Scalar generalized Verma modules

0.17 MB·

by  Helge Maakestad

SCALAR GENERALIZED VERMA MODULES
HELGEMAAKESTAD
1
1
0 Abstract. In this paper we study the Verma module M(µ) associated to a
2 linearformµ∈h∗ wheresl(E)=n−⊕h⊕n+ isatriangulardecompositionof
sl(E). TheSL(E)-module M(µ) has acanonical simplequotient L(µ) witha
n
canonicalgeneratorv. Westudytheleftannihilatoridealann(v)inU(sl(E)).
a
J WealsostudyscalargeneralizedVermamoduleM(ρ)associatedtoacharacter
ρofpwherepisaparabolicsubalgebraofsl(E). WeproveM(ρ)hasacanon-
7
ical simple quotient L(ρ). This simple quotient is in some cases an infinite
1
dimensional SL(E)-module. We get a class of mutually non-isomorphic irre-

ducibleSL(E)-modulescontainingtheclassofallfinitedimensionalirreducible
]
T SL(E)-modules. As aresultwegivean algebraicproofof aclassicalresultof
Smoke on the structure of the jet bundle as P-module on any flag-scheme
R
SL(E)/P whereP isanyparabolicsubgroup.
.
h
t
a
m
[ Contents

1 1. Introduction 1
v
2. Scalar generalized Verma modules 2
4
3. Classical Verma modules and annihilator ideals 6
3
1 References 14
3
.
1
0
1 1. Introduction
1
: We study the scalar generalized Verma module M(ρ) associated to a character
v
i ρ of p where p is a parabolic subalgebra of sl(E). We prove M(ρ) has a canonical
X
simple quotient L(ρ). This simple quotient is in some cases an infinite dimen-
r sionalSL(E)-module. We get a construction of a class of mutually non-isomorphic
a
irreducible SL(E)-modules containing the class of finite dimensional irreducible
SL(E)-modules. This class contains many infinite dimensional SL(E)-modules.
Let V ba an aritrary finite dimensional irreducible SL(E)-module. In this note
λ
we give a construction of V using the enveloping algebra U(sl(E)) and the Verma
λ
modules M(µ). We study the annihilator ideal ann(v) in U(sl(E)) where v is the
highest weight vector v in V We prove the class of simple modules L(ρ) may be
λ
constructed using classical Verma modules M(µ) as done by Dixmier in [1]. As a
Date:March2010.
1991 Mathematics Subject Classification. 20G15,17B35,17B20.
Key words and phrases. annihilator ideal, irreducible representation, highest weight vector,
canonical filtration, canonical basis,generalized Vermamodule, semisimplealgebraicgroup, Lie
algebra,envelopingalgebra.
SupportedbyaresearchscholarshipfromNAV,www.nav.no.
1
2 HELGEMAAKESTAD
result we give an algebraic proof of a classical result of Smoke on the structure of
the jet bundle as P-module on SL(E)/P (see Corollary 3.19).
2. Scalar generalized Verma modules
In this section we construct any scalar generalizedVerma module M(ρ). Here ρ
is a character of p where p is any parabolic subalgebra of sl(E). We prove M(ρ)
has a canonical simple quotient L(ρ). When L(ρ) is finite dimensional we get a
construction of all finite dimensional irreducible SL(E)-modules.
Let G= SL(E) where E is an n-dimensional vector space over an algebraically
closed field K of characteristic zero and let g=sl(E). Let h be the abelian subal-
gebra of sl(E) of diagonal matrices. It follows (sl(E),h) is a split semi simple Lie
algebra and determines a root system R = R(sl(E),h). Let B be a basis for R.
Thisdetermines the positive rootsR andthe negativerootsR . This determines
+ −
a triangular decomposition sl(E)=n ⊕h⊕n of sl(E).
− −
Let 1 ≤ n < n < ··· < n ≤ n−1 be integers where dim(E) = n and let
1 2 k
l ,..,l ∈ Z. Let d = n , d = n −n for i = 2,..,n−1 and d = n−n . Let
1 k 1 1 i i i−1 n k
l = (l ,..,l ) and n = (n ,..,n ). Let e ,..,e be a basis for E as K-vector space
1 k 1 k 1 n
and let E =K{e ,..,e }. Let P in SL(E) be the subgroup fixing the flag
i 1 ni
E :06=E ⊆···⊆E ⊆E
• 1 k
in E. Let p=Lie(P). It follows p consists of matrices on the form
A ∗ ··· ∗
1
 
0 A ··· ∗
2
x= .. 
 0 . ··· ∗ 
 
 0 0 ··· Ak+1
where A is a d ×d -matrix and tr(x) =0. Let L =Kw be a rank one K-vector
i i i ρ
space on the element w. Define the following character ρ =ρ
l
ρ:p→End(L )
ρ
by
k
ρ(x)=η(x)w = l (tr(A )+···+tr(A ))w.
X i 1 i
i=1
Lemma 2.1. The pair (L ,ρ) is a rank one p-module. All rank one p-modules
ρ
arise in this way.
Proof. The proof is an exercise. (cid:3)
Let U (sl(E))⊆U(sl(E)) be the canonical filtration of U(sl(E)).
l
Definition 2.2. Let M(ρ)= U(sl(E))⊗ L be the generalized Verma module
U(p) ρ
associatedtoρ. LetM (ρ)=U (sl(E))⊗ L bethecanonical filtration ofM(ρ).
l l U(p) ρ
Since U(sl(E))⊗ L is a P-submodule of M(ρ) where p=Lie(P) it follows
l U(p) ρ
{M(ρ)} is a filtration of M(ρ) by P-modules.
l l≥0
Since dim (L )=1 wereferto M(ρ)asa scalar generalized Verma module. By
K ρ
definition this construction give all scalar generalized Verma modules for SL(E).
SCALAR GENERALIZED VERMA MODULES 3
Define the following sub Lie algebra of sl(E): n is the subalgebra of matrices x
on the form
A 0 ··· 0
1
 
∗ A ··· 0
2
x= .. 
 ∗ . ··· 0 
 
 ∗ ∗ ··· Ak+1
where A is a d ×d -matrix with zero entries. It follows there is an isomorphism
i i i
n⊕p∼=sl(E) asvectorspaces. Let sl(E)=n−⊕h⊕n+ be the standardtriangular
decomposition of sl(E) as defined in the previous section. It follows there is an
inclusion n ⊆ n of Lie algebras. Let R = R(sl(E),h) be the roots of sl(E) with
−
respect to h. Let B′ ={α ,..,α } be a subset of R such that the set
1 m
X ,..,X
−α1 −αm
is a basis for n.
LetP =(p ,..,p )andletXP =Xp1 ···Xpm . Letα =−p α −···−p α .
1 m −α1 −αm P 1 1 m m
Lemma 2.3. The following holds: The set
{XP ⊗w:p ,..,p ≥0}
1 m
is a basis for M(ρ) as K-vector space. The natural map
φ:U(n)→M(ρ)
defined by
φ(XP)=XP ⊗w
is an isomorphism of left n-modules.
Proof. Since n⊕p=sl(E) there is by definition an isomorphism
U(sl(E))∼=K{XP :p1,..,pm ≥0}U(p)
of free right U(p)-modules. We get
M(ρ)∼={XP :p1,..,pm ≥0}U(p)⊗U(p)Lρ ∼=
K{XP ⊗w :p ,..,p ≥0}.
1 m
Thefirstclaimis proved. One checksthe mapφis amapofleftn-modules andthe
Lemma is proved. (cid:3)
Let ω = L +···+L ∈ h for i = 1,..,n−1 be the fundamental weights for
i 1 i
SL(E) and let λ= k l ω . It follows for all x∈h⊆p that ρ(x)=λ(x).
Pi=1 i ni
Let ρ : U(p) → End(L ) be the associated morphism of ρ. Let L be the
U ρ
following left ideal of U(p):
L=U(p){x−ρ(x)1 :x∈p}.
p
Proposition2.4. LetN =ker(ρ ). ThereisanequalityofidealsinU(p): N =L.
U
Hence L is a two-sided ideal in U(p).
Proof. The proof follows [1], Section 7. There is a short exact sequence of rings
0→N →U(p)→K →0.
4 HELGEMAAKESTAD
Let x ,x ,..,x be a basis with ρ(x ) 6= 0 and ρ(x ) = ··· = ρ(x ) = 0. Assume
1 2 n 1 2 n
x=xv1xv2···xvn ∈U(p). It follows ρ (x)=0 if and only if v +···+v ≥1. We
1 2 n U 2 n
get by the PBW Theorem a direct sum decomposition
U(p)={xv1xv2···xvn :v +···+v ≥1}⊕{xv1 :v ≥0}.
1 2 n 2 n 1 1
It follows there is an inclusion of vector spaces
{xv1xv2···xvn :v +···+v ≥1}⊆N.
1 2 n 2 n
One checks there is an equality of vector spaces
xv1 :v ≥0}={xlρ(x )l :l ≥0}.
1 1 1 1
There is an inclusion
{xv1xv2···xvn :v +···+v ≥1}⊕{xv1 :v ≥1}⊆L
1 2 n 2 n 1 1
hence codim(L,U(p)) ≤ 1. Similarly codim(N,U(p)) = 1. Since L ⊆ N it follows
there is an equality L=N and the Proposition is proved. (cid:3)
Definition 2.5. Letchar(ρ)=U(g){x−ρ(x)1 :x∈p}bethe leftcharacter ideal
g
of ρ in U(g). Let char(ρ)=char(ρ)∩U(g) be the canonical filtration of char(ρ).
l l
Proposition 2.6. The natural map φ : U(g) → M(ρ) defined by φ(x) = x⊗w
defines an exact sequence of left U(g)-modules
0→char(ρ)→U(g)→φ M(ρ)→0.
Proof. Let X ,..,X be the basis for n constructed above and let
−α1 −αm
XP =Xp1 ···Xpm
−α1 −αm
with p ≥0 integers. There is an isomorphism of right U(p)-modules
i
U(g)∼={XP :pi ≥0}U(p)
hence we get an isomorphism of vector spaces
M(ρ)∼={XP ⊗w :pi ≥0}.
Assume X ∈ U(g) is an element. We may write X = XPx with x ∈ U(p).
PP P P
Assume φ(X)=X ⊗w =0. We get
φ(X)=X⊗w = XPx ⊗w = XP ⊗x w=0
X P X P
P P
It follows
x w =0
P
for all P hence x ∈ ker(ρ ) = U(p){y − ρ(y)1 : y ∈ p}. It follows X =
P U p
XPx ∈ char(ρ) hence ker(φ) ⊆ char(ρ). One checks char(ρ) ⊆ ker(φ) and
tPhePPropoPsition is proved. (cid:3)
By Lemma 2.3 it follows
M(ρ)=K{XP ⊗w :p ≥0}.
i
Let
α =−p α −···−p α ∈h∗.
P 1 1 m m
Assume x∈h. We get
x(XP ⊗w)=[x,XP]⊗w+XPx⊗w =
α (x)XP ⊗w+XP ⊗xw =
P
SCALAR GENERALIZED VERMA MODULES 5
α (x)XP ⊗w+λ(x)XP ⊗w=
P
(λ+α )(x)XP ⊗w.
P
Hence
M(ρ) =K{XP ⊗w}.
λ+αP
It follows
M(ρ)=⊕ M(ρ) .
P λ+αP
Define
M(ρ) =⊕ M(ρ) .
+ pi≥1 λ+αP
It follows
M(ρ)=M(ρ) ⊕M(ρ)
λ +
where
M(ρ) =K{1 ⊗w}.
λ g
Let the vector 1 ⊗v ∈M(ρ) be the canonical generator of M(ρ).
g
Theorem 2.7. Let M(ρ) be the scalar generalized Verma module associated to
the character ρ. The following holds: M(ρ) contains a maximal non-trivial sub-
g-module K. The quotient L(ρ) = M(ρ)/K is simple and dim (L(ρ)) ≥ 2. Let
K
v = 1⊗w ∈ L(ρ). The ideal ann(v) is the largest non-trivial left ideal in U(g)
containing char(ρ). The vector v satisfies the following: U(n )v =0. For all x∈h
+
it follows xv =λ(x)v hence v has weight λ
Proof. Consider the element X ⊗w∈M(ρ) and look at the product
−αm +
X X ⊗w.
αm −αm
We get
X X ⊗w =[X ,X ]⊗w+X ⊗X w =
αm −αm αm −αm −αm αm
H ⊗w=1 ⊗H w =
αm g αm
λ(H )(1 ⊗w).
αm g
It follows
X X ⊗w ∈M(ρ)
αm −αm λ
hence M(ρ) is not g-stable.
+
Assume L ( M(ρ) is a non-trivial g-stable module. It follows L∩M(ρ) = 0
λ
hence L⊆M(ρ) . Let K be the sum of all non-trivial sub-g-modules of M(ρ). It
+
follows K ⊆ M(ρ) since M(ρ) is not g-stable. Hence K ( M(ρ) is a maximal
+ +
non-trivial sub-g-module of M(ρ) and L(ρ) = M(ρ)/K is a simple quotient. It
follows dim (L(ρ))≥2.
K
One checks the vector v is annihilated by U(n ) and has weight λ.
+
By Proposition2.6 there is an exact sequence of left U(g)-modules
0→char(ρ)→U(g)→M(ρ)→0
hence there is an isomorphism
M(ρ)∼=U(g)/char(ρ)
of left U(g)-modules. It follows there is a bijection between the set of left sub-g-
modules of M(ρ) and left sub-g-modules of U(g)/char(ρ). This induce a bijection
between the set of left ideals in U(g) containing the ideal char(ρ) and the set of
left sub-g-modules of M(ρ). It follows the submodule K correspondsto a maximal
6 HELGEMAAKESTAD
non-trivial left ideal J in U(g) contaning char(ρ). The ideal J is by definition the
annihilator ideal of v: There is an equality
J =ann(v).
The Theorem is proved. (cid:3)
Corollary 2.8. The following holds:
(2.8.1) L(ρ) is simple for all l∈Zk.
(2.8.2) If l6=l′ it follows L(ρl)6=L(ρl′)
(2.8.3) If l ,..,l <0. It follows dim (L(ρ))=∞
1 k K
Proof. Claim2.8.1: ThisisbydefinitionofL(ρ)sincethesubmoduleK ismaximal.
Claim 2.8.2: Assume φ : L(ρl) →L(ρl′) is a map of g-modules. It follows φ is the
zero map or an isomorphism since the modules are simple. If it is an isomorphism
it follows the weights are equal. This implies l = l′ a contradiction. The claim is
proved. We prove claim 2.8.3: Assume dimK(L(ρ)) < ∞. It follows L(ρ) ∼= Vλ
where V has a highest weight vector v′ with highest weight λ′ = n−1l′ω with
λ Pj=1 j j
l′ ≥0. Sincethe vectorv inL(ρ)hasweightλwithl <0onegetsacontradiction.
j i
The Corollary follows. (cid:3)
Assume p = b = h⊕ n . It follows from [1], Section 7 the SL(E)-module
+ +
L(λ+δ)isisomorphictoV -thefinitedimensionalirreducibleSL(E)-modulewith
λ
highest weight λ= k l ω . Here l ≥ 1 is an integer for i = 1,..,k. Hence the
Pi=1 i ni i
class
{L(ρ ):ρ :p→K,l∈Zk}
l l
is a class of mutually non-isomorphic SL(E)-modules parametrized by a parabolic
subalgebra p ⊆ sl(E) and a character ρ : p → K containing the class of all finite
dimensional irreducible SL(E)-modules.
Let K = K∩M (ρ) ⊆ M(ρ) be the induced filtration on K. There is an exact
l l
sequence of P-modules
0→K →M (ρ)→L (ρ)→0.
l l l
Definition 2.9. Let {L (ρ)} be the canonical filtration of L(ρ).
l l≥0
It follows {L(ρ)} is a filtration of L(ρ) by P-modules.
l l≥0
Corollary 2.10. Assume L(ρ) = V is a finite dimensional irreducible SL(E)-
λ
module with highest weight vector v and highest weight λ. It follows Ll(ρ) ∼=
U(sl(E))v.
l
Proof. The proof is obvious. (cid:3)
3. Classical Verma modules and annihilator ideals
Let b = h⊕n . It follows b is a sub Lie algebra of sl(E). Let (x,n),(y,m)
+ + +
be elements of b . The Lie product on sl(E) induce the following product on b :
+ +
define the following action of h on n .
+
ad:h→End(n )
+
ad(x)(n)=[x,n].
It follows n is a h-module. Let ad(x)(h)=x(h).
+
SCALAR GENERALIZED VERMA MODULES 7
Lemma 3.1. Define
[(x,n),(y,m)]=(0,x(m)−y(n)+[n,m])
where [,] is the bracket on n . It follows the natural injection b →sl(E) is a map
+ +
of Lie algebras.
Proof. The proof is obvious. (cid:3)
Let µ∈h∗ be a linear form on h. Let L =Kw be the free rank one K-module
µ
on w. Define the following map
τ :b →End(L )
µ + µ
by
τ (x,n)(v)=µ(x)v.
µ
Lemma 3.2. The map τ makes L into a b -module.
µ µ +
Proof. Itisclear[τ (x,n),τ (y,m)]=τ ([(x,n),(y,m)])hencetheclaimisproved.
µ µ µ
(cid:3)
It follows L is a left b -module. By definition U(sl(E)) is a right b -module
µ + +
and we may form the tensor product
M(µ)=U(sl(E))⊗ L .
U(b+) µ
It follows M(µ) is a left G-module.
Definition 3.3. TheG-moduleM(µ)istheVermamodule associatedtothelinear
form µ∈h∗.
Let δ = 1 α and let λ be an element in h. Let L = Kv be the free
2Pα∈R+ λ−δ
rank one K-vectorspace on the element v. By the result above we get a character
τ :b →End(L )
λ−δ + λ−δ
defined by
τ (h,n)=(λ−δ)(h)
λ−δ
where(h,n)isanelementinb =h⊕n . Letα ,..,α be themdistinctelements
+ + 1 m
of R . Let X be an element in g−αi. It follows the set
+ −αi
X ,...,X
−α1 −αm
is a basis for n as vector space. There is a decomposition g = n ⊕b and it
− − +
follows U(g) is a free right U(b )-module as follows:
+
U(g)={Xp1 ···Xpm :p ,..,p ≥0}U(b ).
−α1 −αm 1 m +
Lemma3.4. AssumeXP =Xp1 ···Xpm withp ,..,p ≥0 integers. Letx∈h.
−α1 −αm 1 m
It follows
x(XP)=(−p α −···−p α )(x)XP =α (x)XP.
1 1 m m P
Proof. The proof is an easy calculation. (cid:3)
Let P = (p ,..,p ) with p integers and let α = p α + ··· + p α . Let
1 m i P 1 1 m m
XP =Xp1 ···Xpm .
−α1 −αm
8 HELGEMAAKESTAD
Proposition 3.5. Assume x∈h. The following holds:
(3.5.1) M(λ)=K{XP ⊗v :p ,..,p ≥0}
1 m
(3.5.2) x(XP ⊗v)=(λ−δ−α )(x)XP ⊗v
P
(3.5.3) M(λ)=⊕ M(λ)
p1,..,pm≥0 λ−δ−αP
(3.5.4) M(λ) =K(1⊗v)
λ−δ
(3.5.5) M(λ) =K{XP ⊗v :λ−δ−α =µ}
µ P
(3.5.6) U(n )(1⊗v)=M(λ)
−
Proof. We prove Claim 3.5.1: There is a direct sum decomposition
g=n ⊕b
− +
and the set
X ,..,X
−α1 −αm
is a basis for n as K-vector space. It follows by the PBW-theorem that the set
−
{XP :p ,..,p ≥0}
1 m
is a basis for U(n ) as K-vector space. It follows U(g) is isomorphic to
−
{XP :p ,..,p ≥}U(b )
1 m +
as free left U(b )-module. We get
+
M(λ)=U(g)⊗U(b+)Lλ−δ ∼=
{XP :p1,..,pm ≥0}U(b+)⊗U(b+)Lλ−δ ∼=
K{XP ⊗v :p ,..,p ≥0}
1 m
and Claim 3.5.1 is proved.
We prove Claim 3.5.2: By the previous Lemma it follows for all x∈h
[x,XP]=α (x)XP.
P
We get
x(XP ⊗v)=[x,XP]⊗v+XPx⊗v =
α (x)XP ⊗v+XP ⊗(λ−δ)(x)v =
P
(λ−δ−α )(x)XP ⊗v
P
and Claim 3.5.2 is proved. We prove Claim 3.5.3: By Claim 3.5.1 it follows M(λ)
has a basis given as follows:
{XP ⊗v :p ,..,p ≥0}.
1 m
Since for all x∈h it follows
x(XP ⊗v)=(λ−δ−α )(x)XP ⊗v
P
Claim 3.5.3 follows.
We prove Claim 3.5.4: Let XP ⊗v ∈ M(λ) It follows x(XP ⊗v) = (λ−δ −
α )(x)XP ⊗v. Hence XP ⊗v is in M(λ) if and only if
P λ−δ
λ−δ−α =λ−δ.
P
Since p ,..,p ≥ 0 this can only occur when p = ···p = 0 hence α = 0. The
1 m 1 m P
Claim is proved.
Claim 3.5.5 is by definition.
SCALAR GENERALIZED VERMA MODULES 9
We prove Claim 3.5.6: Since the map
φ:U(n )→M(λ)
−
defined by
φ(XP)=XP ⊗v
is an isomorphism of n -modules Claim 3.5.6 follows. The Proposition is proved.
−
(cid:3)
Definition 3.6. The element 1⊗v ∈ M(λ) is called the canonical generator of
M(λ).
Proposition 3.7. There is an isomorphism of n -modules
−
φ:U(n−)∼=M(λ)
defined by
φ(XP)=XP ⊗v.
Proof. Let n have basis X ,..,X and let XP = Xp1 ···Xpm . It follows
− −α1 −αm −α1 −αm
M(λ)hasabasisconsistingofelementsXP⊗v. Hencethemapφisanisomorphism
of vector spaces. One checks it is n -linear and the Proposition follows. (cid:3)
−
Let in the following E =K{e ,..,e } be an n-dimensional vector space over K
1 n
and let g = sl(E) with h in g the abelian sub-algebra of diagonal matrices. Let
E =K{e ,..,e } for i=1,..,n−1. It follows we get a complete flag
i 1 i
E :06=E ⊆E ⊆···⊆E ⊆E
• 1 2 n−1
in E with dim(E )=i. Let l=(l ,..,l ) with l ≥0 integers. Let
i 1 n−1 i
W(l)=Syml1(E)⊗Syml2(∧2E)⊗···⊗Symln−1(∧n−1E).
Letv =∧iE andletv =v l ⊗···⊗vln−1 bealineinW(l). LetP inSL(E)bethe
i i 1 1 n−1
subgroup of elements fixing the flag E . It follows P consists of upper triangular
•
matrices with determinant one. Let p = Lie(P). It follows p consists of upper
triangular matrices with trace zero. Let
ω =L +···+L
i 1 i
where
h∗ =K{L ,..,L }/L +···+l .
1 n 1 n
It follows ω ,..,ω are the fundamental weights for g. Let λ= n−1l ω .
1 n−1 Pi=1 i i
Proposition 3.8. Let x∈p be an element. The following formula holds:
n−1
x(v)= l (a +···+a )v
X i 11 ii
i=1
wherea is thei’thdiagonal element of x. Itfollows thevector v has weight λ. The
ii
SL(E)-module V generated by v is an irreducible finite dimensional SL(E)-module
λ
with highest weight vector v and highest weight λ.
Proof. The Proposition follows from [1], Section 7 and an explicit calculation. (cid:3)
10 HELGEMAAKESTAD
Let L in V be the line spanned by the vector v. It follows the subgroup of
v λ
SL(E) of elements fixing L equals the group P. We get a character
v
ρ:p→End(L )
v
defined by
n−1
ρ(x)v =x(v)= l (a +···+a )v.
X i 11 ii
i=1
We get an exact sequence of Lie algebras
0→p →p→End(L )→0
v v
where p =ker(ρ).
v
Definition 3.9. Let
char(ρ)=U(g){x−ρ(x)1 :x∈p}
g
be the left character ideal of ρ. Let
ann(v)={x∈U(g):x(v)=0}
be the left annihilator ideal of v. Let
char (ρ)=char(ρ)∩U (g)
l l
and
ann (v)=ann(v)∩U (g)
l l
for all l≥1.
If x ∈ h it follows ρ(x) = λ(x) where λ is the highest weight of V . By [1],
λ
Section 7 the following holds: Let for α∈B the integer m be defined as follows:
α
m =λ(H )+1
α α
where
06=H ∈[X ,X ].
α α −α
If α =L −L it follows X =E . It also follows X =E . We get
i i i+1 αi i,i+1 −αi i+1,i
H =E −E .
αi ii i+1,i+1
Lemma 3.10. The following holds for all i=1,..,n−1:
m =l +1.
αi i
Proof. The proof is an easy calculation. (cid:3)
Wegetby[1],Proposisition7.2.7,Section7thefollowingdescriptionoftheideal
ann(v) in U(g). Note that U(g) is a noetherian associative algebra and the ideal
ann(v) is a left ideal in U(g). It follows ann(v) has a finite set of generators. The
following holds: Let
I(v)=U(g)n + U(g)(x−λ(x)1 ).
+ X g
x∈h
It follows I(v)⊆U(g) is a left ideal. It follows
ann(v)=I(v)+ U(g)Xmα.
X −α
α∈B