Normality and non-normality of group compactifications in simple projective spaces

0.27 MB·English

by  Paolo Bravi

NORMALITY AND NON-NORMALITY OF GROUP COMPACTIFICATIONS IN
SIMPLE PROJECTIVE SPACES
PAOLOBRAVI,JACOPOGANDINI,ANDREAMAFFEI,ALESSANDRORUZZI
Abstract. If G is a complex simply connected semisimple algebraic group and if λ is a dominant
weight, we consider the compactification Xλ ⊂ P(cid:0)End(V(λ))(cid:1) obtained as the closure of the G×G-
orbitofthe identity andwegive necessaryand sufficientconditions onthe supportof λsothat Xλ is
normal;aswell,wegivenecessaryandsufficientconditions onthesupportofλsothatXλ issmooth.
1
1
0
2 Introduction

n Consider a semisimple simply connected algebraic group G over an algebraically closed field k of
a
J characteristiczero. If λis a dominantweight(with respectto afixed maximaltorus T anda fixedBorel

7 subgroup B ⊃T) and if V(λ) is the simple G-module of highest weight λ, then End V(λ) is a simple
1
G×G-module. Let I ∈End V(λ) be the identity map and consider the variety X ⊂P End(V(λ))
λ λ(cid:0) (cid:1)

] given by the closure of the G(cid:0)×G-o(cid:1)rbit of [Iλ]. In [Ka], S. Kannan studied for which λ t(cid:0)his variety i(cid:1)s
G
projectively normal, and this happens precisely when λ is minuscule. In [Ti], D. Timashev studied the
A
more generalsituation of a sum of irreducible representations,giving necessary and sufficient conditions
.
h for the normality and smoothness of these compactifications; however the conditions for normality are
t
a not completely explicit. In this paper we give an explicit characterizationof the normality of X , which
λ
m
allows to simplify the conditions for the smoothness as well.
[
To explain our results we need some notation. Let ∆ be the set of simple roots (w.r.t. T ⊂ B) and

2 identify ∆ with the vertices of the Dynkin diagram. Define the support of λ as the set Supp(λ)={α∈
v
∆ : hλ,α∨i6=0}.
8
7
4 TheoremA(seeTheorem13). ThevarietyXλ isnormalifandonlyifλsatisfiesthefollowing property:
2
(⋆) For every non-simply laced connected component ∆′ of ∆, if Supp(λ)∩∆′ contains a long root,
.
5
then it contains also the short root which is adjacent to a long simple root.
0
0
1 In particular, if the Dynkin diagram of G is simply laced then Xλ is normal, for all λ. In the paper
: we will prove the theorem in a more general form, for simple (i.e. with a unique closed orbit) linear
v
i projective compactifications of an adjoint group (see section 1.4). We will make use of the wonderful
X
compactification of G , the adjoint group of G, and of the results on projective normality of these
r ad
a compactificationsprovedbyS.Kannanin[Ka]. Theseresultsholdinthemoregeneralcaseofasymmetric
variety; however our method does not apply to this more general situation (see section 4.2).
Theorem B (see Theorem 22). The variety X is smooth if and only if λ satisfies property (⋆) of
λ
Theorem A together with the following properties:
i) For every connected component ∆′ of ∆, Supp(λ)∩∆′ is connected and, in case it contains a
unique element, then this element is an extreme of ∆′;
ii) Supp(λ) contains every simple root which is adjacent to three other simple roots and at least two
of the latter;
iii) Every connected component of ∆rSupp(λ) is of type A.
Theorem B can be generalized to any simple and normal adjoint symmetric variety. Following a
criterionofQ-factorialityforsphericalvarietiesgivenbyM.Brionin[Br],propertiesi)andii)characterize
1
the Q-factoriality of the normalization of X (see Proposition 20), while property iii) arises from a
λ
criterion of smoothness given by D. Timashev in [Ti] in the case of a linear projective compactification
of a reductive group.
As a corollary of Theorem B, we get that X is smooth if and only if its normalization is smooth.
λ
The paper is organizedas follows. In the first section we introduce the wonderful compactification of
G and the normalization of the variety X . In the second section we prove Theorem A, and in the
ad λ
third section Theorem B. In the last section we discuss some possible generalizations of our results.
1. Preliminaries
1.1. Notation. Recall that G is semisimple and simply connected. Fix a Borel subgroup B ⊂ G, a
maximal torus T ⊂ B and let U denote the unipotent radical of B. Lie algebras of groups denoted
by upper-case latin letters (G,U,L,...) will be denoted by the corresponding lower-case german letter
(g,u,l,...). Let Φ denote the set of roots of G relatively to T and ∆ ⊂ Φ the basis associated to the
choiceofB. Forallα∈∆lete ,α∨,f beansl(2)-tripleofT-weightsα,0,−α. LetΛdenotetheweight
α α
lattice of T and Λ+ the subset of dominant weights. For all α ∈ ∆, denote by ω the corresponding
α
fundamental weight.
If λ∈Λ, recall the definition of its support:
Supp(λ)={α∈∆ : hλ,α∨i6=0}.
If I ⊂∆, define its border ∂I, its interior I◦ and its closure I as follows:
∂I ={α∈∆rI : ∃β ∈I suchthathβ,α∨i6=0};
I◦ =Ir∂(∆rI);
I =I ∪∂I.
For λ∈Λ, denote by L the line bundle on G/B whose T-weight in the point fixed by B is −λ. For
λ
λ dominant, V(λ) = Γ(G/B,L )∗ is an irreducible G-module of highest weight λ; when we deal with
λ
different groups we will use the notation V (λ).
G
Denote by Π(λ) the set of weights occurring in V(λ) and set Π+(λ)=Π(λ)∩Λ+. Let λ7→λ∗ be the
linear involution of Λ defined by (V(λ))∗ ≃V(λ∗), for any dominant weight λ.
The weight lattice Λ is endowed with the dominance order 6 defined as follows: µ6λ if and only if
λ−µ ∈ N∆. If β = n α ∈ Z∆, define its support over ∆ (not to be confused with the previous
α∈∆ α
one) as follows:
P
Supp (β)={α∈∆ : n 6=0}.
∆ α
We introduce also some notations about the multiplication of sections. Notice that, for all λ,µ ∈ Λ,
L ⊗L = L . Therefore, if λ,µ are dominant weights and n ∈ N, the multiplication of sections
λ µ λ+µ
defines maps as follows:
m :V(λ)×V(µ)→V(λ+µ) and mn :V(λ)→V(nλ).
λ,µ λ
We will also write uv for m (u,v) and un for mn(u). Since G/B is irreducible, m and mn induce
λ,µ λ λ,µ λ
the following maps at the level of projective spaces:
ψ :P(V(λ))×P(V(µ))→P(V(λ+µ)) and ψn :P(V(λ))→P(V(nλ)).
λ,µ λ
The following lemma is certainly well known; however we do not know any reference.
Lemma 1. Let λ,µ be dominant weights.
i) If Supp(λ)∩Supp(µ) = ∅, then the map ψ : P(V(λ))×P(V(µ)) → P(V(λ+µ)) is a closed
λ,µ
embedding.
ii) For any n>0, the map ψn :P(V(λ))→P(V(nλ)) is a closed embedding.
λ
2
Proof. i). Fix highest weight vectors v ∈V(λ), v ∈V(µ) and v =v v ∈V(λ+µ).
λ µ λ+µ λ µ
IfV isirreducible,thenP(V)hasauniqueclosedorbit,namelythe orbitofthe highestweightvector.
Consequently, since P(V(λ))×P(V(µ)) has a unique closed orbit, in order to prove the claim it suffices
to prove that ψ is smooth in x = ([v ],[v ]) and that the inverse image of [v ] is x. The second
λ,µ λ µ λ+µ
claim is clear for weight reasons.
In order to prove that ψ is smooth in x, consider T-stable complements U ⊂V(λ), V ⊂V(µ) and
λ,µ
W ⊂V(λ+µ) of kv , kv and kv . So in a neighbourhood of x the map ψ can be described as
λ µ λ+µ λ,µ
ψ :U ×V −→W where ψ(u,v)=uv +v v+uv.
µ λ
The differentialof ψ in x is then givenby the differential of ψ in (0,0),thus it is described as follows:
λ,µ
dψ (u,v)=uv +v v.
x µ λ
Supposethatdψ isnotinjective. SinceitisT-equivariant,consideramaximalweightη ∈Π(λ+µ)r{λ+
x
µ} such that there exists a couple of non-zero T-eigenvectors (u,v) ∈ kerdψ with weights respectively
x
η−µ and η−λ. Suppose that η−µ∈Π(λ)r{λ} is not maximalandtake α∈∆ suchthat η−µ+α∈
Π(λ)r{λ} and e u6=0: then
α
(e u)v +v (e v)=e (uv +v v)=0
α µ λ α α µ λ
and η+α∈Π(λ+µ)r{λ+µ}, againstthe maximality of η. Thus η−µ is maximalin Π(λ)r{λ} and
similarly η−λ is maximal in Π(µ)r{µ}. Therefore, on one hand it must be
η−µ=λ−α
with α∈Supp(λ), while on the other hand it must be
η−λ=µ−β
with β ∈Supp(µ). Since Supp(λ)∩Supp(µ)=∅, this is impossible and shows that, if (u,v)∈kerdψ ,
x
thenitmustbeu=0orv =0. Supposenowthat(u,0)∈kerdψ : thenuv =0andbytheirreducibility
x µ
of G/B also u=0. A similar argument applies if v =0.
ii). Suppose that v,w ∈ V(λ) are such that vn = wn: then v = tw for some t ∈ k. Thus ψn is
λ
injective. Let us show now that ψn is smooth; it is enough to show it in x = [v ] where v ∈ V(λ) is
λ λ λ
a highest weight vector. Let V ⊂ V(λ) be the T-stable complement of kv , identified with the tangent
λ
space T P(V(λ)). If v ∈V, the differential d(ψn) is described as follows
x λ x
d(ψn) (v)=nvn−1v.
λ x λ
Thus d(ψn) is injective and ψn is smooth. (cid:3)
λ x λ
1.2. The variety X . If λ is a dominantweight,denote by E(λ) the G×G-module End(V(λ)) and set
λ
X the closure of the G×G-orbit of [I ]∈P(E(λ)). More generally if λ ,...,λ are dominant weights
λ λ 1 m
we define
X =G×G([I ],...,[I ])⊂P(E(λ ))×···×P(E(λ )).
λ1,...,λm λ1 λm 1 m
Since E(λ) is an irreducible G×G-module of highest weight (λ,λ∗), as a consequence of Lemma 1 we
get that if λ and µ have non-intersecting supports and if n∈N then
X ≃X and X ≃X .
λ+µ λ,µ nλ λ
As a consequence we get the following proposition:
Proposition 2. Let λ,µ be dominant weights. Then X ≃ X as G×G-varieties if and only if λ and
λ µ
µ have the same support. Moreover, if Supp(λ)={α ,...,α } then
1 m
X ≃X .
λ ωα1,...,ωαm
3
Proof. By the discussion above we have to prove only that the condition is necessary. This follows by
noticing that if X and X are G×G-isomorphic then also their closed G×G-orbits are isomorphic,
λ µ
which is equivalent to the fact that λ and µ have the same support. (cid:3)
1.3. The wonderful compactification of G and the normalization of X . When λ is a regular
ad λ
weight(i.e.Supp(λ)=∆)the varietyX iscalledthe wonderfulcompactificationofG andithasbeen
λ ad
studied by C. De Concini and C. Procesi in [DP]. We will denote this variety by M: it is smooth and
the complement of its open orbit is the union of smooth prime divisors with normal crossings whose
intersection is the closed orbit. The closed orbit of M is isomorphic to G/B×G/B and the restriction
of line bundles determines an embedding of Pic(M) into Pic(G/B×G/B), that we identify with Λ×Λ
as before; the image of this map is the set of weights of the form (λ,λ∗). Therefore Pic(M) is identified
with Λ and we denote by M a line bundle on M whose restriction to G/B ×G/B is isomorphic to
λ
Lλ⊠Lλ∗. If D ⊂ M is a G×G-stable prime divisor then the line bundle defined by D is of the form
M , where α is a simple root. The map D 7→α defines a bijection between the set of G×G-stable
αD D D
prime divisors and ∆, and we denote by M the prime divisor which corresponds to a simple root α.
α
We denote by s a section of M whose associated divisor is M ; notice that such a section is G×G-
α α α
invariant. More generally if ν = α∈∆nαα ∈ N∆, set sν = α∈∆snαα ∈ Γ(M,Mν). Then, given any
λ∈Λ, the multiplication by sν injects Γ(M,M ) in Γ(M,M ).
P λ−ν Qλ
Ifλis adominantweight,the mapG −→P(E(λ)) extends to amapq :M −→P(E(λ)) (see[DP])
ad λ
whose image is X and such that M = q∗(O (1)). If we pull back the homogeneous coordinates
λ λ λ P(E(λ))
of P(E(λ)) to M, we get then a submodule of Γ(M,M ) which is isomorphic to E(λ)∗; by abuse of
λ
notation we will denote this submodule by E(λ)∗.
If λ∈Λ, in [DP, Theorem 8.3] the following decomposition of Γ(M,M ) is given:
λ
Γ(M,M )= sλ−µE(µ)∗.
λ
µ∈ΛM+:µ6λ
Consider the graded algebra A(λ) = ∞ A (λ), where A (λ) = Γ(M,M ), and set X =
n=0 n n nλ λ
ProjA(λ). We have then a commutative diagram as follows:
L
e
pλ // //
M X
A λ
A
A
A
qλ AAAA e(cid:15)(cid:15)(cid:15)(cid:15) rλ
X
λ
In[Ka],ithasbeenshownthatA(λ)isgeneratedindegree1andin[D]thatr =r isthenormalization
λ
of X . Notice that the projective coordinate ring of X ⊂ P(E(λ)) is given by the graded subalgebra
λ λ
B(λ)= ∞ B (λ) of A(λ) generated by E(λ)∗ ⊂Γ(M,M ).
n=0 n λ
L
1.4. The variety X . We consider now a generalization of the variety X . Let Σ be a finite set of
Σ λ
dominant weights and denote E(Σ) = E(µ); let x = [(I ) ] ∈ P(E(Σ)) and define X as the
µ∈Σ Σ µ µ∈Σ Σ
closureofthe G×G-orbitofx inP(E(Σ)). IfΣ={λ},thenwe getthe varietyX , whileif Σ=Π+(λ)
Σ L λ
we getits normalizationX . Notice that the diagonalactionofGfixes the pointx so we haveaG×G
λ Σ
equivariantmapG−→X givenbyg 7−→(g,1)x . ThismapinducesamapfromG toX ifandonly
Σ Σ ad Σ
e
if the action of the center of G×G on E(λ) is the same for all λ ∈ Σ or equivalently if Σ is contained
in a coset of Λ modulo Z∆. In this case we say that X is a semi-compactification of G . If G is a
Σ ad ad
simple group and and Σ 6= {0} then X is a compactification of G , while if G is not simple we can
Σ ad ad
only say that is a compactification of a group which is a quotient of G .
ad
We say that Σ is simple if there exists λ ∈ Σ such that Σ ⊂ Π+(λ) or equivalently if Σ contains a
unique maximal element with respect to the dominance order 6. Notice also that if λ∈Σ is such that
for all µ ∈Σ different from λ the vector µ−λ is not in Q [∆] then is easy to construct a cocharacter
>0
4
χ : k∗ −→ G×G such that lim χ(t)x is the highest weight line in P(E(λ)). In particular X is a
t→0 Σ Σ
simple G×G semi-compactification of G if and only if Σ is simple.
ad
By the description of the normalization of X is Σ is simple and λ∈Σ is the maximal element, then
λ
we get
r // //
Xλ XΣ Xλ
In particular, it follows that r=r :X −→X is the normalization of X .
Σ λe Σ Σ
If Σ is simple, denote B(Σ)= ∞ B (Σ) the projective coordinate ring of X ⊂P(E(Σ)): it is the
n=0 n Σ
subalgebra of A(λ) generated by E(Σ)e∗ ⊂Γ(M,M ).
L λ
Remark 3. The discussion above and the fact that in P(E(λ)) there is only one point fixed by the
diagonal action of G (the line of scalar matrices) proves that any G×G linear projective compactifi-
cation of G is of the form X . A projective G×G-variety X is said to be linear if there exists an
ad Σ
equivariant embedding X ⊂P(V) where V is a finite dimensional rational G×G-module. In particular
as a consequence of Sumihiro’s Theorem (see for example [KKLV, Corollary 2.6]) all normal projective
compactifications are linear. In this paper we study only linear compactifications.
2. Normality
In this section we determine for which simple Σ the variety X is normal, proving in particular
Σ
Theorem A. In the following, by λ we will always denote the maximal element of Σ.
Let ϕ ∈ E(λ)∗ be a highest weight vector and set X◦ ⊂ X the open affine subset defined by the
λ Σ Σ
non-vanishingofϕ . Inparticular,wesetX =X andnoticethatX◦ =r−1(X◦). NoticethatX◦
λ λ Π+(λ) λ Σ Σ
is B×B-stable and, since it intersects the closed orbit, it intersects every orbit: therefore X is normal
Σ
if and only if XΣ◦ is normal if and only if thee restriction r Xe◦ :Xλ◦ →XΣ◦eis an isomorphism. Denote by
λ
B¯(Σ) the coordinate ring of X◦ and by A¯(λ) the coordina(cid:12)te ring of X◦; then we have
Σ (cid:12) e λ
ϕ ϕ
A¯(λ)={ : ϕ∈A (λ)}⊃{ : ϕ∈B (Σ)}=B¯(Σ)
ϕn n ϕn n e
λ λ
andX isnormalifandonlyifA¯(λ)=B¯(Σ). TheringsA¯(λ)andB¯(Σ)arenotG×G-modules,however
Σ
since X◦ is an open subset of X we still have an action of the Lie algebra g⊕g on them.
Σ Σ
By [Ka], A¯(λ) is generated by the elements of the form ϕ/ϕ with ϕ∈A (λ). In particular we have
λ 1
the following lemma.
Lemma 4. The variety X is normal if and only if for all µ ∈ Λ+ such that µ 6 λ there exists n > 0
Σ
such that
sλ−µE(µ+(n−1)λ)∗ ⊂B (Σ).
n
Proof. Let ϕ ∈ sλ−µE(µ)∗ be a highest weight vector and suppose that X is normal. Then, by the
µ Σ
descriptions of A¯(λ) and B¯(Σ), for every dominant weight µ6λ there exist n>0 and ϕ∈B (Σ) such
n
that ϕ/ϕn = ϕ /ϕ or equivalently ϕ = ϕ ϕn−1 ∈ B (Σ). Since ϕ is a highest weight vector of the
λ µ λ µ λ n
module sλ−µE(µ+(n−1)λ)∗ the claim follows.
Conversely assume that for every dominant weight µ6λ there exists n such that
sλ−µE(µ+(n−1)λ)∗ ⊂B (Σ);
n
inparticularϕ=ϕ ϕn−1 ∈B (Σ). Let’sprovethatϕ/ϕ ∈B¯(Σ)foreveryϕ∈sλ−µE(µ)∗;thisimplies
µ λ n λ
the thesis since A¯(λ) is generated in degree one. If ϕ = ϕ this is clear. Using the action of the Lie
µ
algebra g⊕g on B¯(Σ), let’s show that if ϕ/ϕ ∈B¯(Σ) then f (ϕ)/ϕ ∈B¯(Σ): indeed we have
λ α λ
f (ϕ) ϕ ϕ f (ϕ )
α α λ
=f ( )+ ·
α
ϕ ϕ ϕ ϕ
λ λ λ λ
and the claim follows since f (ϕ )∈E(λ)∗ ⊂B (Σ). (cid:3)
α λ 1
5
We can describe the set B (Σ) more explicitly. Indeed, as in [D] or in [Ka], it is possible to identify
n
sectionsofalinebundleonM withfunctionsonGandusethedescriptionofthemultiplicationofmatrix
coefficients. RecallthatasaG×G-modulewehavek[G]= E(λ)∗ ≃ V(λ)∗⊗V(λ). More
λ∈Λ+ λ∈Λ+
explicitly if V is a representation of G, define c :V∗⊗V −→k[G] as usual by c (ψ⊗v)(g)=hψ,gvi.
V L L V
If we multiply functions in k[G] of this type then we get
c (ψ⊗v)·c (χ⊗w)=c (ψ⊗χ)⊗(v⊗w) :
V W V⊗W
in particularwe getthat the imageof the multiplicationE(cid:0)(λ)∗⊗E(µ)∗ −→(cid:1)k[G] is the sum ofall E(ν)∗
with V(ν)⊂V(λ)⊗V(µ).
As a consequence we obtain the following Lemma:
Lemma 5 ([Ka, Lemma 3.1] or [D]). Let ν,ν′ be dominant weights, then the image of E(ν)∗ ⊗E(ν′)∗
in Γ(M,Mν+ν′) via the multiplication map is
sν+ν′−µE(µ)∗.
V(µ)⊂VM(ν)⊗V(ν′)
Proof. Indeed let π : G →M be the map induced by the inclusion G ⊂ M. Then any line bundle on
ad
G can be trivializedso that the image of π∗ :E(λ)∗ ⊂Γ(M,M )−→k[G] is the image of c andthe
ν V(λ)
claim follows from previous remarks. (cid:3)
Together with Lemma 4, this gives the following
Proposition 6. The variety X is normal if and only if, for every µ∈Λ+ such that µ6λ, there exist
Σ
n>0 and λ ,...,λ ∈Σ such that
1 n
V(µ+(n−1)λ)⊂V(λ )⊗···⊗V(λ ).
1 n
2.1. Remarks on tensor products. By Proposition 6, in order to establish the normality (or the
non-normality) of X , we need some results on tensor product decomposition.
Σ
Lemma 7. Let λ,µ,ν be dominant weights and let ∆′ ⊂ ∆ be such that Supp (λ+µ−ν) ⊂ ∆′; let
∆
L⊂G be the standard Levi subgroup associated to ∆′. If π ∈Λ+, denote by V (π) the simple L-module
L
of highest weight π. Then
V(ν)⊂V(λ)⊗V(µ) ⇐⇒ V (ν)⊂V (λ)⊗V (µ).
L L L
Proof. If a is any Lie algebra, denote U(a) the corresponding universal enveloping algebra.
Suppose that V (ν) ⊂ V (λ)⊗V (µ); fix maximal vectors v ∈ V (λ) and v ∈ V (µ) for the Borel
L L L λ L µ L
subgroupB∩L⊂Landfix p∈U(l∩u−)⊗U(l∩u−)suchthatp(v ⊗v )∈V (λ)⊗V (µ) isamaximal
λ µ L L
vector of weight ν. Since V (λ)⊗V (µ) ⊂ V(λ)⊗V(µ), we only need to prove that p(v ⊗v ) is a
L L λ µ
maximal vector for B too. If α∈∆′ then we have e p(v ⊗v )=0 by hypothesis. On the other hand,
α λ µ
if α ∈ ∆r∆′, notice that e commutes with p, since by its definition p is supported only on the f ’s
α α
with α∈∆′. Since v ⊗v is a maximal vector for B, then we get
λ µ
e p(v ⊗v )=pe (v ⊗v )=0;
α λ µ α λ µ
thus p(v ⊗v ) generates a simple G-module of highest weight ν.
λ µ
Assume conversely that V(ν) ⊂ V(λ)⊗V(µ) and fix p ∈ U(u−)⊗U(u−) such that p(v ⊗v ) ∈
λ µ
V(λ)⊗V(µ) is a maximal vector of weight ν. Since Supp (λ+µ−ν) ⊂ ∆′, we may assume that the
∆
only f ’s appearing in p are those with α∈∆′; therefore p(v ⊗v )∈V (λ)⊗V (µ) and it generatesa
α λ µ L L
simple L-module of highest weight ν. (cid:3)
Lemma 8. Fix λ,µ,ν ∈Λ+ such that V(ν)⊂V(λ)⊗V(µ). Then, for any ν′ ∈Λ+, it also holds
V(ν+ν′)⊂V(λ+ν′)⊗V(µ).
6
Table 1.
type of Φ highest short root
A α +···+α =ω +ω
r 1 r 1 r
B α +···+α =ω
r 1 r 1
C α +2(α +···+α )+α =ω
r 1 2 r−1 r 2
D α +2(α +···+α )+α +α =ω
r 1 2 r−2 r−1 r 2
E α +2α +2α +3α +2α +α =ω
6 1 2 3 4 5 6 2
E 2α +2α +3α +4α +3α +2α +α =ω
7 1 2 3 4 5 6 7 1
E 2α +3α +4α +6α +5α +4α +3α +2α =ω
8 1 2 3 4 5 6 7 8 8
F α +2α +3α +2α =ω
4 1 2 3 4 4
G 2α +α =ω
2 1 2 1
Proof. Fix a maximal vector vν′ ∈V(ν′) and consider the U-equivariantmap
φ: V(λ)⊗V(µ) −→ V(λ+ν′)⊗V(µ)
w1⊗w2 7−→ mλ,ν′(w1,vν′)⊗w2
Theclaimfollowssince,ifv ∈V(λ)⊗V(µ)isaU-invariantvectorofweightν,thenφ(v )∈V(λ+ν′)⊗
ν ν
V(µ) is a U-invariant vector of weight ν+ν′. (cid:3)
We nowdescribesomemoreexplicitresults. Whenwedealwithexplicitirreduciblerootsystems,un-
lessotherwisestated,wealwaysusethenumberingofsimplerootsandfundamentalweightsofBourbaki
[Bo].
In order to describe the simple subsets Σ ⊂ Λ+ which give rise to a non-normal variety X , we will
Σ
make use of following lemma.
Lemma 9.
(1) Let G be of type B . Then, for any n, V((n−1)ω )6⊂V(ω )⊗n.
r 1 1
(2) Let G be of type G . Then, for any n, V(ω +(n−1)ω )6⊂V(ω )⊗n.
2 1 2 2
Proof. We consider only the first case, the second is similar. Fix a highest weight vector v ∈ V(ω ).
1 1
If α is any simple root and if 1 6 s 6 r, notice that f acts non-trivially on f ···f v if and only
α αs−1 α1 1
if α = α . The T-eigenspace of weight 0 in V(ω ) is spanned by v = f ···f v , and similarly the
s 1 0 αr α1 1
T-eigenspaceof weight (n−1)ω in V(ω )⊗n is spanned by v⊗i−1⊗v ⊗v⊗n−i, where 16i6n. Since
1 1 1 0 1
the vectors
e (v⊗i−1⊗v ⊗v⊗n−i)=v⊗i−1⊗(e v )⊗v⊗n−i
αr 1 0 1 1 αr 0 1
are linearly independent, there exists no maximal vector of weight (n−1)ω in V(ω )⊗n. (cid:3)
1 1
Dual results will be needed to describe the subsets Σ which give rise to a normal variety X , but
Σ
before we need to introduce some further notation.
If Φ is an irreducible root system and ∆ is a basis for Φ we will denote by η the highest root if Φ is
simply laced or the highest short root if Φ is not simply laced. For the convenience of the reader we list
the highest short root of every irreducible root system in Table 1.
Recall the condition (⋆) defined in the introduction: a dominant weight λ satisfies (⋆) if, for every
non-simply laced connected component ∆′ ⊂ ∆, if Supp(λ)∩∆′ contains a long root then it contains
also the short root which is adjacent to a long simple root.
Definition 10. If ∆′ ⊂ ∆ is a non-simply laced connected component, order the simple roots in ∆′ =
{α ,...,α } starting from the extreme of the Dynkin diagram of ∆′ which contains a long root and
1 r
denote α the first short root in ∆′. If λ is a dominant weight such that α 6∈ Supp(λ) and such that
q q
7
Supp(λ)∩∆′ containsalongroot,denoteα thelastlongrootwhichoccursinSupp(λ)∩∆′;forinstance,
p
if ∆′ is not of type G , then the numbering is as follows:
2
q qq qq pppppppppppppppppppp qq q
α1 αp αq αr
The little brother of λ with respect to ∆′ is the dominant weight
λl∆b′ =λ−i=qpαi =( λλ−+ωω1p−+1ω−2ωp+ωq+1 iofthGeriws iosfetype G2
X
where ω is the fundamental weight associated to α if 1 6 i 6 r, while ω = ω = 0. The set of
i i 0 r+1
the little brothers of λ will be denoted by LB(λ); notice that LB(λ) is empty if and only if λ satisfies
condition (⋆) of Theorem A. For convenience, define LB(λ)=LB(λ)∪{λ}, while if ∆ is connected and
non-simply laced set λlb =λlb.
∆
Lemma 11. Assume G to be simple and let λ∈Λ+r{0}. Denote η the highest root of Φ if the latter
is simply laced or the highest short root otherwise.
(1) If λ satisfies the condition (⋆) then
V(λ)⊂V(η)⊗V(λ).
(2) If λ does not satisfy the condition (⋆) and if λlb is the little brother of λ then
V(λ)⊂V(η)⊗V(λlb).
Proof. If ∆ is simply laced, then V(η) ≃ g is the adjoint representation: in this case the claim follows
straightforward by considering the map g⊗V(λ) → V(λ) induced by the g-module structure on V(λ),
which is non-zero since λ is non-zero.
Suppose now that ∆ is notsimply laced. If λ satisfies condition(⋆), then by Lemma 8 it is enoughto
study the case λ=ω where α is a short simple root:
α
Type B : V(ω )⊂V(ω )⊗V(ω ).
r r 1 r
Type C : V(ω )⊂V(ω )⊗V(ω ), with i<r.
r i 2 i
Type F : V(ω )⊂V(ω )⊗V(ω ) and V(ω )⊂V(ω )⊗V(ω ).
4 3 4 3 4 4 4
Type G : V(ω )⊂V(ω )⊗V(ω ).
2 1 1 1
If λ does not satisfy condition (⋆), by Lemma 8 we can assume that λ=ω with α a long root:
α
Type B : V(ω )⊂V(ω )⊗V(ω ), if 1<i<r, and V(ω )⊂V(ω )⊗V(0).
r i 1 i−1 1 1
Type C : V(ω )⊂V(ω )⊗V(ω ).
r r 2 r−2
Type F : V(ω )⊂V(ω )⊗V(ω ) and V(ω )⊂V(ω )⊗V(ω +ω ).
4 1 4 4 2 4 1 4
Type G : V(ω )⊂V(ω )⊗V(ω ).
2 2 1 1
The above mentioned inclusion relations for tensor products are essentially known: let us treat the
case of type C with λ=ω and i<r, the other cases are easier or can be checked directly.
r i
Let v be a highest weight vector of V(ω ) and w be a highest weight vector of V(ω ). Let f be the
0 2 0 i
following product (in the universal enveloping algebra U(u−))
f =f ···f ·f ···f ·f ···f ,
αi α1 αi+1 αr−1 αr α2
and consider all the factorizations f =p·q such that p,q ∈U(u−). If β ,...,β ∈∆, set
1 j
r(f ···f )=(−1)j2δf ···f ,
β1 βj βj β1
where δ equals 0 (resp. 1) if α occurs an even (resp. odd) number of times in {β ,...,β }. Then it is
i 1 j
easy to check that the vector
p.v ⊗ rq.w
0 0
p·q=f
X
is a U-invariant vector in V(ω )⊗V(ω ) of T-weightω . (cid:3)
2 i i
8
IftheDynkindiagramofGisnotsimplylacedwewillneedsomefurtherpropertiesoftensorproducts.
If ∆ is connected but not simply laced, we will denote by α the short simple root that is adjacent
S
to a long simple root α ; moreover, we will denote the associated fundamental weights by ω and ω .
L S L
Finally, define ζ as the sum of all simple roots and notice that ω +ζ is dominant.
S
Lemma 12. Let λ be a non-zero dominant weight.
(1) If G is of type F or C (r >3) and if Supp(λ) contains a long root then
4 r
V(λ+ω )⊂V(ζ +ω )⊗V(λ).
S S
(2) If G is of type G and if λ does not satisfy (⋆) then
2
V(λ+ω )⊂V(ω )⊗V(λlb).
1 2
(3) If G is of type G and if α ∈Supp(λ) then
2 S
V(λ+ω )⊂V(ω )⊗V(λ).
1 2
Proof. By Lemma 8 it is enough to check the statements for λ=ω with α a long root in the first two
α
cases and α=α in the last case.
S
Type C : by Lemma 8 it is enough to check that V(ω )⊂V(ω )⊗V(ω ).
r r−1 1 r
Type F : we have λ=ω or λ=ω and ω +ζ =ω +ω .
4 1 2 S 1 4
Type G : we have λ=ω and λlb =ω in point (2) and λ=ω in point (3). (cid:3)
2 2 1 1
2.2. Normality and non-normality of X . We are now able to state the main theorem.
Σ
Theorem 13. Let Σ be a simple set of dominant weights and let λ be its maximal element. The variety
X is normal if and only if Σ⊃LB(λ).
Σ
Theorem A stated in the introduction follows immediately by considering the case Σ = {λ}. The
remaining part of this section will be devoted to the proof of Theorem 13. The general strategy will be
basedonProposition6andwillproceedbyinductiononthedominanceorderofweights. Theingredients
of this induction will be the results provedin section2.1 together with the descriptionof the dominance
order given by J. Stembridge in [St]: the dominance order between dominant weights is generated by
pairs which differ by the highest short root for a subsystem of the root system.
If K is a subset of ∆, denote Φ ⊂ Φ the associated root subsystem and, in case K is connected,
K
denote by η the correspondinghighest shortroot. Moreover,if β = n α, set β| = n α.
K α∈∆ α K α∈K α
The result of [St] that we will use is the following.
P P
Lemma 14 ([St, Lemma 2.5]). Let λ,µ be two dominant weights with λ > µ; set I = Supp (λ−µ).
∆
Let Φ be an irreducible subsystem of Φ (where K ⊂I).
K I
(a) If h(λ−µ) ,α∨i>0 for all α∈K∩Supp(µ), then µ+η 6λ.
K K
(b) If in addition hµ+η ,α∨i>0 for all α∈IrK, then µ+η ∈Λ+.
(cid:12) K K
(cid:12)
The next two lemmas are the main steps of our induction.
Lemma 15. Suppose that Φ is irreducible; let λ,µ ∈ Λ+ such that λ > µ and Supp (λ−µ) = ∆.
∆
Assume that either Φ is simply laced, or there exists a short root α∈Supp(λ) such that hλ−µ,α∨i>0,
or α ∈/ Supp(µ). Then there exists a connected subset K of ∆ such that
S
i) µ+η 6λ;
K
ii) µ+η ∈Λ+;
K
iii) K∩Supp(λ)6=∅.
Proof. Set K = {α ∈ ∆ : hλ−µ,α∨i > 0}. Since λ > µ we have that K ∩Supp(λ) is non-empty.
1 1
Notice also that Supp(µ)⊃∆rK . Define K as follows:
1
9
a) If Φ is simply laced, let K be a connected component of K which intersects Supp(λ).
1
b) If α ∈ Supp(λ) is a short root such that hλ−µ,α∨i > 0 let K be the connected component of
K containing α.
1
c) If Φ is not simply laced and there does not exist a short root α as in b), let K be a connected
component of K which intersects Supp(λ).
1
Properties i) and iii) are then easily verified by Lemma 14(a) and by construction.
To prove ii) notice that, if Φ is not simply laced, by the construction of K it follows that if α ∈K
L
then α ∈ K as well: indeed, K is a connected component of K and if there is no short root α as in
S 1
b) then α 6∈Supp(µ) implies α ∈K . By the description of highest short roots in Table 1 we deduce
S S 1
that, if α∈KrK◦, then the respective coefficient in η is 1: hence hη ,α∨i=−1 for all α∈∂K and,
K K
since Supp(µ)⊃∆rK ⊃∂K, we get µ+η ∈Λ+. (cid:3)
1 K
In order to proceed with the induction, in the next lemma we will need to consider the condition (⋆)
also for a Levi subgroup of G. If K ⊂∆ let L be the associated standard Levi subgroup; we say that
K
λ ∈ Λ+ satisfies condition (⋆ ) if, for every non-simply laced connected component K′ of K such that
K
Supp(λ)∩K′ contains a long root, Supp(λ)∩K′ contains also the short root adjacent to a long root.
Notice that if λ satisfies (⋆) then it also satisfies (⋆ ) for all K ⊂∆.
K
Similarlywecanalsodefinethelittlebrotherofadominantweightw.r.t.theLevisubgroupL : ifK′
K
is a connectedcomponentofK suchthat λdoes not satisfy(⋆K′), define the little brotherλlKb′ w.r.t. K′
as in Definition 10 and denote by LB (λ) the set of little brothers of λ constructed in this way. Notice
K
that if K′ is a connected component of K such that λ does not satisfy (⋆K′) and if ∆′ is the connected
component of ∆ containing K′, then λ does not satisfy (⋆∆′) as well and λlKb′ = λl∆b′. In particular
LB (λ)⊂LB(λ).
K
Lemma16. AssumeGtobesimpleandletλ,µbetwodominantweights suchthatλ>µandSupp (λ−
∆
µ)=∆. Then there exist µ′ ∈Λ+ and λ′ ∈LB(λ) such that µ<µ′ 6λ and
V(µ+λ)⊂V(µ′)⊗V(λ′).
Proof. Suppose first that either Φ is simply laced or α ∈/ Supp(µ) or there exists a short root α in
S
Supp(λ) such that hλ−µ,α∨i > 0. Take K as in Lemma 15 and set µ′ = µ+η : then by Lemma 11
K
together with Lemma 8 we get
V (µ+λ)⊂V (µ′)⊗V (λ′)
LK LK LK
with λ′ ∈LB (λ). The claim follows by Lemma 7 together with the inclusion LB (λ)⊂LB(λ).
K K
SupposenowthatΦisnotsimplylaced,thatα ∈Supp(µ)andthatthereisnoshortrootα∈Supp(λ)
S
such that hλ−µ,α∨i > 0. Since λ > µ there exists α ∈ Supp(λ) such that hλ−µ,α∨i > 0: therefore,
Supp(λ) contains at least a long root. Set µ′ = µ+ζ; notice that µ′ 6λ and that µ′ is dominant. The
claim follows then by Lemma 11 and by Lemma 8 if Φ is of type B, while if Φ is of type C, F or G it
4 2
follows by Lemma 12 and by Lemma 8. (cid:3)
Proof of Theorem 13. We prove first that the condition is necessary. Assume that there exists a little
brother µ = λlb of λ which is not in Σ. We prove that for every positive n and for every choice of
∆′
weights λ ,...,λ ∈Σ the module V(µ+(n−1)λ) is not contained in V(λ )⊗···⊗V(λ ).
1 n 1 n
We proceed by contradiction. Assume there exist weights λ ,...,λ as above and notice that any of
1 n
them satisfies µ 6 λ 6 λ: indeed, λ−µ = nλ−(µ+(n−1)λ) > nλ− λ > λ−λ for every i.
i i i
Therefore Supp ( λ −(µ+(n−1)λ))⊂Supp (λ−µ). By Definition 10 together with Lemma 7, it
∆ i ∆ P
is enough to analyse the case G of type B and Supp(λ) = {α } or G of type G and Supp(λ) = {α }.
P r 1 2 2
We analyse these two cases separately.
Type B : we have λ = aω , µ = (a −1)ω and µ+ (n−1)λ = (na−1)ω . If a = 1 we notice
r 1 1 1
that there are no dominant weights between λ and µ. So the only possibility is λ = λ = ω for all
i 1
10