Ω2\|V (µ+ε)∗ system is conformally invariant on Ls if and only if in Mq[C−s] Xµ · ( ω2(Y ∗l )⊗ 1−s ) = 0. (2.11) Proof. Observe that, by Proposition 2.1, the Ω2\|V (µ+ε)∗ system is conformally invariant if and only if, for all X ∈ n and Y ∗ ∈ V (µ+ ε)∗, X · ( ω2(Y ∗)⊗ 1−s ) = Xω2(Y ∗)⊗ 1−s = 0. (2.12) Therefore, to prove this proposition, it suffices to show that (2.11) implies (2.12). Since the arguments are similar to ones for Proposition 7.13 with Lemma 3.9 and Lemma 3.12 of [35], we omit the proof. � 3 Parabolic subalgebra of type Dn(n− 2) In this short section we consider the Ω2 systems associated with maximal parabolic subalgebra q = l ⊕ n of type Dn(n − 2). In particular we find the special values for the Ω2 systems and determine the standardness for the maps ϕΩ2 . The parabolic subalgebra q of type Dn(n−2) is the maximal parabolic subalgebra determined by the simple root αn−2; the deleted Dynkin diagram is αn−1◦ ◦ α1 · · · ◦ αn−3 ⊗ αn−2 ◦ αn Systems of Differential Operators and Generalized Verma Modules 13 with subgraphs ◦ α1 ◦ α2 ◦ α3 · · · ◦ αn−3 ◦ αn−1 ◦ αn As the simple root αq that determines the parabolic subalgebra q is αq = αn−2, the fundamental weight λq for αq is λq = λn−2. (For the definition of deleted Dynkin diagrams see, for instance, Section 4.1 of [35].) Recall from Section 2.4 that the Levi subalgebra l may be decomposed as l = CHq ⊕ lγ ⊕ l+nγ ⊕ l−nγ . The unique simple root αγ that is not orthogonal to the highest root γ is αγ = α2. Therefore we have lγ ∼= sl(n − 2,C) and l±nγ ∼= sl(2,C). For convenience we set l+nγ (resp. l−nγ) to be the simple ideal that corresponds to the singleton for αn (resp. αn−1). 3.1 Special constituents and special values We start with finding the special constituents of l ⊗ z(n). As shown in Section 2.4, the tensor product l⊗ z(n) may be decomposed into l⊗ z(n) = ( CHq ⊗ z(n) ) ⊕ ( lγ ⊗ z(n) ) ⊕ ( l+nγ ⊗ z(n) ) ⊕ ( l−nγ ⊗ z(n) ) . With the arguments in [35, Section 6.1], one can easily check that CHq⊗ z(n) and l±nγ ⊗ z(n) are simple l-modules. In fact, if we use the standard realization of roots with αn−1 = εn−1− εn and αn = εn−1 + εn then CHq ⊗ z(n) = V (γ) = V (ε1 + ε2), l+nγ ⊗ z(n) = V (αn + γ) = V (ε1 + ε2 + εn−1 + εn), and l−nγ ⊗ z(n) = V (αn−1 + γ) = V (ε1 + ε2 + εn−1 − εn). On the other hand, the tensor product lγ ⊗ z(n) is reducible. By using the character formula of Klimyk [23, Corollary], it can be shown that lnγ ⊗ z(n) = V (2ε1 + ε2 − εn−2)⊕ V (ε1 + ε2)⊕ V (2ε1). Now one may observe that only V (2ε1) and V (ε1 +ε2 +εn−1±εn) satisfy the conditions (C1) and (C2) in Definition 2.5. Thus these irreducible constituents are the special constituents of l ⊗ z(n). Write εγ and ε±nγ for the roots in ∆(g(1)) so that µ + εγ = 2ε1 and µ + ε±nγ = ε1 + ε2 + εn−1 ± εn, where µ is the highest weight for g(1). Tables 4 and 5 summarize the data for the special constituents. Table 4. Roots µ, εγ , ε+nγ , and ε−nγ . Parabolic q µ εγ ε+nγ ε−nγ Dn(n− 2) ε1 + εn−1 ε1 − εn−1 ε2 + εn ε2 − εn Table 5. Highest weights for special constituents. Parabolic q V (µ+ εγ) V (µ+ ε+nγ) V (µ+ ε−nγ) Dn(n− 2) 2ε1 ε1 + ε2 + εn−1 + εn ε1 + ε2 + εn−1 − εn Observe that µ, εγ , and ε±nγ are all long roots and that neither µ + εγ nor µ + ε±nγ is a root. Thus the special constituents V (µ+ εγ) and V (µ+ ε±nγ) are all of Type 1a (see Definition 2.6). 14 T. Kubo As l ⊗ z(n) contains a special constituent of Type 1a, it follows from the argument in the proof for Proposition 7.3 of [35] that the τ2 map is not identically zero. Also, the argument for Proposition 7.5 of [35] shows that, for V (µ + ε) = V (µ + εγ), V (µ + ε±nγ), the linear map τ̃2\|V (µ+ε)∗ is not identically zero. Now we are going to find the special values of the Ω2 systems arising from the special constituents V (µ + εγ) and V (µ + ε±nγ). If W ⊂ g is an ad(h)-invariant subspace then, for any weight ν ∈ h∗, we write ∆ν(W ) := {α ∈ ∆(W ) \| ν − α ∈ ∆}. (3.1) Theorem 3.1. Let q be the maximal parabolic subalgebra of type Dn(n − 2). If V (µ + ε) = V (µ+ εγ) or V (µ+ ε±nγ) then the Ω2\|V (µ+ε)∗ system is conformally invariant on Ls if and only if s = \|∆µ+ε(g(1))\| 2 − 1, where \|∆µ+ε(g(1))\| is the number of the elements in ∆µ+ε(g(1)). Proof. Since the special constituents V (µ+ ε) and V (µ+ ε±γ ) are of Type 1a, all the statements in [35] for Type 1a special constituents hold for V (µ + ε) and V (µ + ε±γ ). Now the theorem follows from the arguments in [35, Section 7]. � Corollary 3.2. Under the same hypotheses in Theorem 3.1, all Ω2\|V (µ+ε)∗ systems are confor- mally invariant on L1. Proof. By inspection we have \|∆µ+ε(g(1))\| = 4 for each special constituent. Now the results follow from Theorem 3.1. � 3.2 The standardness of the map ϕΩ2 In the rest of this section we determine whether or not the maps ϕΩ2 coming from the Ω2 systems are standard. Observe that V (µ+ εγ)∗ = V (2ε1)∗ = V (−2εn−2) and V ( µ+ ε±nγ )∗ = V (ε1 + ε2 + εn−1 ± εn)∗ = V (−εn−3 − εn−2 + εn−1 ± εn). It follows from Corollary 3.2 that the special value s2 is s2 = 1 for each special constituent. Therefore if we denote by ϕ(Ω2,µ+εγ) (resp. ϕ(Ω2,µ+ε±nγ)) the homomorphism induced by the Ω2\|V (µ+εγ)∗ system (reps. the Ω2\|V (µ+ε±nγ)∗ system) then, by (2.5), we have ϕ(Ω2,µ+εγ) : Mq(−2εn−2 − λn−2 + ρ)→Mq(−λn−2 + ρ) (3.2) and ϕ( Ω2,µ+ε±nγ ) : Mq((−εn−3 − εn−2 + εn−1 ± εn)− λn−2 + ρ)→Mq(−λn−2 + ρ). (3.3) Theorem 3.3. If q is the maximal parabolic subalgebra of type Dn(n − 2) then the standard maps between the generalized Verma modules in (3.2) and (3.3) are zero. Consequently, the maps ϕ(Ω2,µ+εγ) and ϕ(Ω2,µ+ε±nγ) are non-standard. Systems of Differential Operators and Generalized Verma Modules 15 Proof. To prove this theorem, by Proposition 2.4, it suffices to show that there exits αν ∈ Π(l) so that −αν − λn−2 + ρ is linked to ν − λn−2 + ρ for ν = −2εn−2, −εn−3 − εn−2 + εn−1 ± εn. Since the arguments are the same for each case, we only demonstrate a proof for (3.2). To show for (3.2), observe that −2εn−2 = −2(εn−2 − εn−1)− (εn−1 − εn)− (εn−1 + εn) with εn−2 − εn−1 ∈ ∆(g(1)) and εn−1 ± εn ∈ Π(l) (see Appendix A). We claim that (εn−1 − εn, εn−2 − εn−1) links −(εn−1 + εn)− λn−2 + ρ to −2εn−2 − λn−2 + ρ, that is, sεn−2−εn−1sεn−1−εn(−(εn−1 + εn)− λn−2 + ρ) = −2εn−2 − λn−2 + ρ with 〈−(εi+1 − εi+2)− λn−2 + ρ, (εn−1 − εn)∨〉 ∈ Z≥0 and 〈sεn−1−εn(−(εn−1 + εn)− λn−2 + ρ), (εn−2 − εn−1)∨〉 ∈ Z≥0. A direct computation shows that it indeed holds. For (3.3), one may use (εn−3−εn−2, εn−2−εn−1) as a link. Now the proposed statements follow. � 4 Type 1b special constituent In this section we study the Ω2 system associated to the Type 1b special constituent. It follows from Table 1 in the introduction that Type 1b special constituent occurs only when the parabolic subalgebra q is of type Bn(n−1). The simple root αq that determines the parabolic subalgebra q is then αq = αn−1. We write λq = λn−1 for the fundamental weight λq for αq. The deleted Dynkin diagram for q is ◦ α1 ◦ α2 · · · ◦ αn−2 ⊗ αn−1 +3◦ αn with connected subgraphs ◦ α1 ◦ α2 ◦ α3 · · · ◦ αn−2 ◦ αn Since α2 is the unique simple root that is not orthogonal to the highest root γ, it follows from the subgraphs that lγ ∼= sl(n−1,C) and lnγ ∼= sl(2,C). Recall from Table 1 that Type 1b special constituent of l⊗ z(n) is the irreducible constituent V (µ+ εnγ) = lnγ ⊗ z(n). 4.1 The τ̃2\|V (µ+εnγ)∗ map We start with observing the L-intertwining operator τ̃2\|V (µ+εnγ)∗ : V (µ + εnγ)∗ → P2(g(1)). To do so we first fix convenient root vectors for g so that certain computations will be carried out easily. Observe that, as q is of type Bn(n − 1), the Lie algebra g under consideration is g = so(2n+ 1,C). We take h to be the set of block diagonal matrices H(h1, . . . , hn) = diag (( 0 ih1 −ih1 0 ) , ( 0 ih2 −ih2 0 ) , . . . , ( 0 ihn −ihn 0 ) , 0 ) with hj ∈ C and i = √ −1. The positive roots are ∆+ = {εj ± εk \| 1 ≤ j < k ≤ n}∪ {εj \| 1 ≤ j ≤ n} with εj(H(h1, . . . , hn)) = hj . We take the root vectors Xα as follows: 16 T. Kubo 1. α = ±(εj ± εk) (j < k): j k Xα = ( 0 Eα −Etα 0 ) j k with Eεj−εk = 1 2 ( 1 i −i 1 ) , Eεj+εk = 1 2 ( 1 −i −i −1 ) , E−εj+εk = 1 2 ( −1 i −i −1 ) , E−εj−εk = 1 2 ( −1 −i −i 1 ) , where Xα denotes the matrix whose entries are all zero except the jth and kth pairs of indices. 2. α = ±εj : j 2n+ 1 Xα = ( 0 vα −vtα 0 ) j 2n+ 1 with vεj = 1√ 2 ( 1 −i ) , v−εj = 1√ 2 ( −1 −i ) , where Xα denotes the matrix defined similarly to the previous case. For α, β ∈ ∆ with α + β 6= 0, let Nα,β denote the constant so that [Xα, Xβ] = Nα,βXα+β. Table 6 summarizes the values of Nα,β for α+β a positive root. The constant Nα,β satisfies the property that N−α,−β = −Nα,β (see, for instance, [24, Theorem 6.6]). Thus the values of Nα,β for α+β a negative root can also be obtained from Table 6. Here, we would like to acknowledge that, for the cases that α and β are long roots (formulas (1)–(12)), we simply adapt the beautiful multiplication table by Knapp in [25, Section 10]. For α ∈ ∆+ we set Hα = [Xα, X−α]; namely, we have Hεj±εk = diag ( 0, . . . , 0, j( 0 i −i 0 ) , 0, . . . , 0,± k( 0 i −i 0 ) , 0, . . . , 0 ) and Hεj = diag ( 0, . . . , 0, j( 0 i −i 0 ) , 0, . . . , 0, ) . Now observe if T (X,Y ) := Tr(XY ) then T (Xα, X−α) = 2 for all α ∈ ∆+. As the restriction T (·, ·)\|h0×h0 to a real form h0 of h is an inner product on h0, the trace form T (·, ·) is a positive constant multiple of the Killing form κ(·, ·). If b0 is the non-zero constant so that κ(X,Y ) = b0T (X,Y ) then κ(Xα, X−α) = 2b0 for all α ∈ ∆+. Thus the dual vectors X∗α for Xα with respect to the Killing form are X∗α = (1/(2b0))X−α. Now the element ω in (2.7) is given by ω = ∑ γj∈∆(z(n)) X∗γj ⊗Xγj = 1 2b0 ∑ γj∈∆(z(n)) X−γj ⊗Xγj . Systems of Differential Operators and Generalized Verma Modules 17 Table 6. The values of Nα,β for roots α and β for so(2n + 1,C) with indices i < j < k when α + β is a positive root. Formula α β Nα,β (1) εi + εk εj − εk −1 (2) εi − εk εj + εk −1 (3) εi + εk −εj − εk +1 (4) εi − εk −εj + εk +1 (5) εi + εj −εj + εk −1 (6) εi − εj εj + εk +1 (7) εi + εj −εj − εk −1 (8) εi − εj εj − εk +1 (9) εi + εj −εi + εk +1 (10) −εi + εj εi + εk +1 (11) εi + εj −εi − εk +1 (12) −εi + εj εi − εk +1 (13) εj −εj + εk −1 (14) −εj εj + εk −1 (15) εk εj − εk −1 (16) −εk εj + εk +1 (17) εj εk −1 (18) εj −εk +1 Thus the map τ2(X) in (2.8) may be expressed as τ2(X) = 1 2 ( ad(X)2 ⊗ Id ) ω = 1 4b0 ∑ γj∈∆(z(n)) ad(X)2X−γj ⊗Xγj . (4.1) Proposition 7.3 of [35] showed that when q is of type Bn(n − 1), the τ2 map is not identically zero. To construct differential operators from V (µ+ εnγ)∗, it is necessary to show that the linear map τ̃2\|V (µ+εnγ)∗ : V (µ + εnγ)∗ → P2(g(1)) is not identically zero (see Step 2 and Step 3 in Section 2.2). We shall prove this by showing that τ̃2(Y ∗)(X) is a non-zero polynomial on g(1) for some Y ∗ in V (µ + εnγ)∗. To this end observe that, as discussed in Section 2.4, we have V (µ + εnγ) = lnγ ⊗ z(n). Therefore V (µ + εnγ)∗ = l∗nγ ⊗ z(n)∗ = lnγ ⊗ z(n̄). Since γ ∈ ∆(z(n)) and lnγ = spanC{Xαn , Hαn , X−αn}, we have X∗αn ∈ lnγ and X∗γ ∈ z(n̄); therefore X∗αn ⊗ X ∗ γ ∈ lnγ ⊗ z(n̄) = V (µ+ εnγ)∗. Proposition 4.1. If q = l⊕n is the parabolic subalgebra of type Bn(n−1) then the L-intertwining operator τ̃2\|V (µ+εnγ)∗ is not identically zero. Proof. We show that τ̃2(X∗αn ⊗X ∗ γ)(X) is a non-zero polynomial on g(1). As X∗α denotes the dual vector for Xα with respect to the Killing form κ, by (4.1), the operator τ̃2(X∗αn⊗X ∗ γ)(X) = (X∗αn ⊗X ∗ γ)(τ2(X)) is given by τ̃2 ( X∗αn ⊗X ∗ γ ) (X) = (X∗αn ⊗X ∗ γ)(τ2(X)) = 1 4b0 ∑ γj∈∆(z(n)) κ ( X∗αn , ad(X)2X−γj ) κ(X∗γ , Xγj ) = 1 4b0 κ ( X∗αn , ad(X)2X−γ ) = 1 8b20 κ ( X−αn , ad(X)2X−γ ) . (4.2) 18 T. Kubo Write X = ∑ α∈∆(g(1)) ηαXα, where ηα ∈ n∗ is the coordinate dual to Xα with respect to the Killing form. Recall from (3.1) that if W ⊂ g is an ad(h)-invariant subspace then, for any weight ν ∈ h∗, we write ∆ν(W ) = {α ∈ ∆(W ) \| ν − α ∈ ∆}. Then, (4.2) = 1 8b20 κ ( X−αn , ad(X)2X−γ ) = 1 8b20 ∑ β,δ∈∆(g(1)) ηβηδκ(X−αn , [Xδ, [Xβ, X−γ ]]) = 1 8b20 ∑ α,β∈∆(g(1)) ηβηδκ([X−αn , Xδ], [Xβ, X−γ ]]) = 1 8b20 ∑ β∈∆γ(g(1)) δ∈∆αn (g(1)) ηβηδNβ,−γN−αn,δκ(Xδ−αn , Xβ−γ). (4.3) One may observe that κ(Xδ−αn , Xβ−γ) 6= 0 if and only if δ = αn + γ − β ∈ ∆(g(1)). If we write θ(β) = αn + γ − β then (4.3) = 1 8b20 ∑ β∈∆γ(g(1)) δ∈∆αn (g(1)) ηβηδNβ,−γN−αn,δκ(Xδ−αn , Xβ−γ) = 1 8b20 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) ηβηθ(β)Nβ,−γN−αn,θ(β)κ(Xθ(β)−αn , Xβ−γ) = 1 8b20 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) ηβηθ(β)Nβ,−γN−αn,θ(β)κ(Xγ−β, Xβ−γ) = 1 4b0 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) ηβηθ(β)Nβ,−γN−αn,θ(β)κ(X∗β−γ , Xβ−γ) = 1 4b0 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) ηβηθ(β)Nβ,−γN−αn,θ(β) = 1 4b0 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) Nβ,−γN−αn,θ(β)κ(X,X∗β)κ(X,X∗θ(β)). (4.4) As αn = εn and γ = ε1 + ε2, by inspection, we have ∆γ(g(1)) = {ε1 ± εn, ε2 ± εn} ∪ {ε1, ε2} and ∆αn+γ(g(1)) = {ε1 + εn, ε2 + εn} ∪ {ε1, ε2}. In particular, ∆αn+γ(g(1)) ⊂ ∆γ(g(1)). Therefore, (4.4) = 1 4b0 ∑ β∈∆γ(g(1))∩∆αn+γ(g(1)) Nβ,−γN−αn,θ(β)κ(X,X∗β)κ(X,X∗θ(β)) = 1 4b0 ∑ β∈∆αn+γ(g(1)) Nβ,−γN−αn,θ(β)κ(X,X∗β)κ(X,X∗θ(β)). Systems of Differential Operators and Generalized Verma Modules 19 Therefore we obtain τ̃2(X∗αn ⊗X ∗ γ)(X) = 1 4b0 ∑ β∈∆αn+γ(g(1)) Nβ,−γN−αn,θ(β)κ(X,X∗β)κ(X,X∗θ(β)). Since ∆αn+γ(g(1)) = {ε1 + εn, ε2 + εn} ∪ {ε1, ε2}, this reads τ̃2 ( X∗αn ⊗X ∗ γ ) (X) = 1 4b0 ∑ β∈∆αn+γ(g(1)) Nβ,−γN−αn,θ(β)κ(X,X∗β)κ(X,X∗θ(β)) = 1 4b0 ( Nε1+εn,−(ε1+ε2)N−εn,ε2κ(X,X∗ε1+εn)κ(X,X∗ε2) +Nε2,−(ε1+ε2)N−εn,ε1+εnκ(X,X∗ε2)κ(X,X∗ε1+εn) +Nε2+εn,−(ε1+ε2)N−εn,ε1κ(X,X∗ε2+εn)κ(X,X∗ε1) +Nε1,−(ε1+ε2)N−εn,ε2+εnκ(X,X∗ε1)κ(X,X∗ε2+εn) ) = 1 4b0 ( (1)(−1)κ(X,X∗ε1+εn)κ(X,X∗ε2) + (−1)(1)κ(X,X∗ε2)κ(X,X∗ε1+εn) + (−1)(−1)κ(X,X∗ε2+εn)κ(X,X∗ε1)(1)(1)κ(X,X∗ε1)κ(X,X∗ε2+εn) ) = 1 2b0 ( κ(X,X∗ε2+εn)κ(X,X∗ε1)− κ(X,X∗ε1+εn)κ(X,X∗ε2) ) . (4.5) Therefore τ̃2(X∗αn ⊗X ∗ γ)(X) is a non-zero polynomial on g(1). � 4.2 The special value of the Ω2\|V (µ+εnγ)∗ system Now we find the special value of the Ω2\|V (µ+εnγ)∗ system. To do so we use Proposition 2.7. To this end recall from Section 2.3 the linear map ω2\|V (µ+εnγ)∗ : V (µ + εnγ)∗ → U(n̄) defined by ω2\|V (µ+εnγ)∗ = σ ◦ τ̃2\|V (µ+εnγ)∗ , where σ : Sym2(g(−1)) → U(n̄) the symmetrization operator (here we identify P2(g(1)) ∼= Sym2(g(−1))). If Y ∗l := 8b30(X∗αn ⊗X ∗ γ) then, by (4.5), ω2(Y ∗l ) := ω2\|V (µ+εnγ)∗(Y ∗l ) is given by ω2(Y ∗l ) = 4b20 ( σ(X∗ε2+εnX ∗ ε1)− σ(X∗ε1+εnX ∗ ε2) ) . As the dual vector X∗α for Xα with respect to the Killing form is X∗α = (1/2b0)X−α, this amounts to ω2 ( Y ∗l ) = σ(X−(ε2+εn)X−ε1)− σ(X−(ε1+εn)X−ε2). Moreover, since ε1 + ε2 + εn 6∈ ∆, the symmetrization is unnecessary. Therefore we obtain ω2 ( Y ∗l ) = X−(ε2+εn)X−ε1 −X−(ε1+εn)X−ε2 . (4.6) Now we are going to determine the special value of the Ω2\|V (µ+εnγ)∗ system. Theorem 4.2. Let q be the maximal parabolic subalgebra of type Bn(n− 1). The Ω2\|V (µ+εnγ)∗ system is conformally invariant on Ls if and only if s = 1. Proof. Observe that, as Xαn ⊗ Xγ is a highest weight vector for lnγ ⊗ z(n) = V (µ + εnγ), X∗αn ⊗X ∗ γ is a lowest weight vector for V (µ+ εnγ)∗; consequently, Y ∗l is a lowest weight vector for V (µ + εnγ)∗. Therefore, by Proposition 2.7, to find the special value for the Ω2\|V (µ+εnγ)∗ 20 T. Kubo system, it suffices to determine s ∈ C so that Xµ · (ω2(Y ∗l ) ⊗ 1−s) = 0 for µ the highest weight for g(1). By inspection, we have µ = ε1 + εn (see Appendix A). Thus we compute Xε1+εn · (ω2(Y ∗l )⊗ 1−s). It follows from (4.6) that Xε1+εn · (ω2 ( Y ∗l ) ⊗ 1−s) = Xε1+εnX−(ε2+εn)X−ε1 ⊗ 1−s −Xε1+εnX−(ε1+εn)X−ε2 ⊗ 1−s. We observe from the second term. By the standard computation, we have T2 = −Xε1+εnX−(ε1+εn)X−ε2 ⊗ 1−s = −Hε1+εnX−ε2 ⊗ 1−s = sλn−1(Hε1+εn)X−ε2 ⊗ 1−s. Observe that since λn−1 = n−1∑ j=1 εj , we have λn−1(Hε1+εn) = 1. Thus, T2 = sλn−1(Hε1+εn)X−ε2 ⊗ 1−s = sX−ε2 ⊗ 1−s. Similarly the standard computation shows that the first term amounts to T1 = Xε1+εnX−(ε2+εn)X−ε1 ⊗ 1−s = −X−ε2 ⊗ 1−s. Therefore, Xε1+εn · (ω2 ( Y ∗l ) ⊗ 1−s) = T1 + T2 = (s− 1)X−ε2 ⊗ 1−s. Now the assertion follows from Proposition 2.7. � Remark 4.3. Theorem 7.16 in [35] shows that the special values s2 for the Ω2 systems asso- ciated to Type 1a special constituents V (µ + ε) are given by s2 = ( \|∆µ+ε(g(1))\|/2 ) − 1, where \|∆µ+ε(g(1))\| is the number of elements in ∆µ+ε(g(1)). Now since ∆µ+εnγ (g(1)) = ∆αn+γ(g(1)) = {ε1 + εn, ε2 + εn} ∪ {ε1, ε2}, the special value s2 may be expressed as s2 = 1 = \|∆µ+εnγ (g(1))\| 2 − 1. Thus the special value s2 for the Ω2 systems associated to Type 1a and Type 1b special con- stituents can be given in the same formula. 4.3 The standardness of the map ϕΩ2 In the rest of this section we determine whether or not the map ϕΩ2 coming from the Ω2\|V (µ+εnγ)∗ system is standard. Observe that, as V (µ + εnγ) = lnγ ⊗ z(n) and as the highest weights for lnγ and z(n) are εn and −εn−2 − εn−1, respectively, we have V (µ+ εnγ)∗ = lnγ ⊗ z(n̄) = V (−εn−2 − εn−1 + εn). By Theorem 4.2, the special value s2 for the Ω2\|V (µ+εnγ)∗ system is s2 = 1. Therefore, by (2.5), the Ω2\|V (µ+εnγ)∗ system yields a non-zero U(g)-homomorphism ϕΩ2 : Mq((−εn−2 − εn−1 + εn)− λn−1 + ρ)→Mq(−λn−1 + ρ). (4.7) Theorem 4.4. If q is the maximal parabolic subalgebra of type Bn(n − 1) then the standard map ϕstd between the generalized Verma modules in (4.7) is zero. Consequently, the map ϕΩ2 is non-standard. Systems of Differential Operators and Generalized Verma Modules 21 Proof. The idea of the proof is the same as for Theorem 3.3. Namely, first show that there exists αν ∈ Π(l) so that −αν −λn−1 + ρ is linked to (−εn−2− εn−1 + εn)−λn−1 + ρ, then apply Proposition 2.4. Observe that −εn−2 − εn−1 + εn = −2(εn−1 − εn)− (εn−2 − εn−1)− εn with εn−1−εn ∈ ∆(g(1)) and εn−2−εn−1, εn ∈ Π(l) (see Appendix A). By a direct computation one can show that (εn−2−εn−1, εn−1−εn) links −εn−λn−1 +ρ to (−εn−2−εn−1 +εn)−λn−1 +ρ Now the theorem follows from Proposition 2.4. � 5 Type 3 special constituent In this section we study the Ω2 system associated to Type 3 special constituent. It follows from Table 1 in the introduction that Type 3 special constituent occurs only when the parabolic subalgebra q is of type Cn(i) for 2 ≤ i ≤ n−1. The simple root αq that determines the parabolic subalgebra q is then αq = αi. We write λq = λi for the fundamental weight λq for αq. The deleted Dynkin diagram for q is ◦ α1 · · · ◦ αi−1 ⊗ αi ◦ αi+1 · · · ◦ αn−1 ks ◦ αn with connected subgraphs ◦ α1 ◦ α2 ◦ α3 · · · ◦ αi−1 ◦ αi+1 · · · ◦ αn−1 ks ◦ αn Since α1 is the unique simple root that is not orthogonal to the highest root γ, it follows from the subgraphs that lγ ∼= sl(i,C) and lnγ ∼= sp(n− i,C). Recall from Table 1 that Type 3 special constituent of l⊗ z(n) is the irreducible constituent V (µ+ εγ) ⊂ lγ ⊗ z(n). As in Section 4, our first goal is to show the L-intertwining operator τ̃2\|V (µ+εγ)∗ is not identically zero. To do so we again fix convenient root vectors for g. Observe that, as q is of type Cn(i), the Lie algebra g under consideration is g = sp(n,C). For 1 ≤ j ≤ n, we write ĵ = j + n. If Eab denotes the matrix with 1 in the (a, b) entry and 0 elsewhere then we take h to be the set of diagonal matrices H(h1, . . . , hn) = h1(E11 − E1̂1̂) + · · ·+ hn(Enn − En̂n̂) with hj ∈ C. The positive roots are ∆+ = {εj ± εk \| 1 ≤ j < k ≤ n} ∪ {2εj \| 1 ≤ j ≤ n} with εj(h1, . . . , hn) = hj . We take the root vectors Xα as follows: Xεj−εk = Ejk − Ek̂ĵ , Xεj+εk = Ejk̂ + Ekĵ , X−(εj+εk) = Eĵk + Ek̂j , X2εj = Ejĵ , X−2εj = Eĵj . For α, β ∈ ∆ with α + β 6= 0, we again denote by Nα,β the constant so that [Xα, Xβ] = Nα,βXα+β. Table 7 summarizes the values of Nα,β for α+ β a positive root. For α ∈ ∆+ we set Hα = [Xα, X−α]. Namely, Hεj±εk = (Ejj − Eĵĵ)± (Ekk − Ek̂k̂) and H2εj = Ejj − Eĵĵ . Now observe that if T (X,Y ) := Tr(XY ) then, as T (·, ·)\|h0×h0 is an inner product on a real form h0 of h, there exists a positive constant c0 so that κ(X,Y ) = c0T (X,Y ). Since T (Xα, X−α) takes the value of one for α long and two for α short, we have κ(Xα, X−α) = { c0 if α is long, 2c0 if α is short. (5.1) 22 T. Kubo Table 7. The values of Nα,β for roots α and β for sp(2n,C) with indices i < j < k when α + β is a positive root. Formula α β Nα,β (1) εi + εk εj − εk −1 (2) εi − εk εj + εk +1 (3) εi + εk −εj − εk +1 (4) εi − εk −εj + εk +1 (5) εi + εj −εj + εk −1 (6) εi − εj εj + εk +1 (7) εi + εj −εj − εk +1 (8) εi − εj εj − εk +1 (9) εi + εj −εi + εk −1 (10) −εi + εj εi + εk +1 (11) εi + εj −εi − εk +1 (12) −εi + εj εi − εk +1 (13) εi + εj εi − εj −2 (14) εi + εj −εi + εj −2 (15) 2εj −εj + εk −1 (16) 2εj −εj − εk +1 (17) 2εk εj − εk −1 (18) −2εk εj + εk +1 Thus the dual vector X∗α for Xα with respect to the Killing form is given by X∗α = { (1/c0)X−α if α is long, (1/(2c0))X−α if α is short. (5.2) Now if ∆(z(n))long (reps. ∆(z(n))short) is the set of long roots (reps. short roots) in ∆(z(n)) then the element ω in (2.7) is given by ω = ∑ γj∈∆(z(n)) X∗γj ⊗Xγj = 1 c0 ∑ γj∈∆(z(n))long X−γj ⊗Xγj + 1 2c0 ∑ γj∈∆(z(n))short X−γj ⊗Xγj . (5.3) Observe that ∆(z(n)) = {εj + εk \| 1 ≤ j < k ≤ i} ∪ {2εj \| 1 ≤ j ≤ i}. Thus, ∆(z(n))long = {2εj \| 1 ≤ j ≤ i} and ∆(z(n))short = {εj + εk \| 1 ≤ j < k ≤ i}. Therefore, (5.3) reads ω = 1 c0 i∑ j=1 X−2εj ⊗X2εj + 1 2c0 ∑ 1≤j<k≤i X−(εj+εk) ⊗Xεj+εk . Thus the τ2 map in (2.8) may be expressed as τ2(X) = 1 2 ( ad(X)2 ⊗ Id ) ω = 1 2c0 i∑ j=1 ad(X)2X−2εj ⊗X2εj + 1 4c0 ∑ 1≤j<k≤i ad(X)2X−(εj+εk) ⊗Xεj+εk . As for the case that q is of type Bn(n− 1), Proposition 7.3 of [35] showed that the τ2 map is not identically zero. Systems of Differential Operators and Generalized Verma Modules 23 5.1 Lowest weight vector for V (µ+ εγ) ∗ As for the case of Type 1b special constituent it is necessary to show that the linear map τ̃2\|V (µ+εγ)∗ : V (µ + εγ)∗ → P2(g(1)) is not identically zero. We will again achieve it by showing that τ̃2(Y ∗l )(X) is a non-zero polynomial on g(1), where Y ∗l is a lowest weight vec- tor for V (µ+ εγ)∗. When a special constituent is of Type 1b, as V (µ + εnγ) = lnγ ⊗ z(n), it was easy to find a lowest weight vector for V (µ+εnγ)∗. In contrast, in the present case, since V (µ+εγ) ( lγ⊗z(n), we cannot use the same idea. So our first goal is to find an explicit form of a lowest weight vector for V (µ+ εγ)∗. To do so we now observe a highest weight vector for V (µ+ εγ). If prlγ⊗z(n) : l ⊗ z(n) → lγ ⊗ z(n) is the projection map from l ⊗ z(n) onto lγ ⊗ z(n) then we claim that prlγ⊗z(n)(τ2(Xµ + Xεγ )) is a highest weight vector for V (µ + εγ). The following two technical lemmas will simplify the expositions of the proof. Lemma 5.1. For Z ∈ l and X1, X2 ∈ g(1), we have Z · ( ad(X1) ad(X2)⊗ Id ) ω = ( (ad([Z,X1]) ad(X2) + ad(X1) ad([Z,X2]))⊗ Id ) ω, (5.4) where dot (·) denotes the usual Lie algebra action on tensor products. Proof. As this lemma simply follows from the arguments used in the proof for Proposition 7.5 of [35], we omit the proof. � Lemma 5.2. We have prlγ⊗z(n)(τ2(Xµ +Xεγ )) = 1 2 ( prlγ⊗z(n)(ad(Xµ) ad(Xεγ )⊗ Id)ω + prlγ⊗z(n)(ad(Xεγ ) ad(Xµ)⊗ Id)ω ) . Proof. Since τ2(Xµ +Xεγ ) = 1 2 ( ad(Xµ +Xεγ )2 ⊗ Id ) ω = 1 2 ( (ad(Xµ)2 ⊗ Id)ω + (ad(Xµ) ad(Xεγ )⊗ Id)ω + (ad(Xεγ ) ad(Xµ)⊗ Id)ω + (ad(Xεγ )2 ⊗ Id)ω ) , (5.5) we have prlγ⊗z(n)(τ2(Xµ +Xεγ )) = 1 2 ( prlγ⊗z(n)(ad(Xµ)2 ⊗ Id)ω + prlγ⊗z(n)(ad(Xµ) ad(Xεγ )⊗ Id)ω + prlγ⊗z(n)(ad(Xεγ ) ad(Xµ)⊗ Id)ω + prlγ⊗z(n)(ad(Xεγ )2 ⊗ Id)ω ) . (5.6) Observe that prlγ⊗z(n)(ad(Xµ)2⊗ Id)ω) = 0. Indeed, as µ = ε1 +εi+1 and ω = (1/c0) i∑ j=1 X−2εj ⊗ X2εj + (1/2c0) ∑ 1≤j<k≤i X−(εj+εk) ⊗Xεj+εk (see (5.3)), we have ( ad(Xµ)2 ⊗ Id ) ω = 1 c0 i∑ j=1 ad(Xε1+εi+1)2X−2εj ⊗X2εj + 1 2c0 ∑ 1≤j<k≤i ad(Xε1+εi+1)2X−(εj+εk) ⊗Xεj+εk 24 T. Kubo = 1 c0 ad(Xε1+εi+1)2X−2ε1 ⊗X2ε1 = 1 c0 Nε1+εi+1,−2ε1Nε1+εi+1,−ε1+εi+1X2εi+1 ⊗X2ε1 . Clearly, X2εi+1 ⊗ X2ε1 ∈ lnγ ⊗ z(n). Thus, prlγ⊗z(n)(ad(Xµ)2 ⊗ Id)ω) = 0. It can be shown similarly that prlγ⊗z(n)(ad(Xεγ )2⊗ Id)ω) = 0. Now the proposed equality follows from (5.6). � Proposition 5.3. The vector prlγ⊗z(n)(τ2(Xµ +Xεγ )) is a highest weight vector for V (µ+ εγ). Proof. We start with showing that prlγ⊗z(n)(τ2(Xµ+Xεγ )) has weight µ+εγ . Since the l-action commutes with the projection map prlγ⊗z(n), by Lemma 5.2, for H ∈ h ⊂ l, we have H · prlγ⊗z(n)(τ2(Xµ +Xεγ )) = 1 2 ( prlγ⊗z(n)(H · ad(Xµ) ad(Xεγ )⊗ Id)ω + prlγ⊗z(n)(H · ad(Xεγ ) ad(Xµ)⊗ Id)ω ) = 1 2 ( prlγ⊗z(n) (( ad([H,Xµ]) ad(Xεγ ) + ad(Xµ) ad([H,Xεγ ]) ) ⊗ Id ) ω + prlγ⊗z(n) ( (ad([H,Xεγ ]) ad(Xµ) + ad(Xεγ ) ad([H,Xµ]))⊗ Id ) ω ) = (µ+ εγ)(H) 2 ( prlγ⊗z(n)(ad(Xµ) ad(Xεγ )⊗ Id)ω + prlγ⊗z(n)(ad(Xεγ ) ad(Xµ)⊗ Id)ω ) = (µ+ εγ)(H) prlγ⊗z(n)(τ2(Xµ +Xεγ )). Note that Lemma 5.1 is applied from line two to line three. Next we show that prlγ⊗z(n)(τ2(Xµ + Xεγ )) is a highest weight vector. Let α ∈ Π(l). As the l-action commutes with prlγ⊗z(n), we first observe Xα · τ2(Xµ +Xεγ ). It follows from (5.5) that Xα · τ2(Xµ +Xεγ ) is given by Xα · τ2(Xµ +Xεγ ) = 1 2 ( Xα · ( ad(Xµ)2 ⊗ Id ) ω +Xα · (ad(Xµ) ad(Xεγ )⊗ Id)ω +Xα · (ad(Xεγ ) ad(Xµ)⊗ Id)ω +Xα · ( ad(Xεγ )2 ⊗ Id ) ω ) . If Z = Xα in (5.4) then, as [Xα, Xµ] = 0, we obtain Xα · τ2(Xµ +Xεγ ) = 1 2 ( ad(Xµ) ad([Xα, Xεγ ])⊗ Id ) ω + ( ad([Xα, Xεγ ]) ad(Xµ)⊗ Id ) ω + ( ad([Xα, Xεγ ]) ad(Xεγ )⊗ Id ) ω + ( ad(Xεγ ) ad([Xα, Xεγ ])⊗ Id ) ω. (5.7) Recall from Tables 2 and 4 in [35, Section 6] that we have µ+ εγ = ε1 + ε2 with µ = ε1 + εi+1 and εγ = ε2 − εi+1. Since Π(l) = {εj − εj+1 : 1 ≤ j ≤ n − 1 with j 6= i} ∪ {2εn}, it follows that α+ εγ ∈ ∆ if and only if α = ε1 − ε2. So it suffices to consider the case that α = ε1 − ε2. As (ε1 − ε2) + 2(ε2 − εi+1) /∈ ∆, we have ad(Xε1−ε2) ad(Xε2−εi+1) = ad(Xε2−εi+1) ad(Xε1−ε2). Therefore, if α = ε1 − ε2 then, as µ = ε1 + εi+1 and εγ + α = ε1 − εi+1, (5.7) becomes Xε1−ε2 · τ2(Xε1+εi+1 +Xε2−εi+1) = Nε1−ε2,ε2−εi+1 2 (( ad(Xε1+εi+1) ad(Xε1−εi+1)⊗ Id ) ω + ( ad(Xε1−εi+1) ad(Xε1+εi+1)⊗ Id ) ω + ( 2 ad(Xε2−εi+1) ad(Xε1−εi+1)⊗ Id ) ω ) . Systems of Differential Operators and Generalized Verma Modules 25 Table 7 shows that Nε1−ε2,ε2−εi+1 = 1. Therefore, Xε1−ε2 · τ2(Xε1+εi+1 +Xε2−εi+1) = 1 2 (( ad(Xε1+εi+1) ad(Xε1−εi+1)⊗ Id ) ω + ( ad(Xε1−εi+1) ad(Xε1+εi+1)⊗ Id ) ω + ( 2 ad(Xε2−εi+1) ad(Xε1−εi+1)⊗ Id ) ω ) . Now we consider the contribution from each term separately. A direct computation shows that the first term is given by T1 = ( ad(Xε1+εi+1) ad(Xε1−εi+1)⊗ Id ) ω = 1 c0 i∑ j=1 ad(Xε1+εi+1) ad(Xε1−εi+1)X−2εj ⊗X2εj + 1 2c0 ∑ 1≤j<k≤i ad(Xε1+εi+1) ad(Xε1−εi+1)X−(εj+εk) ⊗Xεj+εk = 1 c0 Nε1−εi+1,−2ε1Hε1+εi+1 ⊗X2ε1 + 1 2c0 i∑ k=2 Nε1−εi+1,−(ε1+εk)Nε1+εi+1,−(εk+εi+1)Xε1−εk ⊗Xε1+εk = −1 c0 Hε1+εi+1 ⊗X2ε1 − 1 2c0 i∑ k=2 Xε1−εk ⊗Xε1+εk . Similarly we have T2 = ( ad(Xε1−εi+1) ad(Xε1+εi+1)⊗ Id ) ω = 1 c0 Hε1−εi+1 ⊗X2ε1 + 1 2c0 i∑ k=2 Xε1−εk ⊗Xε1+εk and T3 = ( 2 ad(Xε2−εi+1) ad(Xε1−εi+1)⊗ Id ) ω = 4 c0 X2εi+1 ⊗X−(ε1+ε2). Therefore, Xε1−ε2 · τ2(Xε1+εi+1 +Xε2−εi+1) = T1 + T2 + T3 = −1 c0 Hε1+εi+1 ⊗X2ε1 − 1 2c0 i∑ k=2 Xε1−εk ⊗Xε1+εk + 1 c0 Hε1−εi+1 ⊗X2ε1 + 1 2c0 i∑ k=2 Xε1−εk ⊗Xε1+εk + 4 c0 X2εi+1 ⊗X−(ε1+ε2) = 1 c0 (Hε1−εi+1 −Hε1+εi+1)⊗X2ε1 + 4 c0 X2εi+1 ⊗X−(ε1+ε2). Observe that h∩ lγ is spanned by the elements Hεj−εj+1 = (Ejj−Eĵ,ĵ)−(Ej+1,j+1−Eĵ+1,ĵ+1 ) for 1 ≤ j ≤ i−1. Since Hε1−εi+1−Hε1+εi+1 = −2(Ei+1,i+1−Eî+1,̂i+1 ), it follows that Hε1−εi+1− Hε1+εi+1 /∈ h ∩ lγ . As prlγ⊗z(n)(X2εi+1 ⊗X−(ε1+ε2)) = 0, we then obtain Xε1−ε2 · prlγ⊗z(n)(τ2(Xε1+εi+1 +Xε2−εi+1)) = prlγ⊗z(n)(Xε1−ε2 · τ2(Xε1+εi+1 +Xε2−εi+1)) = 1 c0 prlγ⊗z(n) ( (Hε1−εi+1 −Hε1+εi+1)⊗X2ε1 ) + 4 c0 prlγ⊗z(n) ( X2εi+1 ⊗X−(ε1+ε2) ) = 0. � 26 T. Kubo Now we define the “opposite” τ2 map τ̄2 by τ̄2 : g(−1)→ g(0)⊗ g(−2), X∗ 7→ 1 2 ( ad(X∗)2 ⊗ Id ) ω̄ with ω̄ = ∑ γj∈z(n) Xγ ⊗X∗γ = 1 c0 i∑ j=1 X2εj ⊗X−2εj + 1 2c0 ∑ 1≤j<k≤i Xεj+εk ⊗X−(εj+εk). It follows from the same arguments in the proof for Lemma 3.3 and Proposition 7.3 in [35] that the τ̄2 map is not identically zero and L-equivariant. Let prlγ⊗z(n̄) : l ⊗ z(n̄) → lγ ⊗ z(n̄) be the projection map from l⊗ z(n̄) onto lγ ⊗ z(n̄). Proposition 5.4. The vector prlγ⊗z(n̄)(τ̄2(X−µ+X−εγ )) is a lowest weight vector for V (µ+εγ)∗. Proof. This proposition immediately follows from the arguments used in the proof for Propo- sition 5.3, by replacing positive (resp. negative) roots with negative (resp. positive) roots. � We set Y ∗l := 8c2 0 i+ 1 prlγ⊗z(n̄)(τ̄2(X−µ +X−εγ )). (5.8) It follows from Proposition 5.4 that Y ∗l is a lowest weight vector for V (µ + εγ)∗. In the next subsection we compute τ̃2(Y ∗l )(X). To the end we give an explicit form for Y ∗l . Lemma 5.5. We have Y ∗l = 4c0 i+ 1 Xε1−ε2 ⊗X−2ε1 − 4c0 i+ 1 X−(ε1−ε2) ⊗X−2ε2 + 2c0 i+ 1 i∑ k=3 X−(ε2−εk) ⊗X−(ε1+εk) − 2c0 i+ 1 i∑ k=3 X−(ε1−εk) ⊗X−(ε2+εk) − 2c0 i+ 1 Hε1−ε2 ⊗X−(ε1+ε2). Proof. By using the same arguments for Lemma 5.2, one can obtain prlγ⊗z(n̄)(τ2(X−µ +X−εγ )) = 1 2 ( prlγ⊗z(n̄)(ad(X−µ) ad(X−εγ )⊗ Id)ω̄ + prlγ⊗z(n̄)(ad(X−εγ ) ad(X−µ)⊗ Id)ω̄ ) . (5.9) A direct computation shows that prlγ⊗z(n̄)(ad(X−µ) ad(X−εγ )⊗ Id)ω̄) = 1 c0 i∑ j=1 prlγ⊗z(n̄) ( ad(X−(ε1+εi+1)) ad(X−(ε2−εi+1))X2εj ⊗X−2εj ) + 1 2c0 ∑ 1≤j<k≤i prlγ⊗z(n̄) ( ad(X−(ε1+εi+1)) ad(X−(ε2−εi+1))Xεj+εk ⊗X−(εj+εk) ) = − 1 c0 X−(ε1−ε2) ⊗X−2ε2 − 1 2c0 prlγ⊗z(n̄)(Hε1+εi+1 ⊗X−(ε1+ε2)) − 1 2c0 i∑ k=3 X−(ε1−εk) ⊗X−(ε2+εk). Systems of Differential Operators and Generalized Verma Modules 27 Observe that we have Hε1+εi+1 = 1 i i∑ j=1 (Ejj − Eĵĵ) + 1 i i∑ k=2 Hε1−εk +H2εi+1 . Since (1/i) i∑ j=1 (Ejj − Eĵĵ) ∈ z(l) and H2εi+1 ∈ h ∩ lnγ , it follows that prlγ⊗z(n̄)(Hε1+εi+1 ⊗X−(ε1+ε2)) = 1 i i∑ k=2 Hε1−εk ⊗X−(ε1+ε2). Therefore, prlγ⊗z(n̄)(ad(X−µ) ad(X−εγ )⊗ Id)ω̄) is given by prlγ⊗z(n̄)(ad(X−µ) ad(X−εγ )⊗ Id)ω̄) = − 1 c0 X−(ε1−ε2) ⊗X−2ε2 − 1 2ic0 i∑ k=2 Hε1−εk ⊗X−(ε1+ε2) − 1 2c0 i∑ k=3 X−(ε1−εk) ⊗X−(ε2+εk). (5.10) Similarly we have prlγ⊗z(n̄)(ad(X−εγ ) ad(X−µ)⊗ Id)ω̄) = 1 c0 Xε1−ε2 ⊗X−2ε1 − 1 2ic0 (Hε1−ε2 − i∑ k=3 Hε2−εk)⊗X−(ε1+ε2) + 1 2c0 i∑ k=3 X−(ε2−εk) ⊗X−(ε1+εk). (5.11) By substituting (5.10) and (5.11) into (5.9) and multiplying the resulting equation by 8c2 0/(i+1), one obtains Y ∗l = 4c0 i+ 1 Xε1−ε2 ⊗X−2ε1 − 4c0 i+ 1 X−(ε1−ε2) ⊗X−2ε2 + 2c0 i+ 1 i∑ k=3 X−(ε2−εk) ⊗X−(ε1+εk) − 2c0 i+ 1 i∑ k=3 X−(ε1−εk) ⊗X−(ε2+εk) − 2c0 i(i+ 1) ( i∑ k=2 Hε1−εk +Hε1−ε2 − i∑ k=3 Hε2−εk)⊗X−(ε1+ε2). Now the proposed equality follows from manipulating the elements in the Cartan subalgebra. � 5.2 The τ̃2\|V (µ+εγ)∗ map Now we show that the map τ̃2\|V (µ+εγ)∗ is not identically zero. To do so, we recall several essential ingredients. First observe that, as the duality is with respect to the Killing form κ, if Y ∗ = Xα ⊗ X−γj ∈ lγ ⊗ z(n̄) and Xβ ⊗ Xγk ∈ lγ ⊗ z(n) then Y ∗(Xβ ⊗ Xγk) is given by Y ∗(Xβ ⊗Xγk) = κ(Xα, Xβ)κ(X−γk , Xγj ). As observed in (5.1), we have κ(Xα, X−α) = { c0 if α is long, 2c0 if α is short. Finally we recall from (3.1) that if W ⊂ g is an ad(h)-invariant subspace then, for any weight ν ∈ h∗, we write ∆ν(W ) = {α ∈ ∆(W ) \| ν − α ∈ ∆}. 28 T. Kubo Proposition 5.6. The L-intertwining operator τ̃2\|V (µ+εγ)∗ is not identically zero. Proof. Take lowest weight vector Y ∗l as in (5.8). We show that τ̃2(Y ∗l )(X) is a non-zero poly- nomial on g(1). As τ2(X) = (1/2) ∑ γj∈∆(z(n)) ad(X)2X∗γj ⊗ Xγj , by Lemma 5.5, the polynomial τ̃2(Y ∗l )(X) may express as a sum of five terms. We consider the contribution from each term separately, and start with observing the contribution from the first term T1 = 2c0 i+ 1 ∑ γj∈∆(z(n)) κ ( Xε1−ε2 , ad(X)2X∗γj ) κ(X−2ε1 , Xγj ) = 2c0 i+ 1 κ ( Xε1−ε2 , ad(X)2X−2ε1 ) . (5.12) Write X = ∑ α∈∆(g(1)) ηαXα, where ηα ∈ n∗ is the coordinate dual to Xα with respect to the Killing form. Then, (5.12) = 2c0 i+ 1 κ ( Xε1−ε2 , ad(X)2X−2ε1 ) = 2c0 i+ 1 ∑ α,β∈∆(g(1)) ηαηβκ(Xε1−ε2 , [Xβ, [Xα, X−2ε1 ]]) = 2c0 i+ 1 ∑ α,β∈∆(g(1)) ηαηβκ([Xε1−ε2 , Xβ], [Xα, X−2ε1 ]) = 2c0 i+ 1 ∑ α∈∆2ε1 (g(1)) β∈∆ε1−ε2 (g(1)) Nε1−ε2,βNα,−2ε1ηαηβκ(Xβ+(ε1−ε2), Xα−2ε1), (5.13) where ∆ε1−ε2(g(1)) = {α ∈ ∆(g(1)) \| (ε1−ε2)+α ∈ ∆}. Observe that κ(Xβ+(ε1−ε2), Xα−2ε1) 6= 0 if and only if α ∈ ∆2ε1(g(1)). Indeed, first one may see from (5.13) that κ(Xβ+(ε1−ε2), Xα−2ε1) 6= 0 if and only if β = (ε1+ε2)−α; equivalently, the value of the Killing form is non-zero if and only if α ∈ ∆2ε1(g(1))∩∆ε1+ε2(g(1)). By inspection, we have ∆2ε1(g(1)) = {ε1± εj \| i+ 1 ≤ j ≤ n}. Thus, for any α ∈ ∆2ε1(g(1)), it follows that (ε1 + ε2) − α ∈ ∆. Therefore ∆2ε1(g(1)) ∩ ∆ε1+ε2(g(1)) = ∆2ε1(g(1)). Now, if (α, β) denotes a pair so that κ(Xβ+(ε1−ε2), Xα−2ε1) 6= 0 then (α, β) = (ε1 ± εj , ε2 ∓ εj) for i+ 1 ≤ j ≤ n with respect to the signs. Therefore, (5.13) = 2c0 i+ 1 ∑ α∈∆2ε1 (g(1)) β∈∆ε1−ε2 (g(1)) Nε1−ε2,βNα,−2ε1ηαηβκ(Xβ+(ε1−ε2), Xα−2ε1) = 2c0 i+ 1 ∑ α∈∆2ε1 (g(1)) Nε1−ε2,(ε1+ε2)−αNα,−2ε1ηαη(ε1+ε2)−ακ(X2ε1−α, Xα−2ε1) = 2c0 i+ 1 n∑ j=i+1 Nε1−ε2,ε2−εjNε1+εj ,−2ε1ηε1+εjηε2−εjκ(Xε1−εj , X−(ε1−εj)) + 2c0 i+ 1 n∑ j=i+1 Nε1−ε2,ε2+εjNε1−εj ,−2ε1ηε1−εjηε2+εjκ(Xε1+εj , X−(ε1+εj)) = 2c0 i+ 1 n∑ j=i+1 (1)(1)ηε1+εjηε2−εjκ(Xε1−εj , X−(ε1−εj)) + 2c0 i+ 1 n∑ j=i+1 (1)(−1)ηε1−εjηε2+εjκ(Xε1+εj , X−(ε1+εj)) Systems of Differential Operators and Generalized Verma Modules 29 = 4c2 0 i+ 1 n∑ j=i+1 ηε1+εjηε2−εj − ηε1−εjηε2+εj = 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ). By a similar computation one obtains T2 = −2c0 i+ 1 ∑ γj∈∆(z(n)) κ ( X−(ε1−ε2), ad(X)2X∗γj ) κ(X−2ε2 , Xγj ) = 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ), T3 = c0 i+ 1 ∑ γj∈∆(z(n)) i∑ k=3 κ ( X−(ε2−εk), ad(X)2X∗γj ) κ(X−(ε1+εk), Xγj ) = 2(i− 2)c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ), T4 = −c0 i+ 1 ∑ γj∈∆(z(n)) i∑ k=3 κ ( X−(ε1−εk), ad(X)2X∗γj ) κ(X−(ε2+εk), Xγj ) = 2(i− 2)c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ), and T5 = −c0 i+ 1 ∑ γj∈∆(z(n)) κ ( Hε1−ε2 , ad(X)2X∗γj ) κ(X−(ε1+ε2), Xγj ) = 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ). Therefore τ̃2(Y ∗l )(X) may be given as τ̃2 ( Y ∗l ) (X) = T1 + T2 + T3 + T4 + T5 = 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ) + 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ) + 2(i− 2)c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ) + 2(i− 2)c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ) + 4c2 0 i+ 1 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ) = 4c2 0 n∑ j=i+1 κ(X,X∗ε1+εj )κ(X,X∗ε2−εj )− κ(X,X∗ε1−εj )κ(X,X∗ε2+εj ). (5.14) Hence τ̃2(Y ∗l )(X) is a non-zero polynomial on g(1). � 30 T. Kubo 5.3 The special value Now we are going to find the special value for the Ω2\|V (µ+εγ)∗ system. As for Type 1b case, to find the special value, we use Proposition 2.7. Recall from Section 2.3 the linear map ω2\|V (µ+εγ)∗ : V (µ+ εγ)∗ → U(n̄) defined by ω2\|V (µ+εnγ)∗ = σ ◦ τ̃2\|V (µ+εnγ)∗ , where σ : Sym2(g(−1))→ U(n̄) is the symmetrization operator. If Y ∗l is the lowest weight vector for V (µ + εγ)∗ defined in (5.8) then it follows from (5.14) that ω(Y ∗l ) := ωV (µ+εγ)∗ is given by ω ( Y ∗l ) = 4c2 0 n∑ j=i+1 σ(X∗ε1+εjX ∗ ε2−εj )− σ(X∗ε1−εjX ∗ ε2+εj ). By (5.2), this amounts to ω(Y ∗l ) = n∑ j=i+1 σ ( X−(ε1+εj)X−(ε2−εj ) ) − σ ( X−(ε1−εj)X−(ε2+εj) ) . The following lemma will simplify arguments for a proof for Theorem 5.8 below. Lemma 5.7. For X,Y, Z ∈ g, in U(g), we have X · σ(Y Z) = σ([X,Y ]Z) + σ(Y [X,Z]). Proof. A direct computation. � Now we are ready to determine the special value for the Ω2\|V (µ+εγ)∗ system. Theorem 5.8. Let q be the maximal parabolic subalgebra of type Cn(i) for 2 ≤ i ≤ n− 1. The Ω2\|V (µ+εγ)∗ system is conformally invariant on Ls if and only if s = n− i+ 1. Proof. By Proposition 2.7, to prove this theorem, it suffices to show that Xµ ·(ω(Y ∗l )⊗1−s) = 0 in Mq(C−s) if and only if s = n−i+1, where µ is the highest weight for g(1). It follows from (5.3) that Xµ · (ω(Y ∗l )⊗ 1−s) may be a sum of two terms. As µ = ε1 + εi+1, the first term is T1 = n∑ j=i+1 ( Xε1+εi+1 · σ(X−(ε1+εj)X−(ε2−εj)) ) ⊗ 1−s. By Lemma 5.7, this may be expressed as T1 = n∑ j=i+1 σ([Xε1+εi+1 , X−(ε1+εj)]X−(ε2−εj))⊗ 1−s + n∑ j=i+1 σ(X−(ε1+εj)[Xε1+εi+1 , X−(ε2−εj)])⊗ 1−s = n∑ j=i+1 σ([Xε1+εi+1 , X−(ε1+εj)]X−(ε2−εj)) = σ(Hε1+εi+1X−(ε2−εj))⊗ 1−s + n∑ j=i+2 Nε1+εi+1,−(ε1+εj)σ(X−(εj−εi+1)X−(ε2−εj))⊗ 1−s. (5.15) Since σ(ab) = (1/2)(ab+ ba), we have (5.15) = σ(Hε1+εi+1X−(ε2−εj))⊗1−s+ n∑ j=i+2 Nε1+εi+1,−(ε1+εj)σ(X−(εj−εi+1)X−(ε2−εj))⊗1−s Systems of Differential Operators and Generalized Verma Modules 31 = 1 2 ( Hε1+εi+1X−(ε2−εj) +X−(ε2−εj)Hε1+εi+1 ) ⊗ 1−s (5.16) + 1 2 n∑ j=i+2 Nε1+εi+1,−(ε1+εj) ( X−(εj−εi+1)X−(ε2−εj) +X−(ε2−εj)X−(εj−εi+1) ) ⊗ 1−s. If λi = i∑ j=1 εj is the fundamental weight for αi then, as H · 1−s = λi(H)1−s for H ∈ h, a direct computation shows that (5.16) = 1 2 ( Hε1+εi+1X−(ε2−εj) +X−(ε2−εj)Hε1+εi+1 ) ⊗ 1−s + 1 2 n∑ j=i+2 Nε1+εi+1,−(ε1+εj) ( X−(εj−εi+1)X−(ε2−εj) +X−(ε2−εj)X−(εj−εi+1) ) ⊗ 1−s = −1 2 (ε2 − εi+1)(Hε1+εi+1)X−(ε2−εi+1) ⊗ 1−s − sλi(Hε1+εi+1)X−(ε2−εi+1) ⊗ 1−s + 1 2 n∑ j=i+2 Nε1+εi+1,−(ε1+εj)Nεi+1−εj ,−(ε2−εj)X−(ε2−εi+1) ⊗ 1−s = −1 2 (1)X−(ε2−εi+1) ⊗ 1−s − s(1)X−(ε2−εi+1) ⊗ 1−s + 1 2 n∑ j=i+2 (1)(1)X−(ε2−εi+1) ⊗ 1−s = −1 2 (2s− n+ i)X−(ε2−εi+1) ⊗ 1−s. Similarly, by a direct computation, the second term amounts to T2 = − n∑ j=i+1 ( Xµ · σ(X−(ε1−εj)X−(ε2+εj)) ) ⊗ 1−s = −1 2 Nε1+εi+1,−(ε2+εi+1)Nε1−ε2,−(ε1−εi+1)X−(ε2−εi+1) ⊗ 1−s − 1 2 n∑ j=i+1 Nε1+εi+1,−(εi−εj)Nεi+1+εj ,−(ε2+εj)X−(ε2−εi+1) ⊗ 1−s = −1 2 (1)(−1)X−(ε2−εi+1) ⊗ 1−s − 1 2 (−2)(1)X−(ε2−εi+1) ⊗ 1−s − 1 2 n∑ j=i+2 (−1)(1)X−(ε2−εi+1) ⊗ 1−s = 1 2 (n− i+ 2)X−(ε2−εi+1) ⊗ 1−s. Therefore, Xµ · (ω(Y ∗l )⊗ 1−s) is given by Xµ · (ω ( Y ∗l ) ⊗ 1−s) = T1 + T2 = −1 2 (2s− n+ i)X−(ε2−εi+1) ⊗ 1−s + 1 2 (n− i+ 2)X−(ε2−εi+1) ⊗ 1−s = −1 2 (2s− n+ i− (n− i+ 2))X−(ε2−εi+1) ⊗ 1−s = −(s− (n− i+ 1))X−(ε2−εi+1) ⊗ 1−s. Hence Xµ · (ω(Y ∗l )⊗ 1−s) = 0 if and only if s = n− i+ 1. � 32 T. Kubo 5.4 The standardness of the map ϕΩ2 In the remainder of this section we determine the standardness of the map ϕΩ2 coming from the conformally invariant Ω2\|V (µ+εγ)∗ system. Observe that if w0 is the longest Weyl group element for lγ then the highest weight ν for V (µ+εγ)∗ = V (ε1 +ε2)∗ is ν = −w0(ε1 +ε2) = −εi−1−εi. By Theorem 5.8, the special value s2 for the Ω2\|V (µ+εnγ)∗ system is s2 = n− i+ 1. Therefore, by (2.5), the Ω2\|V (µ+εnγ)∗ system yields a non-zero U(g)-homomorphism ϕΩ2 : Mq((−εi−1 − εi)− (n− i+ 1)λi + ρ)→Mq(−(n− i+ 1)λi + ρ). (5.17) Theorem 5.9. If q is the maximal parabolic subalgebra of type Cn(i) for 2 ≤ i ≤ n− 1 then the standard map ϕstd between the generalized Verma modules in (5.17) is zero. Consequently, the map ϕΩ2 is non-standard. Proof. By using the argument similar to one given in the proof for Theorem 3.3, one can easily show that (εi−1 − εi, εi − εn) links −2εn − (n− i+ 1)λi + ρ to (−εi=1 − εi)− (n− i+ 1)λi + ρ. Now the theorem follows from Proposition 2.4. � A Miscellaneous data This appendix summarizes the miscellaneous data for the maximal parabolic subalgebras q = l⊕ g(1)⊕ z(n) of types Bn(n− 1), Cn(i) (2 ≤ i ≤ n− 1), and Dn(n− 2). For the data for other maximal parabolic subalgebras of quasi-Heisenberg type see, for example, Appendix A of [35]. A.1 Bn(n− 1) 1. The deleted Dynkin diagram: ◦ α1 ◦ α2 · · · ◦ αn−2 ⊗ αn−1 +3◦ αn 2. The subgraph for lγ : ◦ α1 ◦ α2 ◦ α3 · · · ◦ αn−2 3. The subgraph for lnγ : ◦ αn We have αγ = α2. The highest weight µ and the set of weights ∆(g(1)) for g(1) are µ = ε1+εn and ∆(g(1)) = {εj ± εn \| 1 ≤ j ≤ n − 1} ∪ {εj \| 1 ≤ j ≤ n − 1}. The highest weight γ and the set of weights g(z(n)) for z(n)) are γ = ε1 + ε2 and ∆(z(n)) = {εj + εk \| 1 ≤ j < k ≤ n − 1}. The highest root ξγ and the set of positive roots ∆+(lγ) for lγ are ξγ = ε1 − εn−1 and ∆+(lγ) = {εj − εk \| 1 ≤ j < k ≤ n− 1}. The highest root ξnγ and the set of positive roots ∆+(lnγ) for lnγ are ξnγ = εn and ∆+(lnγ) = {εn}. A.2 Cn(i), 2 ≤ i ≤ n− 1 1. The deleted Dynkin diagram: ◦ α1 · · · ◦ αi−1 ⊗ αi ◦ αi+1 · · · ◦ αn−1 ks ◦ αn Systems of Differential Operators and Generalized Verma Modules 33 2. The subgraph for lγ : ◦ α1 ◦ α2 ◦ α3 · · · ◦ αi−1 3. The subgraph for lnγ : ◦ αi+1 · · · ◦ αn−1 ks ◦ αn We have αγ = α1. The highest weight µ and the set of weights ∆(g(1)) for g(1) are µ = ε1 + εi+1 and ∆(g(1)) = {εj ± εk \| 1 ≤ j ≤ i and i+ 1 ≤ k ≤ n}. The highest weight γ and the set of weights ∆(z(n)) for z(n) are γ = 2ε1 ∆(z(n)) = {εj +εk \| 1 ≤ j < k ≤ i}∪{2εj \| 1 ≤ j ≤ i}. The highest root ξγ and the set of positive roots ∆+(lγ) for lγ are ξγ = ε1 − εi and ∆+(lγ) = {εj − εk \| 1 ≤ j < k ≤ i} The highest root ξnγ and the set of positive roots ∆(lnγ) for lnγ are ξnγ = 2εi+1 and ∆+(lnγ) = {εj ± εk \| i+ 1 ≤ j < k ≤ n} ∪ {2εj \| i+ 1 ≤ j ≤ n}. A.3 Dn(n− 2) 1. The deleted Dynkin diagram: αn−1◦ ◦ α1 · · · ◦ αn−3 ⊗ αn−2 ◦ αn 2. The subgraph for lγ : ◦ α1 ◦ α2 ◦ α3 · · · ◦ αn−3 3. The subgraph for l−nγ : ◦ αn−1 4. The subgraph for l+nγ : ◦ αn We have αγ = α2. The highest weight µ and the set of weights ∆(g(1)) for g(1) are µ = ε1 + εn−1 and ∆(g(1)) = {ej ± ek \| 1 ≤ j ≤ n− 2 and k = n− 1, n}. The highest weight γ and the set of weights ∆(z(n)) for z(n) are γ = ε1 + ε2 and ∆(z(n)) = {ej + ek \| 1 ≤ j < k ≤ n− 2}. 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Soc. 251 (1979), 19–37. http://dx.doi.org/10.1016/S0001-8708(03)00012-4 http://arxiv.org/abs/math.RT/0111083 http://dx.doi.org/10.1016/S0001-8708(03)00013-6 http://arxiv.org/abs/math.RT/0111085 http://dx.doi.org/10.1016/S0001-8708(03)00014-8 http://arxiv.org/abs/math.RT/0111086 http://arxiv.org/abs/1305.6040 http://arxiv.org/abs/1301.2111 http://dx.doi.org/10.1007/s002229900030 http://dx.doi.org/10.2140/pjm.2011.253.439 http://arxiv.org/abs/1104.1999 http://arxiv.org/abs/1209.5516 http://arxiv.org/abs/1209.1861 http://dx.doi.org/10.1016/0021-8693(77)90254-X http://dx.doi.org/10.1215/S0012-7094-05-13113-1 http://arxiv.org/abs/math.RT/0309454 http://arxiv.org/abs/1205.6748 http://arxiv.org/abs/1303.7311 http://dx.doi.org/10.2307/1998680 1 Introduction 2 Preliminaries 2.1 A specialization of a vector bundle V M 2.2 The k systems 2.3 The k systems and generalized Verma modules 2.4 The 2 systems associated to maximal parabolic subalgebras of quasi-Heisenberg type 3 Parabolic subalgebra of type Dn(n-2) 3.1 Special constituents and special values 3.2 The standardness of the map 2 4 Type 1b special constituent 4.1 The 2\|V(+n)* map 4.2 The special value of the 2\|V(+n)* system 4.3 The standardness of the map 2 5 Type 3 special constituent 5.1 Lowest weight vector for V(+)* 5.2 The 2\|V(+)* map 5.3 The special value 5.4 The standardness of the map 2 A Miscellaneous data A.1 Bn(n-1) A.2 Cn(i), 2i n-1 A.3 Dn(n-2) References

Systems of Differential Operators and Generalized Verma Modules

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