On the tensor semigroup of affine kac-moody lie algebras

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by  Nicolas Ressayre

ON THE TENSOR SEMIGROUP OF AFFINE
KAC-MOODY LIE ALGEBRAS
NICOLAS RESSAYRE
Abstract. In this paper, we are interested in the decomposition
7
1 of the tensor product of two representations of a symmetrizable
0 Kac-Moody Lie algebra g. Let P+ be the set of dominant integral
2 weights. For λ P+, L(λ) denotes the irreducible, integrable,
n highestweightrep∈resentationofgwithhighestweightλ. LetP+,Q
a be the rational convexcone generated by P+. Consider the tensor
J
cone

1
1 Γ(g):={(λ1,λ2,µ)∈P+3,Q|∃N >1 L(Nµ)⊂L(Nλ1)⊗L(Nλ2)}.
Ifgisfinitedimensional,Γ(g)isapolyhedralconvexconedescribed
]
G in[BK06]byanexplicitfinitelistofinequalities. Ingeneral,Γ(g)is
norpolyhedral,norclosed. Inthisarticlewedescribetheclosureof
A
Γ(g)byanexplicitcountablefamilyoflinearinequalities,whengis
.
h untwisted affine. This solves a Brown-Kumar’s conjecture [BK14]
t in this case.
a
m We also obtain explicit saturation factors for the semigroup of
[ triples(λ1,λ2,µ) P+3 suchthatL(µ) L(λ1) L(λ2). Notethat
∈ ⊂ ⊗
even the existence of such saturation factors is not obvious since

2 the semigroup is not finitely generated. For example, in case A˜ ,
n
v
76 wtheepcraosveeAt˜h1asthaonwynininteg[BerKd104]≥. 2isasaturationfactor,generalizing
1
2
0
.
1
0
7 1. Introduction
1
: Let A be a symmetrizable irreducible GCM of size l + 1. Let h
v ⊃
Xi {α0∨,...,αl∨} and h∗ ⊃ {α0,...,αl} =: ∆ be a realization of A. We
fix an integral form h h containing each α∨, such that h∗ :=
r Z ⊂ i Z
a Hom(hZ,Z) contains ∆ and such that hZ/ ⊕ Zαi∨ is torsion free. Set
h∗ = h∗ Q h∗, P := λ h∗ α∨,λ 0 i , and P =
Q Z ⊗ ⊂ +,Q { ∈ Q |h i i ≥ ∀ } +
h P .
Z +,Q
∩
Let g = g(A) be the associated Kac-Moody Lie algebra with Cartan
subalgebra h. For λ P , L(λ) denotes the irreducible, integrable,
+
∈
highest weight representation of g with highest weight λ. Define the
saturated tensor semigroup as
Γ(A) := (λ ,λ ,µ) P3 N > 1 L(Nµ) L(Nλ ) L(Nλ ) .
{ 1 2 ∈ +,Q |∃ ⊂ 1 ⊗ 2 }
In the case that g is a semisimple Lie algebra, Γ(A) (also denoted
by Γ(g)) is a closed convex polyhedral cone given by an explicit set
1
2 NICOLAS RESSAYRE
of inequalities parametrized by some structure constants of the coho-
mology rings of the flag varieties for the corresponding algebraic group
(see [BK06, Kum14, Kum15, Res10]).
In the general case, Γ(A) is no longer closed or polyhedral. Never-
theless, in this paper we describe the closure of Γ(A) by infinitely many
explicit linear inequalities, if g is affine untwisted. Note that some in-
termediate results are true for any symmetrizable GCM A.
Let G be the minimal Kac-Moody group as in [Kum02, Section 7.4]
and B its standard Borel subgroup. Fix (̟ ,...,̟ ) h be ele-
α∨ α∨ Q
0 l ⊂
ments dual to the simple roots. Let W be the Weyl group of A. To any
simple root α , is associated a maximal standard parabolic subgroup
i
P , its Weyl group W W and the set WPi of minimal length repre-
i Pi ⊂
sentative of elements of W/W . We also consider the partial flag ind-
Pi
variety X = G/P containing the Schubert varieties X = BwP /P ,
i i w i i
for w WPi. Let ǫ H∗(X ,Z) be the Schubert basis dual
∈ { w}w∈WPi ⊂ i
to the basis of the singular homology of X given by the fundamental
i
classes of X . As defined by Belkale–Kumar [BK06, Section 6] in the
w
finite dimensional case, Brown-Kumar defined in [BK14, Section 7] a
deformed product in H∗(X ,Z), which is commutative and associa-
0 i
⊙
tive.
Theorem 1. Let g ba an affine untwisted Kac-Moody Lie algebra with
central element c. Let (λ ,λ ,µ) P3 such that λ (c) > 0 and
1 2 ∈ +,Q 1
λ (c) > 0.
2
Then,
(λ ,λ ,µ) Γ(g)
1 2
∈
if and only if
(1) µ(c) = λ (c)+λ (c),
1 2
and
(2) µ,v̟ λ ,u ̟ + λ ,u ̟
α∨ 1 1 α∨ 2 2 α∨
h i i ≤ h i i h i i
for any i 0,...,l and any (u ,u ,v) (WPi)3 such that ǫ occurs
1 2 v
∈ { } ∈
with coefficient 1 in the deformed product
ǫ ǫ .
u1⊙0 u2
The statement of Theorem 1 is very similar to [BK06, Theorem 22]
that describes Γ(g), if g is finite dimensional. Nevertheless, the proof
is very different. Indeed, in the classical case the main ingredients
are Kempf’s semistability theory and Hilbert-Mumford’s theorem (see
[BK06] or [Res10]). These results have no known generalization in our
situation. We overcome this difficulty by using a new strategy that we
now explain roughly speaking.
ON THE TENSOR SEMIGROUP OF AFFINE KAC-MOODY LIE ALGEBRAS 3
Consider the cone (g) defined by equality (1) and inequalities (2).
C
It remains to prove that, up to the assumption “λ (c) and λ (c) are
1 2
positive”, the cone (g) is equal to Γ(g). The proof proceeds in fives
C
steps.
Step 1. Γ(g) is convex.
Thisisawell-knownconsequenceofBorel-Weil’stheorem(seeLemma4).
Step 2. The set Γ(g) is contained in (g).
C
This step is proved in [BK14] and reproved here. The first ingredi-
ent is the easy implication in the Hilbert-Mumford’s theorem. Indeed
“semistable numerically semistable” is still true for ind-varieties and
⇒
ind-groups. In the finite dimensional case, the second argument is
Kmeiman’s transversality theorem. In [BK14], this is by an argument
inK-theorywhichexpressthestructureconstantsofH∗(G/P ,Z)asthe
i
Euler characteristic of sheaves supported by the intersection of three
translated Schubert orBirkhoff varieties. Here, we refine this argument
by proving a version of Kleiman’s theorem that allows to express these
structure constants as the cardinality of three translated Schubert or
Birkhoff varieties.
Step 3. The cone (g) is locally polyhedral.
C
This is a consequence of Proposition 4 below. We study the in-
equalities (2) defining (g). In particular, we use some consequences of
C
the nonvanishing of a structure constant of the ring H∗(G/P,Z) (see
Lemmas 17 and 18 below or [BK14]).
Step 4. Study of the boundary of (g).
C
Let (λ ,λ ,µ) be an integral point in the boundary of (g). Step 3
1 2
C
implies that some inequality (2) has to be an equality for (λ ,λ ,µ).
1 2
Then, one can use the following Theorem 2 to describe inductively the
multiplicity of L(µ) in L(λ ) L(λ ). Let α be a simple root and
1 2 i
⊗
let L denote the standard Levi subgroup of P . For w WPi and
i i
∈
λ P , w−1λ is a dominant weight for L : we denote by L (w−1λ)
∈ + i Li
the corresponding irreducible highest weight L -module.
i
Theorem 2. Here, g is any symmetrizable Kac-Moody Lie algebra and
α is a simple root. Let (λ ,λ ,µ) P3. Let (u ,u ,v) (WPi)3 such
i 1 2 ∈ + 1 2 ∈
that ǫ occurs with coefficient 1 in the ordinary product ǫ .ǫ . We
v u1 u2
assume that
(3) µ,v̟ = λ ,u ̟ + λ ,u ̟ .
α∨ 1 1 α∨ 2 2 α∨
h i i h i i h i i
Thenthe multiplicityofL(µ) in L(λ ) L(λ ) is equalto the multiplicity
1 2
⊗
of L (v−1µ) in L (u−1λ ) L (u−1λ )
Li Li 1 1 ⊗ Li 2 2
Note that Theorem 5 and its corollary in Section 6 are a little bit
stronger than Theorem 2.
4 NICOLAS RESSAYRE
Step 5. Induction.
Whereas there are numerous technical difficulties the basic idea is
simple. By convexity, it is sufficient to prove that the boundary of (g)
C
is contained in Γ(g). Using Step 4, this can be proved by induction.
More precisely, consider a face of codimension one of (g) as-
F C
sociated to some structure constant of H∗(G/P ,Z) for equal to
i 0
⊙
one. We have to prove that is contained in Γ(g). By Theorem 2,
F
it remains to prove that the points of satisfy the inequalities that
F
characterize Γ(L ). Fix such an inequality associated to a structure
i
constant of H∗(L /(P L ),Z) for equal to one. Consider the flags
i j i 0
∩ ⊙
ind-varieties:
L /(P L ) G/(P P )
i j i i j
∩ ∩
G/P G/P
i j
Proposition 3 that shows a property of multiplicativity for struc-
ture constants of the rings H∗(G/P,Z) gives us a structure constant of
H∗(G/(P P ),Z) equal to one, for the ordinary product. An impor-
i j
∩
tant point is Theorem 6 that proves that, if the considered inequality
of Γ(L ) is “useful” then this structure constant of H∗(G/(P P ),Z)
i i j
∩
is actually nonzero for . Then it gives a structure constant of
0
⊙
H∗(G/(P ),Z) for equal to one. In particular, this gives an in-
j 0
⊙
equality of (g) and the inequality we wanted to prove for the points
C
of .
F
If we prove Theorem 1 only for the untwisted affine case, the general
strategy should works more generally. For this reason, we prove some
intermediate results for any symmetrizable Kac-Moody Lie algebra. In
particular Steps 1, 2 and 4 works with this generality. Proposition 3 of
multiplicativity also holds in this context.
Let Q denote the root lattice of g. Consider the tensor semigroup
Γ (g) := (λ ,λ ,µ) P3 N > 1 L(µ) L(λ ) L(λ ) .
N { 1 2 ∈ +|∃ ⊂ 1 ⊗ 2 }
It is actually a semigroup but it is not finitely generated for g affine.
Despite this, we obtain explicit saturation factors: a positive integer d
0
is called a saturation factor for g if for any (λ ,λ ,µ) Γ(g) (P )3
1 2 +
∈ ∩
such that λ +λ µ Q, L(d µ) is a submodule of L(d λ ) L(d λ ).
1 2 0 0 1 0 2
− ∈ ⊗
Observe that the condition λ +λ µ Q is necessary to have L(µ)
1 2
− ∈ ⊂
L(λ ) L(λ ).
1 2
⊗
To describe our saturation factors, we need additional notation. Up
to now, g is the affine Lie algebra associated to the simple Lie algebra
g˙. Let us define the constant k to be the least common multiple
s
of saturation factors of maximal Levi subalgebras of g. The value of
ON THE TENSOR SEMIGROUP OF AFFINE KAC-MOODY LIE ALGEBRAS 5
k depends on known saturation factors for the finte dimensional Lie
s
algebras. With the actual literature (see Section 10), possible values
for k are given in the following tabular.
s
Type of g˙ A B B B (˜l 5) C (˜l 2) D D (˜l 5)
˜l 3 4 ˜l ≥ ˜l ≥ 4 ˜l ≥
k 1 2 4 2 1 4
s
Type of g˙ E E E F G G
6 7 8 4 2 2
k 36 144 3600 144 2 3
s
˙
Let k be the least common multiple of coordinates of θ written in
g˙
terms of the simple roots. The values of k are
g˙
Type A B (˜l 2) C (˜l 3) D (˜l 5) E E E F G
˜l ˜l ≥ ˜l ≥ ˜l ≥ 6 7 8 4 2
k 1 2 2 2 6 12 60 12 6
g˙
Theorem 3. Let (λ ,λ ,µ) (P )3 such that there exists N > 0
1 2 +
∈
such that L(Nµ) embeds in L(Nλ ) L(Nλ ). We also assume that
1 2
⊗
µ λ λ Q.
1 2
− − ∈
Then,
(i) if k = 1 then any integer d 2, L(dk µ) embeds in L(dk λ )
s g˙ g˙ 1
≥ ⊗
L(dk λ );
g˙ 2
(ii) if k > 1 then L(k k µ) embeds in L(k k λ ) L(k k λ ).
s g˙ s g˙ s 1 g˙ s 2
⊗
Observe that, in type A, k k = 1. The case A was obtained before
g˙ s 1
in [BK14].
Let δ denote the fundamental imaginary root. We also obtain the
following variation.
Theorem 4. Let (λ ,λ ,µ) (P )3 such that there exists N > 0
1 2 +
∈
such that L(Nµ) embeds in L(Nλ ) L(Nλ ). We also assume that
1 2
⊗
µ λ λ Q.
1 2
− − ∈
Then, for any integer d 2, L(k k µ dδ) embeds in L(k k λ )
g˙ s g˙ s 1
≥ − ⊗
L(k k λ ).
g˙ s 2
In Section 11, we collect some technical lemmas used in the paper.
Acknowledgements. IampleasedtothankMichaelBulois,Stéphane
Gaussent, Philippe Gille, Kenji Iohara, Nicolas Perrin, Bertrand Remy
for useful discussions.
The author is partially supported by the French National Agency
(Project GeoLie ANR-15-CE40-0012).
Contents
1. Introduction 1
2. Ind-varieties 6
2.1. Ind-varieties 6
6 NICOLAS RESSAYRE
2.2. Irreducibility 7
2.3. Line bundles 9
3. Using Borel-Weil Theorem 10
3.1. Tensor multiplicities 10
3.2. Multiplicities as dimensions 10
4. Enumerative meaning of structure constants of H∗(G/P,Z) 11
4.1. Richardson varieties 11
4.2. Kleiman’s lemma 12
4.3. The case nv = 1 15
u1u2
5. Inequalities for Γ(A) 15
6. Multiplicities on the boundary 17
7. The Belkale-Kumar-Brown product 25
7.1. Preliminaries of linear algebra 26
7.2. Definition of the BKB product 27
7.3. On Levi movability 29
8. Multiplicativity in cohomology 30
8.1. The multiplicativity 30
8.2. Application to the BKB-product 33
9. The untwisted affine case 34
9.1. Notation 34
9.2. Essential inequalities and BKB-product 35
9.3. About Γ(g) 38
9.4. A cone defined by inequalities 39
9.5. Realisation of as an hypograph 39
C
9.6. The convex set is locally polyhedral 40
C
9.7. An example of a face of codimension 1 43
9.8. The main result 43
10. Saturation factors 46
11. Some technical lemmas 48
11.1. Bruhat and Birkhoff decompositions 48
11.2. Affine root systems 50
11.3. Jacobson-Morozov’s theorem 51
11.4. Geometric Invariant Theory 52
References 54
2. Ind-varieties
2.1. Ind-varieties. Inthissection, wecollectdefinitions, notationand
properties of ind-varieties. The results are certainly well-known, but
we include some proofs for the convenience of the reader.
2.1.1. The category. Let(X ) beasequence ofquasiprojective com-
n n∈N
plex varieties given with closed immersions ι : X X . Con-
n n n+1
→
sider the inductive limit X = limX . A subset F in X is said to be
n
−→
ON THE TENSOR SEMIGROUP OF AFFINE KAC-MOODY LIE ALGEBRAS 7
Zariski closed if F X is closed for any n N. A continuous map
n
∩ ∈
f : X Y = limY between two ind-varieties is a morphism if for
n
−→ −→
anyn N thereexists m N such thatf(X ) Y andtherestriction
n m
∈ ∈ ⊂
f : X Y is a morphism. Let X′ X be closed subsets such
n −→ m n ⊂
that X = X′ and X′ X′ X′ . Then X′ = limX′
∪n∈N n 0 ⊂ 1 ⊂ ··· n ⊂ ··· −→ n
is an ind-variety. The filtrations (X ) and (X′) are said to be
n n∈N n n∈N
equivalentiftheidentity mapsX X′ andX′ X aremorphisms.
−→ −→
Actually, the filtrations on ind-varieties are regarded up to equivalence;
an ind-variety X endowed with a filtration X X X
0 1 n
⊂ ⊂ ··· ⊂ ···
by closed subsets is called a filtered ind-variety and simply denoted by
X = X .
n∈N n
∪
Lemma 1. Let X = X be a filtered ind-variety. Assume, we
n∈N n
∪
have a family (X′) of closed subsets in X such that X′ X′ and
n n∈N n ⊂ n+1
X = X′.
∪n∈N n
Then the two filtrations (X ) and (X′) are equivalent.
n n∈N n n∈N
Proof. Consider an irreducible component C of some X . Then C =
n0
X′ C and X′ C is closed in C. Assume that for any n, X′
∪n n ∩ n ∩ n ∩
C = C. Then, for any n, dim(X′ C) < dimC. Hence C is the
6 n ∩
union of countably many subvarieties of smaller dimension. This is a
contradiction since we are working on the uncountable field of complex
numbers: there exists n such that X′ C = C.
C nC ∩
Since X has finitely many irreducible components, there exists N
n0 0
such that X X′ .
n0 ⊂ N0
Observe that X X′ is closed in X and hence in X′ . Then the
n ∩ m m
same proof as above shows that for any n , there exists N such that
1 1
X′ X . (cid:3)
n1 ⊂ N1
For x X, we denote by T X the tangent space of X at x. By
x
∈
definition T X = limT X .
x x n
−→
2.2. Irreducibility. An ind-variety X is said to be irreducible if it is
as a topological space. A poset is said to be directed if for any two
elements x,y there exists z bigger or equal to x and y. If the poset of
irreducible components of the X ’s is directed for inclusion then X is
n
irreducible. Unless[Sha81,Proposition1],theconverse ofthisassertion
is not true (see [Kam96, Sta12] for examples). Here, the ind-variety X
is said to be ind-irreducible if the poset of irreducible components of
the X ’s is directed for inclusion. The following lemma gives a more
n
precise definition.
Lemma 2. Let X be an ind-variety. The following are equivalent:
(i) there exists a filtration X = X such that the poset of
n∈N n
∪
irreducible components of the X ’s is directed;
n
8 NICOLAS RESSAYRE
(ii) for any equivalent filtration X = X , the poset of irre-
n∈N n
∪
ducible components of the X is directed;
n
(iii) there exists an equivalent filtration X = X with X irre-
n∈N n n
∪
ducible, for any n.
If X satisfies these properties then X is said to be ind-irreducible.
Proof. We prove (ii) (i) (iii) (ii). The first implication is
⇒ ⇒ ⇒
tautological.
Show (i) (iii). Since X has finitely many irreducible components
n
⇒
the assumption implies that
n N and an irreducible component C of X such that X C .
N N n N
∀ ∃ ⊂
Then one can construct by induction an increasing sequence ϕ : N
−→
N and irreducible components C of X such that
ϕ(n) ϕ(n)
(4) n X C X .
ϕ(n) ϕ(n+1) ϕ(n+1)
∀ ⊂ ⊂
Note that the C ’s are closed, satisfy C C and X =
ϕ(n) ϕ(n) ϕ(n+1)
⊂
C . Moreover, by (4), the filtrations by X ’s and C ’s are
n∈N ϕ(n) n ϕ(n)
∪
equivalent.
Show (iii) (ii). Fix a filtration X = X with irreducible
n∈N n
⇒ ∪
closed subsets X . Let X = X′ be an equivalent filtration. Con-
n ∪n∈N n
sider two irreducible components C and C of some X′ and X′ .
n1 n2 n1 n2
There exist N and N in N such that C C X X′ . Since
1 2 n1 ∩ n2 ⊂ N1 ⊂ N2
X is irreducible, there exists an irreducible component C′ of X′
N1 N2 N2
containing X . Hence the poset of irreductible components of the X′
N1 n
is directed. (cid:3)
Examples.
(i) The simplest examples A(∞) = An, P(∞) Pn of ind-
n∈N n∈N
∪ ∪
varieties are ind-irreducible.
(ii) A nonempty open subset of an ind-irreducible ind-variety is
ind-irreducible. A product of two ind-irreducible ind-varieties
is ind-irreducible.
(iii) Consider a surjective morphism f : X Y of ind-varieties.
−→
If X is ind-irreducible then so is Y. Indeed, let X be a filtra-
n
tion of X by irreducible subvarieties. Denote by Y the closure
n
in Y of f(X ). Then, by Lemma 1, Y = Y is an equiv-
n n n
∪
alent filtration of Y by irreducible subvarieties. Hence Y is
ind-irreducible.
(iv) If G is a Kac-Moody group and P a standard parabolic sub-
group then G/P is a projective ind-variety. Indeed, a filtration
of G/P is given by the unions of Schubert varieties of bounded
dimension. Since the Bruhat order is directed (see e.g. [BB05,
Proposition 2.2.9]), G/P is ind-irreducible.
(v) The Birkhoff subvarieties of G/P are ind-varieties. Indeed the
Richardson varieties are irreducible.
ON THE TENSOR SEMIGROUP OF AFFINE KAC-MOODY LIE ALGEBRAS 9
(vi) Let G be the minimal Kac-Moody group as defined in [Kum02,
Section 7.4]. Then G is an ind-variety ind-irreducible. Indeed,
for each real root β, denote by U : C G the radicial
β
−→
subgroup. Consider an infinite word w = β ,...,β ,... in the
1 n
real roots of g such that any finite word in these roots is a
subword of w. Consider the map
θ : A(∞) T G
× −→
((τ ) ,t) ( U (τ ))t
i i∈N 7−→ i βi i
Since G is an ind-group and U Qare morphism of ind-groups,
β
θ is a morphism of ind-varieties. By definition, it is surjective.
By Example (iii), G is ind-irreducible.
Another result we need, is the following.
Lemma 3. LetX bean ind-varietyind-irreducible. LetΩ bea nonempty
open subset of X.
Let (Y ) be a collection of closed subsets of X such that Y Y
n n∈N n n+1
⊂
and Y contains Ω.
n∈N n
∪
Then X = Y is an equivalent filtration of X.
n∈N n
∪
Proof. Fix a filtration X = X by closed irreducible subsets of X
n∈N n
∪
intersecting Ω.
Fix n N. Observe that X Ω (X Y Ω). But
0 ∈ n0 ∩ ⊂ ∪n∈N n0 ∩ n ∩
X Y Ω isa locallyclosed subvariety ofX . Consider thesequence
n0∩ n∩ n0
n dim(X Y Ω). It is nondeacreasing.
7→ n0 ∩ n ∩
Assume that dim(X Y Ω) < dim(X Ω), for any n. Then
n0 ∩ n ∩ n0 ∩
X Ω is the union of contable many strict subvarieties. This is a
n0 ∩
contration since the base field is uncountable. Hence there exists N
such that dim(X Y Ω) = dim(X Ω).
n0 ∩ N ∩ n0 ∩
Then, X Ω being irreducible, it is contained in Y . Since Y is
n0 ∩ N N
closed, it follows that X is contained in Y . In particular, X = Y .
n0 N ∪n n
We conclude using Lemma 1. (cid:3)
2.3. Line bundles. Let X = X be an ind-variety. Denote by
n∈N n
∪
ι : X X the inclusion. A line bundle over X is an ind-variety
n n
−→ L
with a morphism π : X such that ι∗( ) is a line bundle over
L −→ n L
X , for any n.
n
A section of is a morphism σ : X such that π σ = Id .
X
L −→ L ◦
We denote by H0(X, ) the vector space of sections. Given a section
L
σ, we consider the sequence of sections (σ = ι∗(σ)) . Then
n n n∈N
(5) σ = σ .
n+1|Xn n
Conversely, a sequence σ of sections of ι∗( ) on X satisfying condi-
n n L n
tion (5) induces a well defined section σ of .
L
10 NICOLAS RESSAYRE
3. Using Borel-Weil Theorem
3.1. Tensor multiplicities. Recall that g is a symmetrizable Kac-
Moody Lie algebra. For given λ and λ in P , L(λ ) L(λ ) decom-
1 2 + 1 2
⊗
poses as a sum of integrable irreducible highest weights modules (see
[Kum02, Corrolary 2.2.7]), with finite multiplicities:
L(λ1)⊗L(λ2) = ⊕µ∈P+L(µ)⊕cµλ1λ2.
Let M be a g-module in the category ; under the action of h, M
O
decomposes as M with finite dimensional weight spaces M . Set
µ µ µ
⊕
M∨ = M∗: it is a sub-g-module of the dual space M∗.
⊕µ µ
3.2. Multiplicities as dimensions. Recall that G is the minimal
Kac-MoodygroupforagivenirreduciblesymmetrizableGCMA. LetB
bethestandardBorelsubgroupofGandB− betheoppositeBorelsub-
group. Consider G/B and G/B− endowed with the usual ind-variety
structures. Let o = B/B (resp. o− = B−/B−) denote the base point
of G/B (resp. G/B−). For λ h∗ = Hom(T,C∗) = Hom(B,C∗) =
Z
∈
Hom(B−,C∗), we consider the G-linearized line bundle (λ) (resp.
L
(λ)) on G/B (resp. G/B−) such that B (resp. B−) acts on the
−
L
fiber over o (resp. o−) with weight λ (resp. λ). For λ P , we have
+
− ∈
G-equivariant isomorphisms (see [Kum02, Section VIII.3])
H0(G/B, (λ)) Hom(L(λ),C),
L ≃
H0(G/B−, (λ)) Hom(L(λ)∨,C).
−
L ≃
Set
X = (G/B−)2 G/B.
×
A significant part of the following lemma is contained in [BK14, Proof
of Theorem 3.2].
Lemma 4. Let λ , λ , and µ in P . Then the space
1 2 +
H0(X, (λ ) (λ ) (µ))G
− 1 − 2
L ⊗L ⊗L
of G-invariant sections has dimension cµ . In particular, this dimen-
λ1λ2
sion is finite.
Proof. Set = (λ ) (λ ) (µ). We have the following canon-
− 1 − 2
L L ⊗L ⊗L
ical isomorphisms:
H0(X, )G Hom(L(λ )∨ L(λ )∨ L(µ),C)G
1 2
L ≃ ⊗ ⊗
Hom(L(µ),(L(λ )∨ L(λ )∨)∗)G
1 2
≃ ⊗
Hom(L(µ),(L(λ )∨ L(λ )∨)∨)G by h-invariance
1 2
≃ ⊗
Hom(L(µ),L(λ ) L(λ ))G
1 2
≃ ⊗
Thus this space of invariant sections has dimension cµ . We already
mentioned that cµ is finite. Nevertheless, we provλe1λi2ndependently
λ1λ2
that H0(X, )G is finite dimensional, reproving that cµ is finite.
L λ1λ2