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 determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.


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 g 0 a Lie algebra of first-order differential operators acting on smooth sections on V. A linearly independent list D 1 , . . . , D m of differential operators on V is then said to be conformally invariant if, for each X ∈ g 0 , the condition [X, D i ] ∈ Many examples of conformally invariant systems implicitly or explicitly exist in the literature, especially in the case of m = 1. The Laplacian ∆ on R n and wave operator on the Minkowski space R 3,1 are, for instance, two outstanding examples. The Yamabe operator on S n 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 arXiv:1304.1646v5 [math.RT] 24 Jan 2014 agrees with the notion of conformal invariance introduced by Ehrenpreis [12]. As a generalization 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 geometricanalytic 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 representations) 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 homomorphisms 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 = N G (q) = LN . For a real form g 0 of g, define G 0 to be an analytic subgroup of G with Lie algebra g 0 . Set Q 0 = N G 0 (q). We consider a line bundle L s → G 0 /Q 0 for s ∈ C. As the homogeneous space G 0 /Q 0 admits an open dense submanifoldN 0 Q 0 /Q 0 , we restrict our bundle to this submanifold. By slight abuse of notation we refer to the restricted bundle as L s . The systems that we shall construct act on smooth sections of the restricted bundle L s →N 0 .
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 Ω k systems. (We shall describe the construction more precisely in Section 2.2.) An Ω k system is a system of kth-order differential operators. There is no reason to expect that Ω k systems are conformally invariant on L s for arbitrary s ∈ C; the conformal invariance of Ω k systems depends on the complex parameter s for the line bundle L s . We then say that an Ω k system has special value s k if the system is conformally invariant on the line bundle L s k .
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 Ω k systems for k ≥ 5 are zero. We then sought the special values for the Ω 1 system and Ω 2 systems. While the special value s 1 for the Ω 1 system was determined for each parabolic subalgebra q as s 1 = 0, we left three cases open for Ω 2 systems. To describe the open cases, let us now start explaining briefly the classification for the irreducible constituents W that contribute to Ω 2 systems. In [35], we first observed that if irreducible constituents W contribute to Ω 2 systems then their highest weights are of the form µ+ , where µ is the highest weight for g (1) and is some weight for g (1). We called such irreducible constituents special and classified as Type 1a, Type 1b, Type 2, and Type 3 with respect to certain technical conditions for the highest weight µ + . (The precise conditions will be given in Definition 2.6.) Table 1 summarizes the types of special constituents. Here, for example, "B n (i)" indicates the maximal standard parabolic subalgebra of g of type B n , which is determined by the ith simple root α i . In the table V (µ+ γ ), V (µ+ nγ ), and V (µ+ ± nγ ) denote the special constituents with highest weights µ + γ , µ + nγ , and µ + ± 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 + nγ ) indicates that there is no special constituent V (µ + nγ ) in the case.
In [35], under the assumption that q is not of type D n (n − 2), we found the special values s 2 for Ω 2 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.
These are the cases boxed in Table 1. Our goal in this article is to find the special values s 2 of Ω 2 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 D n (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.
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 Ω 2 | V (µ+ ) * denotes the Ω 2 system coming from the special constituent V (µ + ) then Table 2 exhibits the special values for all the Ω 2 systems under consideration. Parabolic subalgebra Here λ i denotes the fundamental weight for the simple root α i and L(sλ i ) := L s , the restricted line bundle overN 0 . 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 otherwise [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 ϕ Ω k between generalized Verma modules arising from the conformally invariant Ω 1 and Ω 2 systems. While the map ϕ Ω 1 was shown to be standard for each parabolic subalgebra q, the classification for the map ϕ Ω 2 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 ϕ Ω 2 coming from the conformally invariant Ω 2 systems in the three cases are standard.
The classification results are given in Theorem 3.3 (type D n (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 ϕ Ω 2 is nonstandard. With the results from [34], Table 3 summarizes the classification of the standardness of the maps ϕ Ω 2 . Table 3. The classification of ϕ Ω2 .
Parabolic subalgebra Recall that, for each parabolic subalgebra q, the special value s 1 for the Ω 1 system is s 1 = 0 and that the map ϕ Ω 1 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 ϕ Ω k for k = 1, 2 is non-standard if and only if the special value s k of the Ω k system is a positive integer.
Here we wish to note that there is another interesting relationship between the special values of Ω k 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 Ω k systems were accounted by the special values of the systems. In this case, as of quasi-Heisenberg case, the Ω k systems for k ≥ 5 are zero. To obtain the account the special values of Ω 3 systems and Ω 4 systems as well as these of the Ω 1 system and Ω 2 systems were used. In the quasi-Heisenberg case the results in [35] and in this paper for the Ω 1 system and Ω 2 systems do not give an account for the first reducibility points. The special values of Ω 3 systems and Ω 4 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 understand that this introduction is already long enough. The first is on the technique to determine the special values of Ω 2 systems. In [35], to determine the special values, we used some reduction techniques. Although the techniques significantly reduced the amount of computations, as several 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.
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 Ω 2 systems as ones whose highest weights satisfy certain conditions. On the other hand, in [34], we redefined such constituents for any Ω k systems as irreducible constituents that contribute to Ω k systems. The reason why we redefined special constituents as in [34] is that the later definition works not only for Ω 2 systems but also for any Ω k systems, and also that the irreducible constituents with highest weights satisfying the technical conditions are highly expected to contribute to Ω 2 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 D n (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 Ω 2 systems.
We now outline the rest of this paper. This paper consists of five sections (with this introduction) and one appendix. In Section 2 we review the works [35] and [34]. In particular we give the precise construction of Ω k systems. We also review about maximal parabolic subalgebras q of quasi-Heiseberg type in this section. In Section 3, when q is of type D n (n − 2), we find the special values of the Ω 2 systems and determine the standardness of the map ϕ Ω 2 . The special values are given in Corollary 3.2 together with Theorem 3.1. The standardness of ϕ Ω 2 is determined in Theorem 3.3. Sections 4 and 5 are devoted to the Ω 2 systems arising from Type 1b special constituent and Type 3 special constituent, respectively. The special values and the standardness of ϕ Ω 2 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.

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.

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 = N G (q) then write Q = LN for the Levi decomposition of Q. Let g 0 be a real form of g in which the complex parabolic subalgebra q has a real form q 0 . Let G 0 be the analytic subgroup of G with Lie algebra g 0 . Define Q 0 = N G 0 (q) ⊂ Q, and write Q 0 = L 0 N 0 . We will work with G 0 /Q 0 for a class of maximal parabolic subgroup Q 0 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 g α for the root space for α ∈ ∆. For each subset S ⊂ Π, let q S be the corresponding standard parabolic subalgebra. Write q S = l S ⊕n S with Levi factor l S = h⊕ α∈∆ S g α and nilpotent radical n S = α∈∆ + \∆ S g α , where ∆ S = {α ∈ ∆ | α ∈ span(Π\S)}. If q is a maximal parabolic subalgebra of g then there exists a unique simple root α q ∈ Π so that q = q {αq} . Let λ q be the fundamental weight of α q . The weight λ q is orthogonal to any roots α with g α ⊂ [l, l]. Hence it exponentiates to a character χ q of L. For s ∈ C, we set χ s := |χ q | s , an analytic character for L 0 . As χ q takes real values on L 0 , we have dχ s = sλ q . Let C χ s be the one-dimensional representation of L 0 with character χ s . The representation χ s is extended to a representation of Q 0 by making it trivial on N 0 . It then deduces a line bundle L s on G 0 /Q 0 with fiber C χ s .
The group G 0 acts on the space arising from this action is given by for Y ∈ g 0 , 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). LetN 0 be the unipotent subgroup opposite to N 0 . The natural infinitesimal action of g on the image of the restriction map C ∞ gives an action of g on the whole space C ∞ (N 0 , C χ s ) (see, for instance, Section 2 of [35]). The line bundle We shall work with the trivial bundle onN 0 . By slight abuse of notation, we refer to the trivial bundle overN 0 as L s .

The Ω k systems
A systematic construction of systems of differential operators for the line bundle L s →N 0 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 ∈ ∆ | g α ⊂ 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) = 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 g 0 . 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 where ω is the element in g(−r) ⊗ g(r) defined by Here X γ j are root vectors for γ j , and X * γ j are the vectors dual to X γ j with respect to the Killing form κ, namely, X * γ j (X γt ) := κ(X * γ j , X γt ) = δ j,t with δ i,t the Kronecker delta.
Step 2: Next, for an L-irreducible constituent W of g(−r + k) ⊗ g(r), consider the associated L-intertwining operatorτ k | W * ∈ Hom L (W * , P k (g)) defined bỹ where W * is the dual space for W with respect to the Killing form κ. Take an irreducible

T. Kubo
Step 3: Last, to the space W * dual to the space W taken in Step 2, apply the following algebraic procedure: Here, P k (g (1)) is the space of homogeneous polynomials on g(1) of degree k, the map σ : Sym k (g(−1)) → U(n) is the symmetrization operator, and D(L s )n is the space of n-invariant differential operators for L s .
Let Ω k | W * : W * → D(L s )n be the composition of the linear maps described in (2.2), namely, We call the system of operators the Ω k | W * system. When the irreducible constituent W * is not important, we simply refer to each Ω k | W * system as an Ω k system. We may want to note that Ω k systems are independent of a choice for a basis for W * up to some natural equivalence (see Definition 3.5 of [35]).
The Ω k systems are not conformally invariant for arbitrary complex parameter s ∈ C for the line bundle L s . We say that an Ω k system has special value s k if the system is conformally invariant on the line bundle L s k .

The Ω k systems and generalized Verma modules
Conformally invariant systems yield non-zero U(g)-homomorphisms between appropriate generalized Verma modules. Since the theory simplifies computations to find the special values of Ω k systems, in this subsection, we review how conformally invariant Ω k 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 M q [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 → G 0 /Q 0 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 M q [E * ], where E * is the dual space for E (see, for example, [7] and [13]). Via this natural pairing, associated to the line bundle L s is the generalized then we obtain the following diagram: 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 ω k (Y * 1 ), . . . , ω k (Y * m ) in ω k (W * ). It can be seen from (2.3) that constructing the Ω k | W * system is equivalent to constructing the L-submodule ω k (W * ) ⊗ C −s . Proposition 2.1 below shows that there is a further relationship. Theorem 19]). The Ω k | W * system is conformally invariant on the line bundle L s k if and only if The following proposition shows that this is indeed a generalized Verma module.

Now it follows from Propositions 2.1 and 2.2 that a conformally invariant Ω
The inclusion map ι then induces a non-zero U(g)-homomorphism that is given by 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 ϕ Ω k in (2.4), we give a simple criterion to determine whether or not the standard homomorphism ϕ std : To do so, it is convenient to parametrize generalized Verma modules by their infinitesimal characters. Therefore we write where ρ is half the sum of the positive roots. Similarly, if W * has highest weight ν then, by Proposition 2.2, we write 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 .
denotes the set of simple roots α ∈ Π so that g α ⊂ l then the criterion is given as follows.

2.4
The Ω 2 systems associated to maximal parabolic subalgebras of quasi-Heisenberg type In Sections 3, 4, and 5, we study Ω 2 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 ϕ Ω 2 . Then, in this subsection, we recall from [35] some observation on the Ω 2 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 α q be a simple root, so that the maximal parabolic subalgebra q = q {αq} = l ⊕ n determined by α q 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 {α i } determined by α i then the maximal parabolic subalgebras q = l ⊕ n of quasi-Heisenberg type are classified as follows: and Here, the Bourbaki conventions [6] are used for the labels of the simple roots. Note that, in type A n , any maximal parabolic subalgebra has abelian nilpotent radical, and also that, in type G 2 , 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 λ q is the fundamental weight for α q then, for all β ∈ ∆, the quotient 2 λ q , β /||α q || 2 takes the values of 0, ±1, or ±2 (see for example Section 4.1 of [35]). Therefore, if H q is the element in h so that β(H q ) = 2 λ q , β /||α q || 2 for all β ∈ ∆, and if g(j) is the j-eigenspace of ad(H q ) on g then the adjoint action of H q 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 subalgebran, the nilpotent radical opposite to n, is given bȳ n = g(−1) ⊕ g(−2). Here we have g(2) = z(n) and g(−2) = z(n), where z(n) (resp. z(n)) is the center of n (resp.n). Thus we denote the 2-grading on g by with parabolic subalgebra Now, for 1 ≤ k ≤ 4, the maps τ k associated to the grading (2.6) are given by In particular when k = 2, we have The Ω 2 systems, the systems of differential operators that we study, are constructed from the τ 2 map.
By construction the Ω 2 systems arise from irreducible constituents W of l⊗z(n) so thatτ 2 | 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 Ω 2 systems, for the remainder of this section, we recall from [35] the observation on l and irreducible constituents W for which τ 2 | W * ≡ 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 α q , we have z(l) = CH q . 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 A n then there is exactly one simple root not orthogonal to γ. It is well known that if α γ 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 α γ is in Π(l) = Π\{α q }. In particular there is a unique simple ideal of [l, l] containing the root space g αγ for α γ . We denote by l γ the unique simple ideal containing g αγ . Similarly, when [l, l] consists of two (resp. three) simple ideals, we denote the other simple ideal(s) by l nγ (resp. l + nγ and l − nγ ). The three simple factors occur only when q is of type D n (n − 2). So, when q is not of type D n (n − 2), the Levi subalgebra l may decompose into Similarly, when q is of type D n (n − 2), one may write Note that when [l, l] is a simple ideal, we have l nγ = {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 To build an Ω 2 system, it is necessary to choose an irreducible constituent W in l ⊗ z(n) so that the L-intertwining map τ 2 | W * ∈ Hom L W * , P 2 (g (1)) is not identically zero. Now we give a necessary condition for irreducible constituents W so that τ 2 | W * ≡ 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] .
It is shown in [35,Section 6] that, for q not of type D n (n − 2), there are exactly one or two special constituents of l ⊗ z(n); one is an irreducible constituent of l γ ⊗ z(n) and the other is equal to l nγ ⊗ z(n). We denote by V (µ + γ ) and V (µ + nγ ) the special constituents so that V (µ + γ ) ⊂ l γ ⊗ z(n) and V (µ + nγ ) = l nγ ⊗ z(n). It will be shown in Section 3 that if q is of type D n (n − 2) then there are three special constituents, namely, V (µ + γ ) ⊂ l γ ⊗ z(n) and V (µ + ± nγ ) = l ± nγ ⊗ z(n). To compute the special values of Ω 2 systems efficiently, the special constituents V (µ + ) 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 + ) is of (1) Type 1a if µ + is not a root with = µ and both µ and are long roots, (2) Type 1b if µ + is not a root with = µ and either µ or is a short root, (3) Type 2 if µ + = 2µ is not a root, or (4) Type 3 if µ + is a root. Table 1 in the introduction shows the types of special constituents. A dash in the column for V (µ + nγ ) indicates that l nγ = {0} for the case. (So there is no special constituent V (µ + nγ ).) In Sections 4 and 5, we find the special values of the Ω 2 systems coming from special constituents of Type 1b and Type 3, respectively. The following proposition will play a key role to determine the special values.
Proof . Observe that, by Proposition 2.1, the Ω 2 | V (µ+ ) * system is conformally invariant if and only if, for all X ∈ n and Y * ∈ V (µ + ) * , 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 D n (n − 2) In this short section we consider the Ω 2 systems associated with maximal parabolic subalgebra q = l ⊕ n of type D n (n − 2). In particular we find the special values for the Ω 2 systems and determine the standardness for the maps ϕ Ω 2 .
The parabolic subalgebra q of type D n (n−2) is the maximal parabolic subalgebra determined by the simple root α n−2 ; the deleted Dynkin diagram is α n−1 • α n As the simple root α q that determines the parabolic subalgebra q is α q = α n−2 , the fundamental weight λ q for α q is λ q = λ n−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 The unique simple root α γ that is not orthogonal to the highest root γ is α γ = α 2 . Therefore we have l γ ∼ = 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 α n (resp. α n−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 With the arguments in [35, Section 6.1], one can easily check that CH q ⊗ z(n) and l ± nγ ⊗ z(n) are simple l-modules. In fact, if we use the standard realization of roots with α n−1 = ε n−1 − ε n and α n = ε n−1 + ε n then On the other hand, the tensor product l γ ⊗ z(n) is reducible. By using the character formula of Klimyk [23,Corollary], it can be shown that Now one may observe that only V (2ε 1 ) and V (ε 1 + ε 2 + ε n−1 ± ε n ) satisfy the conditions (C1) and (C2) in Definition 2.5. Thus these irreducible constituents are the special constituents of l ⊗ z(n). Write γ and ± nγ for the roots in ∆(g(1)) so that µ + γ = 2ε 1 and µ + ± nγ = ε 1 + ε 2 + ε n−1 ± ε n , where µ is the highest weight for g (1). Tables 4 and 5 summarize the data for the special constituents.
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 τ 2 map is not identically zero. Also, the argument for Proposition 7.5 of [35] shows that, for V (µ + ) = V (µ + γ ), V (µ + ± nγ ), the linear map τ 2 | V (µ+ ) * is not identically zero. Now we are going to find the special values of the Ω 2 systems arising from the special constituents V (µ + γ ) and V (µ + ± nγ ). If W ⊂ g is an ad(h)-invariant subspace then, for any weight ν ∈ h * , we write (3.1) Theorem 3.1. Let q be the maximal parabolic subalgebra of type D n (n − 2). If V (µ + ) = V (µ + γ ) or V (µ + ± nγ ) then the Ω 2 | V (µ+ ) * system is conformally invariant on L s if and only if where |∆ µ+ (g (1))| is the number of the elements in ∆ µ+ (g (1)).

The standardness of the map ϕ Ω 2
In the rest of this section we determine whether or not the maps ϕ Ω 2 coming from the Ω 2 systems are standard.

Type 1b special constituent
In this section we study the Ω 2 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 B n (n−1). The simple root α q that determines the parabolic subalgebra q is then α q = α n−1 . We write λ q = λ n−1 for the fundamental weight λ q for α q . The deleted Dynkin diagram for q is Since α 2 is the unique simple root that is not orthogonal to the highest root γ, it follows from the subgraphs that l γ ∼ = sl(n − 1, C) and l nγ ∼ = sl(2, C). Recall from Table 1 that Type 1b special constituent of l ⊗ z(n) is the irreducible constituent V (µ + nγ ) = l nγ ⊗ z(n).
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 B n (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(h 1 , . . . , h n )) = h j . We take the root vectors X α as follows: where X α denotes the matrix whose entries are all zero except the jth and kth pairs of indices.
2. α = ±ε j : where X α denotes the matrix defined similarly to the previous case.
Thus the map τ 2 (X) in (2.8) may be expressed as Proposition 7.3 of [35] showed that when q is of type B n (n − 1), the τ 2 map is not identically zero.

Type 3 special constituent
In this section we study the Ω 2 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 C n (i) for 2 ≤ i ≤ n − 1. The simple root α q that determines the parabolic subalgebra q is then α q = α i . We write λ q = λ i for the fundamental weight λ q for α q . The deleted Dynkin diagram for q is Since α 1 is the unique simple root that is not orthogonal to the highest root γ, it follows from the subgraphs that l γ ∼ = sl(i, C) and l nγ ∼ = sp(n − i, C). Recall from Table 1 that Type 3 special constituent of l ⊗ z(n) is the irreducible constituent V (µ + γ ) ⊂ l γ ⊗ z(n).
As in Section 4, our first goal is to show the L-intertwining operatorτ 2 | V (µ+ γ ) * is not identically zero. To do so we again fix convenient root vectors for g. Observe that, as q is of type C n (i), the Lie algebra g under consideration is g = sp(n, C). For 1 ≤ j ≤ n, we writê j = j + n. If E ab denotes the matrix with 1 in the (a, b) entry and 0 elsewhere then we take h to be the set of diagonal matrices with h j ∈ C. The positive roots are ∆ + = {ε j ± ε k | 1 ≤ j < k ≤ n} ∪ {2ε j | 1 ≤ j ≤ n} with ε j (h 1 , . . . , h n ) = h j . We take the root vectors X α as follows: For α, β ∈ ∆ with α + β = 0, we again denote by N α,β the constant so that [X α , X β ] = N α,β X α+β . Table 7 summarizes the values of N α,β for α + β a positive root.
For α ∈ ∆ + we set H α = [X α , X −α ]. Namely, Now observe that if T (X, Y ) := Tr(XY ) then, as T (·, ·)| h 0 ×h 0 is an inner product on a real form h 0 of h, there exists a positive constant c 0 so that κ(X, Y ) = c 0 T (X, Y ). Since T (X α , X −α ) takes the value of one for α long and two for α short, we have (5.1) Table 7. The values of N α,β for roots α and β for sp(2n, C) with indices i < j < k when α + β is a positive root.
Thus the dual vector X * α for X α with respect to the Killing form is given by 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 Therefore, (5.3) reads Thus the τ 2 map in (2.8) may be expressed as As for the case that q is of type B n (n − 1), Proposition 7.3 of [35] showed that the τ 2 map is not identically zero. 5.1 Lowest weight vector for V (µ + γ ) * As for the case of Type 1b special constituent it is necessary to show that the linear map τ 2 | V (µ+ γ ) * : V (µ + γ ) * → P 2 (g(1)) is not identically zero. We will again achieve it by showing thatτ 2 (Y * l )(X) is a non-zero polynomial on g(1), where Y * l is a lowest weight vector for V (µ + γ ) * .
When a special constituent is of Type 1b, as V (µ + nγ ) = l nγ ⊗ z(n), it was easy to find a lowest weight vector for V (µ+ nγ ) * . In contrast, in the present case, since V (µ+ γ ) l γ ⊗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 + γ ) * . To do so we now observe a highest weight vector for V (µ + γ ).
If pr lγ ⊗z(n) : l ⊗ z(n) → l γ ⊗ z(n) is the projection map from l ⊗ z(n) onto l γ ⊗ z(n) then we claim that pr lγ ⊗z(n) (τ 2 (X µ + X γ )) is a highest weight vector for V (µ + γ ). The following two technical lemmas will simplify the expositions of the proof.
Lemma 5.1. For Z ∈ l and X 1 , X 2 ∈ g(1), we have 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.

T. Kubo
Now we define the "opposite" τ 2 mapτ 2 bȳ It follows from the same arguments in the proof for Lemma 3.3 and Proposition 7.3 in [35] that theτ 2 map is not identically zero and L-equivariant. Let pr lγ ⊗z(n) : l ⊗ z(n) → l γ ⊗ z(n) be the projection map from l ⊗ z(n) onto l γ ⊗ z(n).
Proof . This proposition immediately follows from the arguments used in the proof for Proposition 5.3, by replacing positive (resp. negative) roots with negative (resp. positive) roots.
We set It follows from Proposition 5.4 that Y * l is a lowest weight vector for V (µ + γ ) * . In the next subsection we computeτ 2 (Y * l )(X). To the end we give an explicit form for Y * l .
Lemma 5.5. We have Proof . By using the same arguments for Lemma 5.2, one can obtain pr lγ ⊗z(n) (τ 2 (X −µ + X − γ )) A direct computation shows that
Lemma 5.7. For X, Y, Z ∈ g, in U(g), we have

Proof . A direct computation.
Now we are ready to determine the special value for the Ω 2 | V (µ+ γ ) * system.
Theorem 5.8. Let q be the maximal parabolic subalgebra of type C n (i) for 2 ≤ i ≤ n − 1.
The Ω 2 | V (µ+ γ ) * system is conformally invariant on L s if and only if s = n − i + 1.
If λ i = i j=1 ε j is the fundamental weight for α i then, as H · 1 −s = λ i (H)1 −s for H ∈ h, a direct computation shows that Similarly, by a direct computation, the second term amounts to Therefore, X µ · (ω(Y * l ) ⊗ 1 −s ) is given by Hence X µ · (ω(Y * l ) ⊗ 1 −s ) = 0 if and only if s = n − i + 1.
(5.17) Theorem 5.9. If q is the maximal parabolic subalgebra of type C n (i) for 2 ≤ i ≤ n − 1 then the standard map ϕ std between the generalized Verma modules in (5.17) is zero. Consequently, the map ϕ Ω 2 is non-standard.

A Miscellaneous data
This appendix summarizes the miscellaneous data for the maximal parabolic subalgebras q = l ⊕ g(1) ⊕ z(n) of types B n (n − 1), C n (i) (2 ≤ i ≤ n − 1), and D n (n − 2). For the data for other maximal parabolic subalgebras of quasi-Heisenberg type see, for example, Appendix A of [35].