Table Of ContentOn the Reducibility of Scalar Generalized Verma Modules
of Abelian Type
5
1 Haian HE
0
2
g
u Abstract
A
2 A parabolic subalgebra p of a complex semisimple Lie algebra g is called a parabolic
1 subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a
completecharacterizationoftheparametersforscalargeneralizedVermamodulesattached
]
T toparabolic subalgebras ofabelian typesuchthat themodulesare reducible. The proofs
R use Jantzen’s simplicity criterion, as well as the Enright-Howe-Wallach classification of
. unitary highest weight modules.
h
t
a
m
[ 1 Introduction
7
v
4 Let g be a complex semisimple Lie algebra, and fix a Cartan subalgebra h. Denote by Φ
8
(respectively, Φ+) the set of roots (respectively, positive roots) of (g,h). Let p be a maximal
8
1 parabolicsubalgebraofgwithstandardLevidecompositionp=l+u withrespectto(h,Φ+),
+
0
where l is a Levifactor and u is the nilpotent radical. Let Φ be the set of roots of (l,h), and
. + l
1 putΦ+ =Φ ∩Φ+. Ifλ∈h∗ isΦ+-dominantintegral,letF bethefinite-dimensionalcomplex
0 l l l λ
5 simple l-module with highest weightλ. By letting the nilpotent radical u+ actby 0, Fλ is also
1 a module of the parabolic subalgebra p. Now the generalized Verma module of g attached to
:
v p with the parameter λ is defined to be
i
X
Mg(λ):=U(g)⊗ F
r p U(p) λ
a
whereU(g)(respectively,U(p))istheuniversalenvelopingalgebraofg(respectively,p). When
dimCFλ =1,Mpg(λ)iscalledascalargeneralizedVermamodule;whenpisaBorelsubalgebra,
it is just a Verma module. As is known, generalized Verma modules form a fundamental and
distinguished class of objects in the parabolic BGG category Op. A universalproperty is that
each highest weight module in Op can be covered by a generalized Verma module Mg(λ) for
p
some λ. More details about generalized Verma modules can be found in [H].
The reducibility of generalized Verma modules is an interesting problem, which is much more
complicatedthantheproblemforVermamodules. Thestudyofthisproblemhasalonghistory.
In 1977, a simplicity criterion for generalized Verma modules was shown by J. C. Jantzen in
[J],anditremainsoneofthe mostwell-knownandwidely usedresultsalongtheselines. After
that,T.Enright,R.Howe,andN.Wallachworkedoutthe parametersofreduciblegeneralized
Vermamodulesrelatedtounitaryhighestweightmodulesin1983[EHW].Recently,A.Kamita
1
used b-functions to describe the reducibility of generalized Verma modules in [Ka]. Apart
fromdirectlystudyingthereducibilityproblem,manymathematiciansstudiedhomomorphisms
between generalized Verma modules, which gave some results on reducibility of generalized
Verma modules indirectly, e.g., [B], [F], [G], and [M].
AparabolicsubalgebrapofacomplexsemisimpleLiealgebragiscalledaparabolicsubalgebra
of abelian type if its nilpotent radical is abelian. We now explain how to characterize the
parabolic subalgebras of abelian type in simple Lie algebras in terms of Hermitian symmetric
pairs. Suppose that G is a connected real simple Lie group with center Z, and let K be a
closed maximal subgroup of G with K/Z compact. Let g be the complexified Lie algebra of
G. A unitary representation (π,V) of G such that the underlying (g,K)-module is a simple
quotient of a Verma module of g is called a unitary highest weight module. Harish-Chandra
showed that G admits non-trivial unitary highest weight modules precisely when (G,K) is a
Hermitian symmetric pair ([HC1] and [HC2]). Now denote by l the complexified Lie algebra
ofK. Fix aCartansubalgebrahofl; thenourassumptionsonGimply thathisalsoaCartan
subalgebraofg. Since(g,l)isaHermitiansymmetricpair,wemaychooseasimplerootsystem
∆ of (g,h) such that the standard Borel subalgebra b satisfying that p := l+b is a parabolic
subalgebra of g. According to the classification of the Hermitian symmetric pairs [W], the
nilpotent radical of p must be abelian and hence p is a parabolic subalgebra of abelian type.
Conversely, suppose that g is a complex simple Lie algebra and p is a parabolic subalgebra of
abeliantype ofg,andthen pis automaticallyamaximalparabolicsubalgebra. Itfollowsfrom
[RRS] that there exists a real form gR of g such that GR/(GR∩P) is a Hermitian symmetric
space, where GR and P are the subgroups of the adjoint group G = Int(g) with Lie algebras
gR andp. The groupK :=GR∩P isamaximalcompactsubgroupofGR,andits complexified
Lie algebra gives the Levi factor, denoted by l, of p.
If p is a parabolic subalgebra of abelian type, we call a generalized Verma module Mg(λ)
p
attached to p a generalized Verma module of abelian type. In [EHW], T. Enright, R. Howe,
andN.WallachworkedoutalltheparametersλsuchthatthegeneralizedVermamoduleMg(λ)
p
of abelian type is reducible and its unique simple quotient L(λ) is unitarizable. However, if
L(λ) is not unitarizable, it is not known for which λ Mg(λ) is simple. We shall answer this
p
questionwhenMg(λ) isascalargeneralizedVermamodule,i.e., F =C isaone-dimensional
p λ λ
complex l-module, which is equivalent to saying that (λ,α) = 0 for all α ∈ Φ , where (−,−)
l
is the inner product on h∗ induced by the Killing form of g. We shall recall the techniques in
[EHW] in Section 2.2.
Let γ denote the unique maximal root in Φ+. Denote by ζ the unique element in h∗ such
that (ζ,α) = 0 for all α ∈ Φ and 2(ζ,γ) = 1. It is obvious that the parameters λ of scalar
l (γ,γ)
generalized Verma modules must satisfy λ=cζ for some c∈C.
Write Φ := Φ\Φ , and denote by ∆ the simple root system of Φ+. Then each standard
u l
maximal parabolic subalgebra with respect to (h,∆) is determined by the unique simple root
in ∆ :=Φ ∩∆. Now we can state our main result.
u u
Theorem 1.1. Let g be a complex simple Lie algebra, and let p be a parabolic subalgebra of
abelian type of g. Fix a Cartan subalgebra h⊆p, and choose a simple root system ∆ for (g,h)
such that p contains the standard Borel subalgebra with respect to (h,∆). Then the parameters
λ∈h∗ such that Mg(λ) is a reducible scalar generalized Verma module are precisely given case
p
2
by case, according to the Hermitian symmetric pairs of compact type, as follows.
(i) (SU(p+q),S(U(p)×U(q))) for p,q ≥ 1: ∆ = {e −e | 1 ≤ i ≤ p+q−1}; ∆ =
i i+1 u
p p+q
q p
{e −e }; ζ = e − e ; λ=cζ with c∈1−min{p,q}+Z .
p p+1 p+q X i p+q X j ≥0
i=1 j=p+1
n
(ii) (Sp(n),U(n)) for n≥2: ∆={e −e |1≤i≤n−1}∪{2e }; ∆ ={2e }; ζ = e ;
i i+1 n u n X i
i=1
λ=cζ with c∈ 1−n + 1Z .
2 2 ≥0
(iii) (SO(2n+1),SO(2)×SO(2n−1)) for n ≥ 2: ∆ = {e −e | 1 ≤ i ≤ n−1}∪{e };
i i+1 n
∆ ={e −e }; ζ =e ; λ=cζ with c∈Z ∪(3 −n+Z ).
u 1 2 1 ≥0 2 ≥0
(iv) (SO(2n),SO(2)×SO(2n−2)) for n≥2: ∆={e −e |1≤i≤n−1}∪{e +e };
i i+1 n−1 n
∆ ={e −e }; ζ =e ; λ=cζ with c∈2−n+Z .
u 1 2 1 ≥0
(v) (SO(2n),U(n))forn≥2: ∆={e −e |1≤i≤n−1}∪{e +e };∆ ={e +e };
i i+1 n−1 n u n−1 n
n
1
ζ = e ; λ=cζ withc∈2[3−n]+Z where[x]denotesthelargestintegernotgreater
2X i 2 ≥0
i=1
than x∈R.
(vi) (E ,SO(2)×SO(10)): ∆={α |1≤i≤6} with α = 1(e −e −e −e −e −e −
6(−78) i 1 2 1 2 3 4 5 6
e +e ), α =e +e , α =e −e (3 ≤i≤6); ∆ ={α }; ζ = 2(−e −e +e );
7 8 2 1 2 i i−1 i−2 u 1 3 6 7 8
λ=cζ with c∈−3+Z .
≥0
(vii) (E ,SO(2)×E ): ∆ = {α | 1 ≤ i ≤ 7} with α (1 ≤ i ≤ 6) same as in (vi)
7(−78) 6(−78) i i
and α =e −e ; ∆ ={α }; ζ =e − 1e + 1e ; λ=cζ with c∈−8+Z .
7 6 5 u 7 6 2 7 2 8 ≥0
Remark 1.2. An observation from Theorem 1.1 is that in each case, the set of parameters
for which the corresponding scalar generalized Verma modules of abelian type are reducible
constitutes a finite set and a semi-infinite arithmetic progression.
Remark 1.3. Although Theorem 1.1 only discusses complex simple Lie algebras, people may
deduce the results for complex semisimple Lie algebras immediately. In fact, suppose that
g = g denotes the decomposition of a finite-dimensional complex semisimple Lie algebra
M j
j
into its simple ideals, and p is a parabolic subalgebra of g. Then
• p= p , with each p a parabolic subalgebra of g .
M j j j
j
• p has abelian (respectively, nilpotent) radical if and only if each p has abelian (respec-
j
tively, nilpotent) radical.
• λ = λ is a dominant integral highest weight with respect to p if and only if each λ
X j j
j
is dominant integral highest weight with respect to p .
j
• For the above data, Mg(λ)= Mgj(λ ).
p O pj j
j
• Mg(λ) is a scalar (simple) generalized Verma module if and only if each Mgj(λ ) is a
p pj j
scalar (simple) generalized Verma module.
3
From these points, we actually obtain the parameters of all the reducible scalar generalized
Verma modules of abelian type for complex semisimple Lie algebras.
The reducibility problem can be studied for scalar generalized Verma modules attached to
arbitraryparabolicalgebras. Concretely,letp=l+u beaparabolicsubalgebraofasemisim-
+
ple Lie algebra g, where l is a Levi factor and u is the nilpotent radical. Put u = u and
+ 0 +
u =[u ,u ] for positive integer k. We call u the kth-step of u for a nonnegative integer
k + k−1 k +
k. The nilpotent Lie algebra u is called k-step nilpotent if u 6= 0 and u = 0. If the
+ k−1 k
nilpotent radical u of the parabolic subalgebra p is k-step nilpotent, then we say that p is
+
a parabolic subalgebra of k-step nilpotent type. In particular, if u = u = 0, then p = g,
0 +
and hence generalized Verma modules attached to p are just finite-dimensional complex sim-
ple modules of g, whose reducibility problems are well-known [Kn, Theorem 5.5]. If p is a
parabolic subalgebra of 1-step nilpotent type, then it is just a parabolic subalgebra of abelian
type,thescalargeneralizedVermamodulesattachedtowhicharewhatthispaperhandles. As
foraparabolicsubalgebrapof2-stepnilpotenttype,ifdimCu1 =1,thenpiscalledaparabolic
subalgebraof2-stepnilpotentHeisenbergtype;else,itiscalledaparabolicsubalgebraof2-step
nilpotent non-Heisenberg type. T. Kubo investigated and solved the reducibility problem for
scalar generalized Verma modules associated to exceptional simple Lie algebras and maximal
parabolicsubalgebrasof2-stepnilpotentnon-Heisenbergtypein[Ku]. Moreover,accordingto
T. Kubo, generalized Verma modules associated to exceptional simple Lie algebras and maxi-
mal parabolic subalgebras of 2-step nilpotent Heisenberg type were studied by R. Zierau, but
the results were not published.
The paper is organized as follows. In Section 2, we recall Janzten’s criterion for simplicity of
generalized Verma modules attached to maximal parabolic subalgebras, and then recall the
techniques in [EHW], both of which offer us powerful tools to deal with our problem. In
Section 3, we prove Theorem 1.1 in a case-by-casefashion for all seven cases.
2 Janzten’s Criterion and Special Lines
2.1 Janzten’s Criterion
Inthis section,we recallthe irreducibility criteriondue toJ.C.Jantzenfor generalizedVerma
modules. Because we focus on parabolic subalgebras of abelian type, which are automatically
maximal parabolic subalgebras, we only state a specialization of Jantzen’s criterion for scalar
generalized Verma modules attached to maximal parabolic subalgebras p.
First of all, we fix some notations for the paper. All Lie algebras and modules considered in
this paper are over the complex number field C unless we make any declaration. Denote by
Z and Z the set of nonnegative integers and the set of positive integers respectively. Let
≥0 >0
g be a finite-dimensional complex semisimple Lie algebra. Choose a Cartan subalgebra h of
g, and a Borel subalgebra b containing h. Denote by Φ, Φ+, and ∆ the root system, the set
of positive roots, and the simple root system with respect to (g,b,h) respectively. Let p be
a maximal parabolic subalgebra such that b ⊆ p ⊆ g, and denote by p = l+u the Levi
+
decomposition with respect to (h,∆), where l is the Levi factor with h ⊆ l and l+b = p,
4
and u is the nilpotent radical. Now we may denote by Φ the root system for (l,h), and set
+ l
Φ+ :=Φ ∩Φ+ and∆ :=Φ ∩∆. Also,defineΦ+ :=Φ+\Φ+. Let(−,−)betheinnerproduct
l l l l u l
on h∗ induced by the Killing form of g, and for µ,ν ∈ h∗ define hµ,νi := 2(µ,ν). Now the set
(ν,ν)
of Φ+-dominant integral weights is defined as Λ+ := {λ ∈ h∗ | hλ,αi ∈ Z for all α ∈ Φ+}.
l l ≥0 l
Moreover,denote by ρ half the sum of the positive roots in Φ+. For α∈Φ, define a reflection
s : h∗ → h∗ given by s (λ) = λ−hλ,αiα for λ ∈ h∗, and then denote by W (respectively,
α α
W ) the Weyl group of (g,h) (respectively, (l,h)) generated by s for α ∈ ∆ (respectively, for
l α
α ∈ ∆). Let U(g) (respectively, U(p)) be the universal enveloping algebra of g (respectively,
l
p).
If λ ∈ h∗ is Φ+-dominant integral, let F be the finite-dimensional complex simple l-module
l λ
with highest weight λ. By letting the elements in u act by 0, F is induced to a p-module.
+ λ
Now the generalized Verma module of g attached to p with the parameter λ is defined to be
Mg(λ):=U(g)⊗ F .
p U(p) λ
When dimCFλ = 1, i.e., (λ,α) = 0 for all α ∈ ∆l, it is called a scalar generalized Verma
module.
Theorem 2.1 ([H, Theorem 9.12]). Let λ ∈ Λ+. If hλ+ρ,βi ∈/ Z for all β ∈ Φ+, then
l >0 u
Mg(λ) is simple. The converse also holds if λ+ρ is regular.
p
The following notation is important in the statement of Jantzen’s criterion. For λ∈h∗, define
Y(λ):=D−1 (−1)l(ω)eωλ (2.1.1)
X
ω∈Wl
where l(ω) denotes the length of ω ∈W , eµ is a function on h∗ which takes values 1 at µ and
l
0 elsewhere, and D = eρ (1−e−α) is the Weyl denominator. It is clear that Y(λ) is the
Y
α∈Φ+
character formula of Mg(λ−ρ) if λ is Φ+-dominant integral.
p l
Proposition 2.2 ([Ku, Corollary A.1.5], [M, Corollary 2.2.10]). We have the following two
properties:
(1) If λ ∈ h∗ satisfies (λ,α) = 0 for some α ∈ Φ , i.e., λ is Φ -singular, then Y(λ) = 0.
l l
Conversely, if λ∈h∗ satisfies hλ,αi∈Z\{0} for all α∈Φ , i.e., λ is Φ -regular integral,
l l
then Y(λ)6=0.
(2) For λ∈h∗ and ω ∈W , we have Y(ωλ)=(−1)l(ω)Y(λ).
l
Denote by s the reflection corresponding to β ∈Φ in W. Set
β
S :={β ∈Φ+ |hλ+ρ,βi∈Z }. (2.1.2)
λ u >0
Then Jantzen’s criterion for the scalar generalized Verma modules attached to a maximal
parabolic subalgebra p is stated as follows.
Theorem 2.3 ([J, Satz 3], [M, Theorem 2.2.11]). Let p be a maximal parabolic subalgebra.
Then the scalar generalized Verma module Mg(λ) is irreducible if and only if
p
Y(s (λ+ρ))=0.
X β
β∈Sλ
5
To use Jantzen’s criterion, we need to determine whether Y(s (λ+ρ)) is 0 or not. This
X β
β∈Sλ
is answered by the following proposition.
Proposition2.4([Ku,PropositionA.2.4]). Thesum Y(s (λ+ρ))is nonzeroifandonly
X β
β∈Sλ
if there is β ∈S satisfying the following two conditions: (a) Y(s (λ+ρ))6=0; and (b) there
0 λ β0
do not exist β ∈S \{β } and ω ∈W of odd length such that s (λ+ρ)=ωs (λ+ρ).
λ 0 l β0 β
2.2 Special Lines
Let us recall the construction of special lines in [EHW]. Henceforth g is a finite-dimensional
complex simple Lie algebra, and p is a parabolic subalgebra of abelian type. Recall from
Section 1 that ζ ∈h∗ is the unique weight such that (ζ,α)=0 for all α∈Φ and hζ,γi=1.
l
Definition 2.5. If λ ∈h∗ satisfies (λ +ρ,γ)= 0, then λ +zζ for z ∈C is called a special
0 0 0
line in h∗.
Obviously,everyelementλ∈h∗ canbe expresseduniquely inthe formλ +zζ withz ∈C and
0
(λ +ρ,γ) = 0. Hence, every element λ ∈ h∗ lies in some special line. In fact, given λ ∈ h∗,
0
if λ = λ +zζ with z ∈ C and (λ +ρ,γ) = 0, then z = hλ+ρ,γi and λ = λ−zζ. This
0 0 0
shows uniqueness. Moreover, hλ +ρ,γi=hλ−zζ+ρ,γi=hλ+ρ,γi−z =0, and existence
0
is showed.
The following result can be found in [EHW], which we state as a theorem here.
Theorem 2.6. For each special line λ +zζ, there exist three real numbers A(λ ), B(λ ), and
0 0 0
C(λ ) with C(λ )>0, A(λ )≤B(λ ), and C(λ )−1(B(λ )−A(λ ))∈Z satisfying:
0 0 0 0 0 0 0 ≥0
(1) If z ∈R and z <A(λ ), then Mg(λ +zζ) is simple.
0 p 0
(2) If z =A(λ )+iC(λ ) for 0≤i≤C(λ )−1(B(λ )−A(λ )), then Mg(λ +zζ) is reducible.
0 0 0 0 0 p 0
Proof. See Theorem 2.4 and Proposition 3.1(b) in [EHW].
We shall make full use of the values of A(λ ), B(λ ), and C(λ ) listed in [EHW] to do
0 0 0
computations for the Lie algebra pairs (g,p) with g simple and p a parabolic subalgebra of
abelian type, case by case according to the Hermitian symmetric pairs.
3 Reducibility of Scalar Generalized Verma Modules of
Abelian Type
In this section, we shall do computations case by case. According to [W], up to isomorphism
there are seven complex Lie algebra pairs (g,p) with g simple and p a parabolic subalgebra of
abelian type, which correspond to the Hermitian symmetric pairs of compact type:
6
(i) (SU(p+q),S(U(p)×U(q))) for p,q≥1,
(ii) (Sp(n),U(n)) for n≥2,
(iii) (SO(2n+1),SO(2)×SO(2n−1)) for n≥2,
(iv) (SO(2n),SO(2)×SO(2n−2)) for n≥2,
(v) (SO(2n),U(n)) for n≥2,
(vi) (E ,SO(2)×SO(10)),
6(−78)
(vii) (E ,SO(2)×E ).
7(−133) 6(−78)
Ineachcase,itis easilyverifiedthatthe MinkowskisumΦ++Φ+ :={α+β |α,β ∈Φ+}⊆h∗
u u u
does not intersect Φ+, whence u is abelian.
u +
Weseebelowthatthecomputationsforthefirsttwopairsarealmosttrivialbecausewedonot
needtouseJantzen’scriterion,whilethecomputationsforthelasttwopairsaremoreinvolved
because the root structures and Weyl groups of exceptional types are complicated. Moreover,
forthe pair(SO(2n),U(n)), wehaveto discussseparatelyaccordingtothe parityofn. Retain
all the notations and settings in the previous two sections.
3.1 (SU(p+q),S(U(p)×U(q))) for p,q ≥ 1
Let g = sl(p + q,C) and let p = l + u be a parabolic subalgebra of abelian type with
+
l=s(gl(p,C)⊕gl(q,C)). We may choose a Cartan subalgebra h⊆l and a simple root system
∆ = {e −e | 1 ≤ i ≤ p+q−1}, such that p is standard with respect to (h,∆). Then
i i+1
∆ = {e −e | 1 ≤ i ≤ p+q−1,i 6= p} and Φ+ = {e −e | i ≤ p,j > p}. Moreover, we
l i i+1 u i j
p+q
p+q+1
have ρ= ( −i)e .
X 2 i
i=1
p p+q
aq ap
If Mg(λ) is of scalar type, an easy computation shows that λ = e − e
p p+q X i p+q X j
i=1 j=p+1
for some a∈C.
p p+q
aq ap
Lemma 3.1. Suppose λ= e − e for some a∈C.
p+q X i p+q X j
i=1 j=p+1
(1) If a∈/ 2−p−q+Z , then Mg(λ) is simple.
≥0 p
(2) If a∈Z , then Mg(λ) is reducible.
≥0 p
Proof. It is obvious that λ+ρ is always Φ+-dominant integral. For i ≤ p and j > p, one
l
computes that hλ+ρ,e −e i = j −i+a. Now the conclusion follows from Theorem 2.1
i j
immediately.
7
p p+q
q p
Themaximalrootγ inΦ+ ise −e ,anditfollowsthatζ = e − e . Ifwe
1 n p+q X i p+q X j
i=1 j=p+1
p p+q
q p
writeλ=λ +zζinthespecialline,thenweobtainλ =( −q) e +(p− ) e
0 0 p+q X i p+q X j
i=1 j=p+1
and a=z−p−q+1. Now we may restate Lemma 3.1.
p p+q
aq ap
Lemma 3.1′. Suppose λ = e + e for some a ∈ C. Write λ = λ +zζ
p+q X i p+q X j 0
i=1 j=p+1
in the special line.
(1) If z ∈/ 1+Z , then Mg(λ) is simple.
≥0 p
(2) If z ∈p+q−1+Z , then Mg(λ) is reducible.
≥0 p
Proof. Because a=z−p−q+1, the conclusion follows from Lemma 3.1 immediately.
By Theorem 2.3, Lemma 7.3, and Theorem 7.4 in [EHW], one may check that A(λ ) =
0
max{p,q}, B(λ )=p+q−1, and C(λ )=1 in this case.
0 0
p p+q
aq ap
Theorem 3.2. Suppose λ = e + e for some a ∈ C. Then Mg(λ) is
p+q X i p+q X j p
i=1 j=p+1
reducible if and only if a∈1−min{p,q}+Z .
≥0
Proof. Write λ = λ +zζ in the special line, and then Lemma 3.1′(1) and Theorem 2.6(1)
0
show that Mg(λ) is reducible only if z ∈ max{p,q} + Z . On the other hand, Lemma
p ≥0
3.1′(2) and Theorem 2.6(2) show that if z ∈ max{p,q}+Z , then Mg(λ) is reducible. It
≥0 p
follows that Mg(λ) is reducible if and only if z ∈ max{p,q}+Z , which is equivalent to
p ≥0
a∈max{p,q}−p−q+1+Z =1−min{p,q}+Z .
≥0 ≥0
3.2 (Sp(n),U(n)) for n ≥ 2
Letg=sp(2n,C)andletp=l+u beaparabolicsubalgebraofabeliantypewithl=gl(n,C).
+
We may choose a Cartan subalgebra h⊆l and a simple root system ∆={e −e | 1≤i≤
i i+1
n−1}∪{2e }, such that p is standard with respect to (h,∆). Then ∆ = {e −e | 1 ≤
n l i i+1
i ≤ n−1} and Φ+ = {e +e | 1 ≤ i < j ≤ n}∪{2e | 1 ≤ k ≤ n}. Moreover, we have
u i j k
n
ρ= (n−i+1)e .
X i
i=1
n
If Mg(λ) is of scalar type, an easy computation shows that λ=a e for some a∈C.
p X i
i=1
n
Lemma 3.3. Suppose λ=a e for some a∈C.
X i
i=1
(1) If a∈/ 1−n+ 1Z , then Mg(λ) is simple.
2 ≥0 p
8
(2) If a∈ 1Z , then Mg(λ) is reducible.
2 ≥0 p
Proof. Itisobviousthatλ+ρisalwaysΦ+-dominantintegral. For1≤i<j ≤nand1≤k ≤n,
l
one computes that hλ+ρ,e +e i=2a+2n−i−j+2 and hλ+ρ,2e i=a+n−k+1. Now
i j k
the conclusion follows from Theorem 2.1 immediately.
n
The maximalrootγ in Φ+ is 2e , andit followsthatζ = e . Ifwe write λ=λ +zζ inthe
1 X i 0
i=1
n
special line, then we obtain λ =−n e and a=z−n. Now we may restate Lemma 3.3.
0 X i
i=1
n
Lemma 3.3′. Suppose λ=a e for some a∈C. Write λ=λ +zζ in the special line.
X i 0
i=1
(1) If z ∈/ 1+ 1Z , then Mg(λ) is simple.
2 ≥0 p
(2) If z ∈n+ 1Z , then Mg(λ) is reducible.
2 ≥0 p
Proof. Because a=z−n, the conclusion follows from Lemma 3.3 immediately.
By Theorem 2.3, Lemma 8.3, and Theorem 8.4 in [EHW], one may check that A(λ )= n+1,
0 2
B(λ )=n, and C(λ )= 1 in this case. Unlike the case in [EHW, Section 7], it is not trivial
0 0 2
to obtain these three numbers in this case. In fact, we have to show q = r in Lemma 8.3
andTheorem8.4in[EHW].AccordingtotheconstructionofR(λ )andQ(λ )under[EHW,
0 0
Theorem2.4],wehaveR(λ )=Q(λ )whichis the fullrootsystemofsp(n), andthen[EHW,
0 0
Equation (8.1)] shows q = r = n in our case. The three numbers A(λ ), B(λ ), and C(λ )
0 0 0
in the cases from Section 3.3 to Section 3.7 can be obtained by similar computations, but we
shall only mention the three numbers and the details are left to the reader.
n
Theorem 3.4. Suppose λ = a e for some a ∈ C. Then Mg(λ) is reducible if and only if
X i p
i=1
a∈ 1−n + 1Z .
2 2 ≥0
Proof. Write λ = λ +zζ in the special line, and then Lemma 3.3′(1) and Theorem 2.6(1)
0
show that Mg(λ) is reducible only if z ∈ n+1 + 1Z . On the other hand, Lemma 3.3′(2) and
p 2 2 ≥0
Theorem 2.6(2) show that if z ∈ n+1 + 1Z , then Mg(λ) is reducible. It follows that Mg(λ)
2 2 ≥0 p p
is reducible if and only if z ∈ n+1 + 1Z , which is equivalent to a∈ 1−n + 1Z .
2 2 ≥0 2 2 ≥0
3.3 (SO(2n+1),SO(2)×SO(2n−1)) for n ≥ 2
Let g = so(2n + 1,C) and let p = l+ u be a parabolic subalgebra of abelian type with
+
l = so(2,C)⊕so(2n−1,C). We may choose a Cartan subalgebra h ⊆ l and a simple root
system ∆={e −e |1≤i≤n−1}∪{e }, such that p is standard with respect to (h,∆).
i i+1 n
Then∆ ={e −e |2≤i≤n−1}∪{e }andΦ+ ={e ±e |2≤j ≤n}∪{e }. Moreover,
l i i+1 n u 1 j 1
n
1
we have ρ= (n−i+ )e .
X 2 i
i=1
9
If Mg(λ) is of scalar type, an easy computation shows that λ=ae for some a∈C.
p 1
Lemma 3.5. Suppose λ=ae for some a∈C.
1
(1) If a∈/ (3−2n+Z )∪(3 −n+Z ), then Mg(λ) is simple.
≥0 2 ≥0 p
(2) If a∈Z ∪(3 −n+Z ), then Mg(λ) is reducible.
≥0 2 ≥0 p
Proof. It is obvious that λ+ρ is always Φ+-dominant integral. For 2 ≤j ≤ n, one computes
l
thathλ+ρ,e +e i=a+2n−jandhλ+ρ,e −e i=a+j−1. Moreover,hλ+ρ,e i=2a+2n−1.
1 j 1 j 1
Now the conclusion follows from Theorem 2.1 immediately.
The maximalrootγ inΦ+ is e +e ,andit followsthatζ =e . Ifwe write λ=λ +zζ inthe
1 2 1 0
special line, then we obtain λ =(2−2n)e and a=z−2n+2. Now we may restate Lemma
0 1
3.5.
Lemma 3.5′. Suppose λ=ae for some a∈C. Write λ=λ +zζ in the special line.
1 0
(1) If z ∈/ (1+Z )∪(n− 1 +Z ), then Mg(λ) is simple.
≥0 2 ≥0 p
(2) If z ∈(n− 1 +Z )∪(2n−2+Z ), then Mg(λ) is reducible.
2 ≥0 ≥0 p
Proof. Because a=z−2n+2, the conclusion follows from Lemma 3.5 immediately.
ByTheorem2.3,Lemma11.3,andTheorem11.4in[EHW],onemaycheckthatA(λ )=n−1,
0 2
B(λ )=2n−2, and C(λ )=n− 3 in this case.
0 0 2
Proposition 3.6. Suppose λ=ae for some a∈C. Write λ=λ +zζ in the special line. If
1 0
z ∈Z and A(λ )<z <B(λ ), then Mg(λ) is simple.
0 0 p
Proof. Because a=z−2n+2, in fact we already computed that hλ+ρ,e +e i=z−j+2,
1 j
hλ+ρ,e −e i = z−2n+j +1 for 2 ≤ j ≤ n, and hλ+ρ,e i = 2z−2n+3. If z ∈ Z and
1 j 1
A(λ )=n− 1 <z <B(λ )=2n−2, then
0 2 0
S ={e +e |2≤j ≤n}∪{e −e |2n−z−1<j ≤n}∪{e }.
λ 1 j 1 j 1
n
1
Since λ+ρ=(z−n+ 3)e + (n−i+ )e , we have that (s (λ+ρ),e +e )=0
2 1 X 2 i e1+ej 2n−z−1 j
i=2
for2≤j ≤n andj 6=2n−z−1. Itfollowsif 2≤j ≤nandj 6=2n−z−1,thens (λ+ρ)
e1+ej
is Φ -singular, and hence Y(s (λ + ρ)) = 0 by Proposition 2.2(1). Similarly, we have
l e1+ej
(s (λ+ρ),e −e )=0for2n−z−1<j ≤n,soY(s (λ+ρ))=0for2n−z−1<
e1−ej 2n−z−1 j e1−ej
j ≤ n. On the other hand, it is easy to check that s (λ+ρ) and s (λ+ρ) are Φ -
e1+e2n−z−1 e1 l
regular integral, and hence Y(s (λ+ρ)) 6= 0 and Y(s (λ+ρ)) 6= 0 by Proposition
e1+e2n−z−1 e1
2.2(1). Moreover,because s (λ+ρ)=s s (λ+ρ) and s ∈W is of odd
e1+e2n−z−1 e2n−z−1 e1 e2n−z−1 l
length,itfollowsfromProposition2.2(2)thatY(s (λ+ρ))+Y(s (λ+ρ))=0. Now
e1+e2n−z−1 e1
Y(s (λ+ρ))=Y(s (λ+ρ))+Y(s (λ+ρ))=0.
X β e1+e2n−z−1 e1
β∈Sλ
By Theorem 2.3, Mg(λ) is simple.
p
10
