On the composition series of the standard Whittaker (g,K)-modules

For a real reductive linear Lie group G, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker (g,K)-modules, which are K-admissible submodules of the space of Whittaker functions. We first determine the structures of them when the infinitesimal characters characterizing them are generic. As an example of the integral case, we determine the composition series of the standard Whittaker (g,K)-module when G is the group U(n,1) and the infinitesimal character is regular integral.


Introduction
One of the most basic problems in representation theory is to study the composition series of a standard representation. In the category of highest weight modules, Verma modules play the role of standard representations, and in the category of Harish-Chandra modules, principal series representations do. The composition series problem is called the Kazhdan-Lusztig conjecture.
In this paper, the author propose a Whittaker version of standard (g, K)-modules and study their composition series problem.
Let G be a real reductive linear Lie group in the sense of [11] and G = KAN be an Iwasawa decomposition of it. Let η : N −→ C × be a unitary character of N and denote the differential representation n 0 → √ −1R of it by the same letter η. We assume η is non-degenerate, i.e. it is non-trivial on every root space corresponding to a simple root of ∆ + (g 0 , a 0 ). Define −→ C | f (gn) = η(n) −1 f (g), g ∈ G, n ∈ N } and call it the space of Whittaker functions on G. This is a representation space of G by the left translation, which is denoted by L. Let C ∞ (G/N ; η) K be the subspace of C ∞ (G/N ; η) consisting of K-finite vectors. As for the subrepresentations of this space, there are many deep and interesting results, called the theory of Whittaker models. On the other hand, it is not too much to say that the structure of the whole space is not known at all. Though our ultimate goal is to determine the structure of C ∞ (G/N ; η), this space is too large to analyze. So we need to cut off a submodule of suitable size from it. Let, as usual, M be the centralizer of A in K, and let be the stabilizer of η in M . This subgroup acts naturally on C ∞ (G/N ; η) K by the right translation. Consider the subspace of C ∞ (G/N ; η) K consisting of those functions f which satisfy the following conditions: (1) f is a joint eigenfunction of Z(g) (the center of the universal enveloping algebra U (g)) with eigenvalue χ Λ : L(z)f = χ Λ (z)f , z ∈ Z(g).
(2) For an irreducible representation (σ, V M η σ ) of M η , f is in the σ * -isotypic subspace (σ * is the dual of σ) with respect to the right action of M η .
Therefore, the space of functions f satisfying the above conditions (1)-(3) is isomorphic to I η,Λ,σ := {f ∈ I • η,Λ,σ | f grows moderately at the infinity}. We call these the standard Whittaker (g, K)-modules. Note that these are not the "standard Whittaker module" defined in [5]. It is easy to show that these are K-admissible and then have finite length (Corollary 2.4).
Though the composition series problem of standard Whittaker (g, K)-modules is an interesting problem by itself, we may hope to apply the result of it to the analysis of principal series representations. I η,Λ,σ is induced from M η N and the behavior of f ∈ I η,Λ,σ on A is controlled by the infinitesimal character and the asymptotic behavior, so we may think this module is near to the principal series representation. Therefore, it is significant to compare the structure of this module and that of a principal series representation. According to the theory of Whittaker models, an irreducible Harish-Chandra module π can be a submodule of I η,Λ,σ only if the Gelfand-Kirillov dimension of π is equal to dim N ( [8]). On the other hand, any irreducible Harish-Chandra module can be a submodule of some principal series representation. This difference comes from the difference of the structures of I η,Λ,σ and principal series. So if you understand the common features and the different points of these modules, then new insights of Whittaker models and principal series are expected to be obtained.
For example, assume G = SL(2, R) and the infinitesimal character Λ is regular dominant integral. In this case M = M η ≃ {±1}, so we identify an irreducible representation of M and that of M η , which is denoted by σ. There are two irreducible representations of {±1}, one is trivial, denoted by 1, and the other is the signature representation, denoted by −1. Let ρ A = 1 2 tr(ad a | Lie(N ) ) ∈ Lie(A) * . Then the principal series representation Ind G MAN (σ ⊗ e Λ+ρA ) is reducible if and only if Λ ≡ σ + 1 mod 2. There are four equivalence classes of irreducible Harish-Chandra modules with the infinitesimal character Λ. The irreducible principal series is denoted by π − 01 , the irreducible finite dimensional representation by π + 01 and two discrete series are denoted by π 0 , π 1 . It is well known that the composition series of reducible principal series are For the meaning of these diagrams, see Definition 3.3. On the other hand, if σ ∈ M η = M corresponds to the reducible (resp. irreducible) principal series, then the composition series of the standard Whittaker (g, K)-modules are for an appropriately chosen η. This result can be obtained by direct computation.
In this paper, we first determine the structure of I η,Λ,σ when Λ is generic. Let X P (δ, ν) be the Harish-Chandra module of the C ∞ -induced principal series representation C ∞ -Ind G P (δ ⊗ e ν+ρA ). Here δ is an irreducible representation of M . The Weyl group of g is denoted by W . Let H Λ be the set of equivalence classes of irreducible Harish-Chandra modules with the infinitesimal character Λ. We call Λ generic if every principal series representation with the infinitesimal character Λ is irreducible. The main theorem on the generic case is Theorem 1.1 (Theorem 2.2). Let G be a real reductive linear Lie group. Suppose Λ is generic and σ is an irreducible representation of M η . Then I • η,Λ is completely reducible. Moreover, the irreducible decomposition of I η,Λ,σ is given by For the non-generic case, there is little result that can be applied to general groups. Therefore, we examine the case G = U (n, 1) in the second half of this paper so that it becomes a springboard to the study of general cases. Let π i,j be the irreducible Harish-Chandra module of U (n, 1) defined in § 3.2. The main result on this case is Theorem 1.2 (Theorem 5.16). Suppose G = U (n, 1) and the infinitesimal character Λ is regular integral. If the highest weight of σ ∈ M η ≃ U (n − 2) × U (1) satisfies (4.1) for some i = 1, . . . , n − 1, j = 2, . . . , n + 1 − i, then the composition series of I η,Λ,σ is This paper is organized as follows. The generic case is treated in §2. The main result of this section is Theorem 2.2. From §3 later, we put G = U (n, 1) and examine the composition series of I η,Λ,σ when the infinitesimal character is regular integral. §3 recalls the structure of U (n, 1) and the classification of irreducible Harish-Chandra modules of it. In §4, we first show that I η,Λ,σ has a unique irreducible submodule if it is non-zero. Also determined are the possible irreducible modules appearing in the composition series of it. In §5, the composition series of I η,Λ,σ is completely determined. For this step, we use the explicit form of K-type shift operators and the central elements of the universal enveloping algebra. The key lemmas for our calculation are Lemma 5.7 and 5.11, and the main theorem of the latter half of this paper is Theorem 5.16. In §6, another formulation of our problem is discussed.
Before going ahead, we introduce notation used in this paper. For a real Lie group L, the Lie algebra of it is denoted by l 0 and its complexification by l = l 0 ⊗ R C. This notation will be applied to groups denoted by other Roman letters in the same way without comment. For a compact Lie group L, the set of equivalence classes of irreducible representations of L is denoted by L. The representation space of π ∈ L is denoted by V L π . When L is connected and π is the irreducible representation whose highest weight is λ, we also denote it by V L λ . For π ∈ L, the contragredient representation is denoted by π * , and if λ is the highest weight of π, then the highest weight of π * is denoted by λ * .
Suppose that K is a maximal compact subgroup of a real reductive group G. For a (g, K)-module π, the K-spectrum {τ ∈ K | τ ⊂ π| K } is denoted by K(π).
For a numerical vector a = (a 1 , . . . , a ℓ ) ∈ C ℓ or R ℓ , write |a| := ℓ i=1 a i . This notation will be applied for an element of the dual of a Cartan subalgebra when this space is identified with numerical vector space by using some fixed basis.
The author would like to thank Hiroshi Yamashita, Kyo Nishiyama, Noriyuki Abe and Hisayosi Matumoto for helpful discussion on this problem. He also thanks Tôru Umeda, Minoru Itoh and Akihito Wachi for useful advice on the determinant type central element of the universal enveloping algebra. This research is partially supported by JSPS Grant-in-Aid Scientific Research (C) # 19540226.

The generic case
In this section, we first write down the differential equations characterizing I • η,Λ,σ . After that, we determine the structure of the standard Whittaker (g, K)-modules when Λ is generic. This is the first main theorem of this paper. As a corollary to the proof of this theorem, the K-admissibility of any standard Whittaker (g, K)-module is obtained.
The K-type decomposition of I • η,Λ,σ is given by By Iwasawa decomposition, an element of Hom Therefore, we may identify φ 1 with an element The g action on φ 1 can be transferred to φ 2 , which we denote by X · φ 2 : (X · φ 2 )(a)(v) = L(X)(φ 1 (v))(a). It follows that Hom K (V K τ , I • η,Λ,σ ) is isomorphic to (2.1) and the subspace I η,Λ,σ (τ ) of I • η,Λ,σ (τ ) consisting of functions which grow moderately at the infinity is isomorphic to Hom K (V K τ , I η,Λ,σ ). We write down the action of z ∈ Z(g) on φ 2 . Firstly, the U (k) action is Secondly, consider the action of U (a). Denote by Π = {α 1 , . . . , α l } the set of simple roots of ∆ + (g 0 , a 0 ). Let H 1 , . . . , H l be the basis of a 0 dual to α 1 , . . . , α l : For u = n c n H n1 1 · · · H n l l ∈ U (a), n = (n 1 , . . . , n l ) ∈ (Z ≥0 ) l , c n ∈ C, let . Lastly, we write down the action of U (n). Denote by (g 0 ) α the root space corresponding to a root α. Let {N α,j | α ∈ ∆ + (g 0 , a 0 ), 1 ≤ j ≤ dim(g 0 ) α } be a basis of n 0 such that it satisfies η(N α,j ) = 0 if α ∈ Π and j = 1, and η(N α,j ) = 0 otherwise. We define and extend it to an algebra homomorphism U (n) → C[t 1 , . . . , t l ]. Then for u ∈ U (n). Therefore, U (g) acts on Choose a Cartan subalgebra t m of m. Fix a positive system ∆ + (m, t m ) of the root system ∆(m, t m ). Let δ ∈ M . Its highest weight with respect to ∆ + (m, t m ) is denoted by µ δ . Note that since we assume every Cartan subgroup of G is commutative [11, (0.1.2) f)], the highest weight µ δ of the restriction of δ to the identity component of M is well defined even if M is not connected. Let P = M AN be the minimal parabolic subgroup of G corresponding to our Iwasawa N . For δ ∈ M and ν ∈ a * , let X P (δ, ν) be the Harish-Chandra module of the smooth principal series representation C ∞ -Ind G P (δ ⊗ e ν+ρA ). Definition 2.1. An infinitesimal character Λ is called generic if every principal series representation X P (δ, ν) which admits the infinitesimal character Λ is irreducible.
Choose a Cartan subalgebra h := t m + a of g. Let W = W (g, h) and W m = W (m, t m ) be the Weyl groups of g and m, respectively. The little Weyl group is denoted by W (G, A). It is well known that the infinitesimal character of X P (δ, ν) is Λ ∈ h * if and only if (µ δ + ρ m , ν) is in the orbit W · Λ. It is also well known that two principal series representations X P (δ, ν) and X P (δ ′ , ν ′ ) have the same composition factors if and only if there exists w ∈ W (G, A) such that (δ ′ , ν ′ ) = (w · δ, w · ν). We denote by A Λ the set of (δ, ν) ∈ M × a * satisfying (µ δ + ρ m , ν) ∈ W · Λ. The set of equivalence classes of irreducible Harish-Chandra modules is denoted by H Λ . Note that, if Λ is generic, every member of H Λ is a principal series representation. Therefore, H Λ is parametrized by W (G, A)\A Λ , the set of W (G, A) -orbits in A Λ .
The first main result of this paper is the following theorem.
Theorem 2.2. Suppose Λ is generic and σ is an irreducible representation of M η . Then I • η,Λ is completely reducible. Moreover, the irreducible decomposition of I η,Λ,σ is given by Proof. We first count the dimension of I • η,Λ,σ (τ ). Letn be the nilpotent subalgebra opposite to n. We denote by u m andū m the nilpotent subalgebras in m corresponding to ∆ + (m, t m ) and −∆ + (m, t m ), respectively. Then u := u m +n is the nilradical of a Borel subalgebra h + u. Let ρ m be half the sum of elements in ∆ + (m, t m ). We define non-shifted Harish-Chandra maps γ ′ 1 , , respectively. Then Harish-Chandra maps are given by are characterized by the system of differential equations We know that the system of partial differential equations z ·φ 2 = χ Λ (z)φ 2 , z ∈ Z(g), has regular singularity at t = 0 (see the definitions (2.2), (2.3) of ∂ t (u) and η t ).
Suppose there exists a non-zero solution φ 2 of this system. Then its leading term φ 0 2 satisfies the system of differential equations (2.7) is equivalent to z modulo nU (g), it is the same as γ ′ 1 (z). Therefore, we may assume that, if η 0 (u Since Λ is regular, wΛ (w ∈ W ) are all different. Therefore, all the solutions of equation (2.7) are In the fundamental paper [7], Lynch gives the dimension of the space of dual Whittaker vectors of principal series representations. His result, together with Theorem C in [8], says that, Here, we first used the Frobenius reciprocity Hom K (τ, X P (δ, ν)) ≃ Hom M (τ | M , δ), and used the fact that every W (G, A)-orbit in A Λ consists of #W (G, A) elements, since Λ is regular. It follows that every composition factor of I • η,Λ is a submodule of it. In other words, the modules I • η,Λ , I • η,Λ,σ and I η,Λ,σ are completely reducible. It also follows that the equality in (2.10) holds.
In order to complete the proof, we recall a result of Wallach's ( [13]). Let dn be the Haar measure of N and w 0 be the longest element of W (G, A) (with respect to N ). Recall the Jacquet integral is an isomorphism of vector spaces. Here, , δ is the pairing of V M δ and its dual.
The image of a continuous dual Whittaker vector is characterized by the moderate growth condition ( [12]). From this theorem and the map (2.12), we know that there are m δ (σ) = dim Hom M η (δ| M η , σ) copies of X P (δ, ν) in the socle of I η,Λ,σ , if X P (δ, ν) ∈ H Λ . As we noted before the theorem, every member of H Λ is a principal series. So the socle of I η,Λ,σ is the right hand side of (2.5). Since I η,Λ,σ is completely reducible, the theorem is shown.
Corollary 2.4. The standard Whittaker (g, K)-modules are K-admissible and they have finite length.
Proof. If Λ is regular, the multiplicity of each K-type is finite because of (2.10). Such estimate is possible even if Λ is not regular. The second assertion is clear since these modules admit an infinitesimal character and K-admissible.

The group U (n, 1) and its irreducible Harish-Chandra modules
Up to now very little is known about the properties of I η,Λ,σ with non-generic Λ, so no smart technique can be used for the analysis of it. Therefore, we will choose a group G such that the structure of Harish-Chandra modules of it is well know and simple (for example K-multiplicity free), and we determine the (g, K)-module structure of I η,Λ,σ for non-generic Λ by direct calculation. Such an example is expected to be a good guide to general cases.
For such reasons, we assume G = U (n, 1) and Λ is regular integral hereafter.
3.1. Structure of U (n, 1). Denote by E ij the standard generators of gl n+1 (C) and define I n,1 = n p=1 E pp − E n+1,n+1 . Let G = U (n, 1) be the subgroup of GL(n + 1, C) consisting of the matrices g satisfying tḡ I n,1 g = I n,1 . The Lie algebra g 0 = u(n, 1) consists of those matrices X ∈ gl n+1 (C) which satisfy tX I n,1 +I n,1 X = O. Let θg = I n,1 gI n,1 be a Cartan involution of G. The corresponding maximal compact subgroup K of G is Let g 0 = k 0 + s 0 be the corresponding Cartan decomposition of g 0 . Then Then a 0 is a maximal abelian subspace of s 0 . The restricted root system ∆(g 0 , a 0 ) is Choose a positive system ∆ + (g 0 , a 0 ) = {f, 2f }, and denote the corresponding nilpotent subalgebra α∈∆ + (g0,a0) (g 0 ) α by n 0 . One obtains an Iwasawa decomposition where A = exp a 0 and N = exp n 0 . Let (3.1) 1} is a basis of (g 0 ) f , and {Z} is a basis of (g 0 ) 2f . In our U (n, 1) case, M is isomorphic to U (n − 1) × U (1). It acts on the space of non-degenerate unitary characters of N by η → η m (n) := η(m −1 nm), m ∈ M . Therefore, we may choose a manageable unitary character when we calculate Whittaker modules. We use the non-degenerate character η defined by It is easy to see that M η is isomorphic to U (n − 2) × U (1).

5)
and each K-type occurs in π i,j with multiplicity one.

Composition factors of I η,Λ,σ
In this section we first determine the submodules of I η,Λ,σ . For this purpose, we need some results on the Whittaker models.

4.2.
Unique simple submodule. Let (π, V ) be an irreducible Harish-Chandra module with DimV = dim N . Suppose that it is an composition factor of some principal series representation X P (δ, ν). By Theorem 4.1(2) and Theorem 2.3, every continuous embedding of V ∞ into C ∞ (G/N ; η) is a composition of (i) a realization of V ∞ as a subquotient of C ∞ -Ind G P (δ ⊗ e ν+ρA ) and (ii) a Jacquet integral. Since a Jacquet integral is right M η -equivariant, V can be a submodule of I η,Λ,σ only if σ ⊂ δ| M η . Let {X P (δ p , ν p ) | p = 1, . . . , k} be the set of principal series representations which contain (π, V ) as a subquotient. If (π, V ) is a submodule of I η,Λ,σ , then by the above discussion σ ⊂ ∩ k p=1 δ p | M η , i.e. σ is a submodule of δ i | M η for every p = 1, . . . , k.
Especially, I η,Λ,σ is non-zero if and only if the highest weight of σ satisfies the condition (4.1) for some i, j. In this case, π i,j is the unique simple submodule of I η,Λ,σ .
and only if the highest weight γ of σ satisfies (4.1). This proves the "only if" part of proposition for the case i + j ≤ n. The case i + j = n+ 1 is shown analogously.
We will show that the condition is sufficient and that the multiplicity in the socle is one.
Let ( M η ) i,j be the set of σ ∈ M η whose highest weight γ satisfies the condition (4.1). Then it is easy to see that ( M η ) i,j ∩ ( M η ) k,l = ∅ if (i, j) = (k, l). It follows that if σ ∈ ( M η ) i,j , then every irreducible factor in the socle of I η,Λ,σ is isomorphic to π i,j . Let m σ be the multiplicity of such factors. Then dim Wh −∞ On the other hand, it is easy to see from Theorem 2.3 and (4.1) that Since m σ ≥ 1 for any σ ∈ ∪ a,b=0,1 ( M η ) i+a,j+b , they are all one. This completes the proof of proposition.

Composition factors.
Hereafter, we denote I η,Λ,σ by I η,Λ,γ if the highest weight of σ is γ. We also denote by σ γ the irreducible representation of M η whose highest weight is γ. We first determine the irreducible representations appearing in the composition series of I η,Λ,γ . Proposition 4.3. Suppose that Λ is regular integral and that γ satisfies (4.1), so π i,j is the unique simple submodule of I η,Λ,γ . In this case, an irreducible module π k,l is a composition factor of I η,Λ,γ only if (k, l) = (i + a, j + b) with a = 0, ±1 and b = 0, ±1.
Proof. We have seen in § 2 that, if π k,l is a composition factor of I η,Λ,γ , each K-type of it must contain the representation σ γ . By   , j + b), a, b = 0, ±1. Then the multiplicity of π i ′ ,j ′ in I η,Λ,γ is at least one.
Proof. By Theorem 3.4, π i ′ ,j ′ is a composition factor of π k,l = X P (δ k,l , ν k,l ), with (k, l) = (i + a, j + b), a, b = 0, −1. Consider the principal series representation X P (δ k,l , ν) with ν ∈ a * . If ν is generic, then I η,(µ k,l +ρm,ν),γ is isomorphic to X P (δ k,l , ν) by Theorem 2.2. Here we used the fact m δ k,l (σ γ ) = 1. Choose a Ktype τ of π i ′ ,j ′ . This is also a K-type of X P (δ k,l , ν). By Frobenius reciprocity, the multiplicity of τ in X P (δ k,l , ν) is one. Therefore, the space of moderately growing solutions of (2.6), with Λ replaced by (µ k,l + ρ m , ν) and σ by σ γ , is one dimensional. Let f ν be a non-zero moderately growing solution. Then by Theorem 2.3, this function is expressed by the Jacquet integral and it is holomorphic in ν. Suppose the order of zero of f ν at ν = ν k,l is m. Let g ν := f ν /(ν − ν k,l ) m . Then g ν k,l is non-zero. It satisfies the equation (2.6) (with σ replaced by σ γ ) and grows moderately at the infinity, so it is an element of I η,Λ,γ .

Determination of the composition series
In this section, we determine the composition series of I η,Λ,γ in the case when Λ is integral. For this purpose, we need to write down the actions of Z(g) and s on this space explicitly. The former is achieved by the determinant type central element of U (gl n+1 ), and the latter by the K-type shift operators.
Then it is easy to see thatψ is an element of Hom K (V K λ ⊗ s, I η,Λ,σ ). Here, we regard s as a representation of K by the adjoint action Ad. Denote by ∆ s the set of weights on s with respect to a fixed Cartan subalgebra of k. In our case, the irreducible decomposition of V K λ ⊗ s is ⊕ α∈∆s m(α) V K λ+α , m(α) = 0 or 1. When m(α) = 1, let ι α be the embedding of Thenψ α is an element of Hom K (V K λ+α , I η,Λ,σ ), and the correspondence ψ →ψ α is a K-type shift in I η,Λ,σ coming from the s-action.

Gelfand-Tsetlin basis.
In order to write down the K-type shift operators explicitly, we realize the space Hom M η (V K τ , V M η σ ) by using the Gelfand-Tsetlin basis ( [3]).
(2) The numbers q i,j are all integers.

Theorem 5.2 ([3]). For a dominant integral weight
) be the irreducible representation of U (n) with the highest weight λ. Then GT (λ) is identified with a basis of (τ λ , V U(n) λ ).
The action of elements E ij ∈ gl(n, C) is expressed as follows. Let l i,j := q i,j − i and |q j | := j i=1 q i,j . Let σ ± i,j be the shift operators on GT (λ), sending q j to q j + (0, . . . , i ±1, 0, . . . , 0). Define a i,j (Q) and b i,j (Q) by (5.1)

Theorem 5.3 ([3]). For Q ∈ GT (λ), the action of the Lie algebra is given by
Remark 5.4. The Gelfand-Tsetlin basis is compatible with the restriction to smaller unitary groups U (k), k = 1, . . . , n − 1. More precisely, the restriction of τ λ to U (n−1) is multiplicity free, and the highest weights of the irreducible representation appearing in τ λ | U(n−1) are the above q n−1 's.

5.4.
Central elements of U (gl n+1 ). In order to show Lemma 5.11 below, we use the explicit forms of the elements in Z(g). One of the most useful forms of the central elements of U (gl n+1 ) is the determinant type one ( [1]). For the standard generator E ij of gl n+1 and a parameter u ∈ C, let E ij (u) := E ij + uδ ij (Kronecker's delta). We define Then C n+1 (u) is an element of Z(gl n+1 ) for any u, and we obtain all the generators of Z(gl n+1 ) by specializing u. Since C n+1 (u) ≡ n+1 p=1 (E pp + u + p − 1) modulo the left ideal generated by strictly lower triangular matrices, the infinitesimal character is Lemma 5.8. C n+1 (u) acts on I η,Λ,γ by the scalar n+1 p=1 (u + Λ p + n/2). The exterior calculus is very useful for the manipulation of non-commutative determinants. We use the method developed in [4].
Lemma 5.12. Suppose that γ is given by (4.1). If a pair V 1 and V 2 satisfies one of the following conditions, then there is no non-zero g-action in I η,Λ,γ which sends V 1 to V 2 : (1) V 1 ≃ π i,j , V 2 ≃ π i+a,j , π i,j+b , a, b = ±1.
Corollary 5.13. The multiplicity of π i,j in I η,Λ,γ is one.
Proof. We know that the socle of I η,Λ,γ is isomorphic to π i,j (Proposition 4.3). Assume that there exists a composition factor V 1 which is isomorphic to π i,j but is not in the socle. By Proposition 4.3, π a,b is a composition factor of I η,Λ,γ only if a = i, i ± 1 and b = j, j ± 1. By Theorem 3.2, it is adjacent to V 1 if and only if |a − a ′ | + |b − b ′ | = 1. Therefore, there exits a composition factor V 2 which is isomorphic to one of π i±1,j , π i,j±1 such that V 1 → V 2 . But we have shown in Lemma 5.12 that this is impossible.
(1) As we stated in the proof of the previous corollary, a composition factor V is adjacent to π i,j only if V is isomorphic to one of π i±1,j , π i,j±1 . So only π i±1,j , π i,j±1 can be a simple submodule of I η,Λ,γ /π i,j . We know from Proposition 4.4 that the multiplicity of each of them in I η,Λ,γ is at least one. Choose a composition factor V isomorphic to, say, π i−1,j . Other cases can be shown analogously. Recall the proof of Lemma 5.12. Let φ be the function which characterizes the non-zero K-type λ ′ of V ≃ π i−1,j . Assume that there is no nonzero s-action from V to the unique simple submodule which is isomorphic to π i,j . Since the multiplicity of π i,j in I η,Λ,γ is one, P + n+1−i φ = 0. But we have seen in the proof of Lemma 5.12 that this implies φ = 0. This is a contradiction, so V → π i,j .
Assume that there are two composition factors V 1 , V 2 in the socle of I η,Λ,γ /π i,j , both of which are isomorphic to π i−1,j . Let φ k , k = 1, 2, be the functions which characterize the non-zero K-type λ ′ of V k , respectively. Then both of P + n+1−i φ k characterize the same K-type λ of the unique simple submodule, and the multiplicity of this K-type is one, there are constants c k such that c 1 P + n+1−i φ 1 = c 2 P + n+1−i φ 2 .
(2) Assume that there is a composition factor V 1 isomorphic to, say, π i−1,j but not in the second floor of I η,Λ,γ . The irreducible modules which are adjacent to π i−1,j are π i,j and π i−1,j±1 . Therefore, there exists a composition factor V 2 in the third or higher floor such that it is isomorphic to one of the above and V 1 → V 2 . But this is impossible since (i) the multiplicity of π i,j is one and it is located in the bottom, and (ii) π i−1,j → π i−1,j±1 by Lemma 5.12 (2). Lemma 5.15. Suppose that γ is given by (4.1).
Proof. The proof is almost the same as that of Lemma 5.14.
Since (i) the multiplicities of π i,j±1 , π i±1,j and π i,j are one, and (ii) they are in the first or second floor, the third floor is a direct sum of π i±1,j±1 's, and there is no higher floor in I η,Λ,γ .
For each π i±1,j±1 , there exists at least one factor isomorphic to it in I η,Λ,γ . Suppose, say, V 1 ≃ π i−1,j+1 and V 2 ≃ π i,j+1 , the latter is in the second floor. Let φ be the function which characterizes a K-type of V 1 adjacent to π i,j+1 . The shift operator sending φ to a K-type of V 2 is P + n+1−i . The proof of Lemma 5.12 says that this is injective. Therefore, V 1 → V 2 . The uniqueness of the factor isomorphic to π i−1,j+1 is shown in the same way as in the proof of the previous lemma.
We have obtained the following second main theorem of this paper.