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On the multiplicity spaces for branching to a spherical
subgroup of minimal rank
Luca Francone, Nicolas Ressayre
To cite this version:
On the multiplicity spaces for branching to a spherical subgroup
of minimal rank
Luca Francone and Nicolas Ressayre
April 28, 2021
Abstract
Letbg be a complex semi-simple Lie algebra and g be a semisimple subalgebra of bg. Consider the branching problem of decomposing the simple bg-representations bV as a sum of simple g-representationsV . When bg = g × g, it is the tensor product decomposition. The multiplicity spaceMult(V, bV ) satisfies
b
V = ⊕VMult(V, bV ) ⊗ V,
where the sum runs over the isomorphism classes of simple g-representations. In the case when g is spherical of minimal rank, we describeMult(V, bV ) as the intersection of kernels of powers of root operators in some weight space of the dual spaceV∗ ofV . Whenbg = g × g, we recover
by geometric methods a well known result.
1
Introduction
Let G be a connected reductive subgroup of a complex semisimple group bG. The branching problem consists in decomposing irreducible representations of bG as sum of irreducible G-representations.
Fix maximal tori T ⊂ bT and Borel subgroups B ⊃ T and bB ⊃ bT of G and bG respectively. Let X(T ) denote the group of characters of T and let X(T )+ denote the set of dominant characters.
For ν ∈ X(T )+, V
ν denotes the irreducible representation of highest weight ν. Similarly, we use
notation X( bT ), X( bT )+, Vbν relatively to bG. For any G-representation V , the subspace of G-fixed
vectors is denoted by VG. Given ν ∈ X(T )+ and bν ∈ X( bT )+, set
Mult(ν,ν) = Hom(Vb ν, Vνb∗)G = (Vν∗⊗ Vνb∗)G, (1)
where V∗
ν and Vbν∗denote the dual representations of Vν and Vνbrespectively. The branching problem
is equivalent to the knowledge of these spaces. Indeed, there is a natural G-equivariant isomorphism: L
ν∈X(T )+Hom(Vν, V∗
b
ν)G⊗ Vν −→ Vbν∗
f ⊗ v 7−→ f (v).
Let bG/ bB denote the complete flag variety of bG. In this article, we are interested in the case when the pair ( bG, G) is spherical of minimal rank. In other words, we assume that there exists x in bG/ bB such that the orbit G.x of x is open in bG/ bB and the stabilizer Gx of x in G contains a
containing T . More generally, the spherical pairs of minimal rank have been classified by the second author in [Res10]. The complete list, assuming in addition that bG is semisimple simply connected, G is simple and G 6= bG is:
1. G is simple, simply connected and diagonally embedded in G = G × G; 2. (SL2n, Sp2n) with n ≥ 2;
3. (Spin2n, Spin2n−1) with n ≥ 4;
4. (Spin7, G2);
5. (E6, F4).
Our aim is to present a uniform description of the multiplicity spaces Mult(ν, bν) for given ( bG, G) in this list. Let us first fix some notation. Recall that T ⊂ bT and denote by ρ : X( bT )−→X(T ) the restriction map. Let ∆ (resp. b∆) denote the set of simple roots of G (resp. bG). For each positive root α of G, we fix an sl2-triple (Xα, Hα, X−α) such that Hα (resp. Xα) belongs to the Lie algebra
of T (resp. B). Given µ ∈ X(T ), we denote by Vν(µ) = {v ∈ Vν : ∀t ∈ T tv = µ(t)v} the
corresponding multiplicity space for the action of the maximal torus T .
Let cW denote the Weyl group of bG. Fix also yb0 ∈ cW such that Gyb0B/ bb B is dense in bG/ bB and
such that by0 has minimal length with this property (see Section 2.3 for details).
For cases 2 to 5, we also denote by Φ1 the set of long roots of G. In case 1, we define Φ1 to be
the empty set. Set
D = {α ∈ bb ∆ : ρ(by0α) 6∈ Φb 1}.
Theorem 1. For ν ∈ X(T )+ and bν ∈ X( bT )+, there is a natural isomorphism from Mult(ν,ν) ontob the subspace of Vν∗(ρ(yb0bν)) consisting in the vectors v such that
1. Xα· v = 0, for any α ∈ Φ+∩ Φ1;
2. Xm
−ρ(by0α)b · v = 0 ∀ m > hbν,αb
∨i, for anyα ∈ D.b
Actually, the conditions 2 are not pairwise independant. For example, in the case of the tensor product, the conditions associated to (α, 0) ∈ D and (0, −w0α) ∈ D are equivalent for any simple
root α. Here w0 denotes the longest element of the Weyl group. For each example, we describe in
Section 4 an explicit subset of D giving an irredundant set of inequalities.
In the case of the tensor product, Theorem 1 is well known. See [PRRV67, Theorem 2.1, p. 392] or [Ž73, Theorem 5, p. 384]. The usual proofs are algebraic, based on properties of the enveloping algebra. Our geometric proof seems to be new even in this case.
The branching rule for (Spin2n, Spin2n−1) is multiplicity free and hence easy to determine (see
e.g. [FH91]). That of (SL2n, Sp2n) has been the subject of much attention in the literature. The
first positive rule in terms of dominos was obtained by Sundaram [Sun90]. Naito-Sagaki conjectured a rule in terms of Littelmann’s patches [NS05]. Later, B. Schumann and J. Torres proved this conjecture by obtaining a bijection with Sundaram’s model. A nonpositive rule for (Spin7, G2)
2
Reminder and complements on spherical homogeneous spaces of
minimal rank
2.1 Roots of G and bG
Fix a spherical pair of minimal rank (G, bG) with G and bG reductive. Choose a maximal torus T in G and a Borel subgroup T ⊂ B ⊂ G. Denote by Φ (resp. Φ+) the set of roots (resp. positive
roots). Recall that ∆ denotes the set of simple roots. Fix also a maximal torus bT of bG containing T and let ρ : X( bT )−→X(T ) denote the restriction map. The set bΦ of roots of bG maps onto Φ (see [Res10, Lemma 4.2]). Let ¯ρ denote the restriction of ρ to bΦ. By setting bΦ+= ¯ρ−1(Φ+), one gets a
choice of positive roots for bG. Let bB denote the corresponding Borel subgroup. Then bB contains B. By [Res10, Lemma 4.6], ρ( b∆) = ∆, where b∆ denote the set of simple roots of bG.
On the following diagrams the restriction of ¯ρ to the set of simple roots is the vertical projection.
A2n−1 Cn Dn Bn−1 E6 ˜ F4 B3 G2
By [Res10, Lemma 4.4], for any α ∈ Φ, ¯ρ−1(α) has cardinality one or two. Hence, Φ splits in
the following two subsets
Φ1 := {α ∈ φh : ♯¯ρ−1(α) = 1} and Φ2 := {α ∈ φh : ♯¯ρ−1(α) = 2}.
Set also bΦ1 = ¯ρ−1(Φ1) and bΦ2 = ¯ρ−1(Φ2). It is easy to check using the pictures, that this set Φ1,
coincides with that defined in the introduction.
2.2 Isotropies
Assume, in addition, that G is connected.
Lemma 2. Let x ∈ bG/ bB. Then the isotropy Gx is connected and contains a maximal torus of G.
Proof. The fact that Gx contains a maximal torus T of G follows from the monotonicity properties
of the rank of orbit closures of G in bG/ bB (see [Kno95, Theorem 2.2]).
Let L be a Levi subgroup (maximal reductive subgroup) of Gx. Then L is isomorphic to the
quotient of Gx by its unipotent radical. A reference for the existence of L is [OV90, Theorem 4.
But L is a reductive subgroup of bGx = bB′, that is a Borel subgroup of bG. Hence L maps
injectively into bB′/ bU′≃ bT and L is abelian. The torus T being its own centralizer in G, we deduce
that L = T .
Now Gx is connected as the product of T and its unipotent radical.
2.3 Orbits of G in bG/ bB
We are now interested in the set G\ bG/ bB of G-orbits in bG/ bB. Let W and cW be the Weyl groups of G and bG respectively. Since T is a regular torus in bG (see [Bri99, Lemma 2.3]), W naturally embeds in cW .
Lemma 3. The map
W \cW −→ G\ bG/ bB b
w 7−→ Gw bbB/ bB is a well-defined bijection.
Proof. It is clear that the G-orbit Gw bbB/ bB does not depend on the representative w in its classb Ww. Hence, the map of the lemma is well-defined.b
Since T is a regular torus in bG, its fixed points in bG/ bB are the points w bbB/ bB forw ∈ cb W . But Lemma 2 implies that each G-orbit in bG/ bB contains a T -fixed point. The surjectivity follows.
Let now bw and wb′ in cW such that Gw bbB/ bB = Gwb′B/ bb B. To get the injectivity and finish the
proof, it remains to prove that W bw = Wwb′. Choose g ∈ G such that g bw bB/ bB = wb′B/ bb B. Let H and H′ denote the isotropies in G of the points bw bB/ bB and wb′B/ bb B, respectively. Observe that T is a maximal torus of both H and H′. Then, T and gT g−1 are two maximal tori of H′, and
there exists h′ ∈ H′ such that h′T h′−1 = gT g−1. Thus, n := h′−1g normalizes T and satisfies
nw bbB/ bB = h′−1wb′B/ bb B =wb′B/ bb B. It follows thatwb′ ∈ Ww.b
Let ℓ : cW −→N denote the length function. It is well known that ℓ(w) = ♯(bb Φ+∩wb−1Φb−), where b
Φ−= −bΦ+.
We fix, once for all, by0 ∈ cW such that Gyb0B/ bb B is the open G-orbit in bG/ bB and such that yb0
has minimal length with this property. Let H0 denote the stabilizer of by0B/ bb B in G.
Given a root α (resp. bα) of g (resp. bg), denote by gα (resp. gαb) the corresponding root space.
Lemma 4. The Lie algebra of H0 is
Lie(T ) ⊕ M
α∈Φ+∩Φ1
gα.
Proof. It is clear that H0 contains bT ∩ G = T . Then
Lie(H0) = Lie(T ) ⊕
M
α∈S
gα,
for some subset S of Φ.
Let now α ∈ Φ1∩ Φ+. We want to prove that α ∈ S. Let bα ∈ bΦ such that ρ(bα) = α. We
have g±bα = g±α. But, exactly one between gαb and g−bα is contained in the Borel Lie algebra
Set bβ = −by0−1α andb yb0′ = sαbyb0. Since bβ is positive and yb0β is negative, ℓ(b yb0′) < ℓ(yb0). But,
g±bα = g±α implies that sα = sαb and that by0′ ∈ Wyb0. This contradicts the minimality assumption
on the length of by0.
At this point, we proved that the Lie algebra of the lemma is contained in Lie(H0). But
dim(G/H0) = dim( bG/ bB), hence
dim(H0) = dim(G) − dim( bG/ bB)
= dim(T ) + 2♯Φ+− ♯bΦ+
= dim(T ) + ♯(Φ+∩ Φ1) and we can conclude.
Lemma 5. Fix β ∈ Φ2. Then there is exactly one root in ¯ρ−1(β) which is sent in bΦ+ by the action
of yb0−1.
Proof. Write ¯ρ−1(β) = { bβ1, bβ2}. Because of [Res10, Lemma 4.4], we have that gβ ⊆ gβb1⊕ gβb2 and
gβ 6= gβbi for i = 1 , 2.
Lemma 4 implies that gβ 6⊆ Lie(H0). But since H0 = H ∩ yb0Bbyb0−1, this means that by0−1β1 and
b
y0−1βb2 are not both positive roots, equivalently: {by0−1βb1,yb0−1βb2} 6⊆ bΦ+. The same argument
applied to −β implies that {−by0−1βb1, −by0−1βb2} 6⊆ bΦ+. The lemma follows.
Lemma 5 allows us to distinguish between the roots in the fiber of ¯ρ by means of the action of b
y0−1. In particular we introduce the following notation. If β ∈ Φ2 ∩ Φ+, then bβ+ (resp. bβ− ) is
the unique element in ¯ρ−1(β) that satisfies: yb
0−1βb+ ∈ bΦ+ (resp. by0−1βb−∈ bΦ−).
2.4 The graph of G-orbits in bG/ bB
We recall the definition given in [Res04] of a graph Γ( bG/G) whose vertices are the elements of G\ bG/ bB. The original construction of Γ( bG/G) due to M. Brion is equivalent but slightly different (see [Bri01]).
If bα belongs to b∆, we denote by Pαb the associated minimal standard parabolic subgroup of bG. Consider the unique bG-equivariant map παb : bG/ bB−→ bG/Pαb which is a P1-bundle.
Let O ∈ G\ bG/ bB andα ∈ bb ∆. Consider πbα−1(παb(O)). Two cases occur.
• G acts transitively on παb−1(παb(O)).
• παb−1(παb(O) contains two G-orbits, one closed V and one open V′. Then we says that bα raises
V to V′. In this case, dim(V′) = dim(V ) + 1, and for any x ∈ V , π b
α−1(παb(x)) ∩ V = {x}.
Definition. Let Γ( bG/G) be the oriented graph with vertices the elements of G\ bG/ bB and edges labeled by b∆, where V is joined to V′ by an edge labeled by bα if bα raises V to V′.
Lemma 6. Let w ∈ cb W . We have
dim Gw bbB/ bB − dim G/B ≤ ℓ(w).b
Proof. We consider the bB-orbits in bG/G. The closed orbit is bB/B. Choose a reduced expression of b
w = sαbs· · · sαb1. Consider the quotient Pαb1×Bb· · · ×BbPαbs×BbB/B of Pb αb1× · · · × Pαbs× bB/B by the
action of Bs given by (b
1, . . . , bs).(p1, . . . , ps, bB/B) = (p1b1−1, . . . , bs−1psb
−1
s , bsB/B) (with obviousb
notation). Consider the regular map
Pαb1 ×Bb· · · ×BbPαbs×BbB/B −→b G/Gb
(p1, . . . , ps, x) 7−→ p1· · · psx.
The dimension of the left hand side is ℓ( bw) + dim( bB/B). The right hand side is a bB-orbit closure containing bBwb−1G/G. The first inequality follows.
This equality is reached when the expression is obtained by reading the labels on some path from the closed orbit to O in the graph Γ( bG/G). Such a path exists by [Res10, Proposition 2.2].
Set
ℓm(w) = min{ℓ(wb w) : w ∈ W }.b
By Lemma 6, ℓm(w) = dim(Gb w bbB/ bB) − dim(G/B). If an element inw ∈ cb W satisfies ℓ(w) = ℓb m(w),b
we say that bw has minimal length.
Lemma 7. 1. There are ♯∆2 codimension one G-orbits in bG/ bB.
2. Let α ∈ bb ∆. Then Gby0sbαB/ bb B has codimension one if and only if yb0α ∈ bb Φ2.
3. Let α ∈ bb ∆ such thatyb0α ∈ bb Φ2. Then, the Lie algebra of the stabilizer of by0sαbB/ bb B is
Lie(H0) ⊕ g−ρ(by0α)b .
Proof. By [Res10, Proposition 2.3], the number of codimension one G-orbits in bG/ bB is the number of G-orbits O of dimension dim(G/B) + 1. This is also
♯({W sαb∈ W \ cW ;α ∈ bb ∆} − {W }).
But sαb ∈ W if and only if α ∈ bb ∆1. Moreover, if bα1 6= αb2 ∈ b∆2 then sαb1sαb2 ∈ W if and only if
ρ(αb1) = ρ(αb2).
Since Gby0B/ bb B is dense, either Gyb0sαbB/ bb B = Gyb0B/ bb B or Gby0sαbB/ bb B has codimension one. But
b
y0sαbB/ bb B belongs to the open G-orbit if and only if by0sαb ∈ Wyb0 if and only if by0α ∈ bb Φ1.
Let bα be like in the last assertion. Then b
y0sαbΦb+= (yb0Φb+∪ {−yb0α}) − {b yb0α}.b
Now, the fact that Lie(H0) is contained in the Lie algebra of the stabilizer ofyb0sαbB/ bb B follows from
Lemma 4.
The root by0α belonging to bb Φ2, there exists a second root bβ in the fiber ¯ρ−1(ρ(−yb0α)). But,b
Lemma 5 implies that bβ belongs to yb0Φb+ and hence to by0sbαΦb+. We deduce that
gρ(−by0α)b ⊂ gβb⊕ g−by0αb ⊂ Lie(yb0sαbB(yb0sαb)−1).
In particular, gρ(−by0α)b is contained in the Lie algebra of the stabilizer of by0sαbB/ bb B.
2.5 Compatible sl2-triples
Recall that we have fixed an sl2-triple (Xβ, Hβ, Yβ) associated to any positive root β of G.
Lemma 8. If β ∈ Φ2∩ Φ+ then there exist sl
2-triples (Xβb+, Hβb+, X− bβ+) and (Xβb−, Hβb−, X
− bβ−)
for bβ+ and bβ− such that:
• Xβ = Xβ+ + Xβ−; Hβ= Hβ+ + Hβ−; X−β = X−β++ X
−β−.
• exp(tXβ) = exp(tXβb+) · exp(tXβb−) for any t ∈ C
Proof. Since gβ ⊆ gβb+ ⊕ gβb− and is not equal to any of the 2 root spaces on the right hand side,
Xβ = Xβb++Xβb−for nonzero Xβb± ∈ gβb±. Then there exist sl2-triples of the form (Xβb+, Hβb+, X− bβ+)
and (Xβb−, Hβb−, X
− bβ−) for bβ+and bβ−. Moreover, X−β = aX
− bβ++ bX
− bβ− with nonzero constants
a and b, and
Hβ = [Xβ, X−β] = [Xβb++ Xβb−, aX
− bβ+ + bX− bβ−] = aHβb++ bHβb−.
For the last equality we use that [Xβb+, X
− bβ−] = 0 and [Xβb−, X
− bβ+] = 0. Indeed, if bβ+− bβ− (reps.
b
β−− bβ+) was a root in bΦ, then ρ( bβ+− bβ−) = 0 (resp. ρ( bβ−− bβ+) = 0). But this is absurd since ρ
sends bΦ in Φ.
Using that ρ(bβ+) = ρ( bβ−) = β, we get that:
2 = β(Hβ) = bβ+(aHβb+ + bHβb−) = a bβ+(aHβb+) = 2a.
Where for the second-last inqueality we used that bβ+ and bβ− are ortogonal, hence bβ+(Hβb−) = 0.
Then a = 1, and similarly we get b = 1. This proves the first point.
For the second one notice that [Xβb+, Xβb−] = 0. In fact if bβ++ bβ−was a root, then ρ(bβ++ bβ−) = 2β.
But this is absurd since 2β is not a root. Hence for any t ∈ C, exp(tXβb+ + tXβb−) = exp(tXβb+) ·
exp(tXβb−), and the second point follows immediately.
Remark. For the proof we could also used [Res10, Lemma 4.1] and [Res10, Lemma4.3] to reduce the problem to the case of PSL2 diagonally embedded in PSL2× PSL2.
3
Proof of Theorem 1
3.1 An embedding of the multiplicity space
Fix dominant weights ν ∈ X(T )+ and bν ∈ X( bT )+. Observe that H
0 is a subgroup of by0Bbyb0−1 and
b
y0ν is a character of this last group. In particular,b yb0ν restricts as a character of Hb 0.
Lemma 9. In the G-representation Vν∗, the subspaces (Vν∗)(H0)y0 bbν = {v ∈ V∗
ν : ∀h ∈ H0 h.v = (yb0ν)(h)v}b
and
{v ∈ Vν∗(yb0bν) : ∀α ∈ Φ+∩ Φ1 Xα· v = 0}
Proof. Since T is a maximal torus of H0, the first subspace is contained in Vν∗(yb0bν). Furthermore
H0= Ru(H0)T , where Ru(H0) denotes the unipotent radical. Hence
(Vν∗)(H0)y0 bbν = V∗ ν(by0bν)R u(H 0)= V∗ ν(yb0ν)bLie(R u(H)) . Now, Lemma 4 allows to conclude.
Consider on G/B the G-linearized line bundle Lν such that B acts on the fiber over B/B with
weight −ν. Similarly, define Lbν. By the Borel-Weyl theorem, the space
H0(G/B × bG/ bB, Lν⊗ Lνb)G≃ (Vν∗⊗ Vbν∗)G
of G-invariant sections identifies with Mult(ν, bν). The orbit G.by0B/ bb B being open, the restriction
map
H0(G/B × bG/ bB, Lν⊗ Lbν)G−→H0(G/B × G.by0B/ bb B, Lν ⊗ Lνb)G
is injective. Moreover
H0(G/B × G.yb0B/ bb B, Lν⊗ Lbν)G≃ H0(G/B, Lν)(H0)y0 bbν ≃ (Vν∗)(H0)y0 bbν.
For ϕ ∈ (V∗
ν)(H0)y0 bbν and denote by ˜σ ∈ H0(G/B × G.yb0B/ bb B, Lν ⊗ Lbν)G the associated section.
To describe the image of H0(G/B × bG/ bB, L
ν ⊗ Lbν)G in (Vν∗)(H0)y0 bbν, we have to understand what
sections ˜σ extend, and hence the order of the poles of ˜σ along the divisors of (G/B × bG/ bB)−(G/B × G.yb0B/ bb B).
3.2 Local description along the divisors
Fix bα ∈ b∆ that is a label of some edge descending from the open orbit in Γ( bG/G). Then Dαb :=
Gyb0sαbB/ bb B is a divisor of bG/ bB. By Lemma 7 this happens if and only ifyb0sαb∈ bΦ2. Set β = ±ρ(by0α)b
the sign being chosen to get β ∈ Φ+.
Lemma 5 allows to distinguish two situations that lead to two slightly different local descriptions of the corresponding divisor.
1) by0α is negative. Then β = −ρ(b yb0α) andb yb0α = − bb β−.
2) by0α is positive. Then β = ρ(b yb0α) andb yb0α = bb β+.
Given a root α (resp. bα) of G (resp. bG), we denote by Uα(resp. Uαb) the corresponding subgroup
isomorphic to (C, +). The aim of this section is to describe an open subset of bG/ bB intersecting the divisor Dαb. In particular we will prove that Uαb is a common transverse slice to Dbα at any point of
this subset. Before doing so we need some preparatory work.
Lemma 10. Let α, β and Db αb be as above. Set S(α) := (Φb +∩ Φ2) \ {β}. Index the elements of
1) If yb0α is negative then the mapb iαb : U−×Qγ∈S(bα)Uγ −→ Gyb0sαbB/ bb B (u−, (uγ)γ) 7−→ (u− s Y i=1 uγi)yb0sαbB/ bb B is an open immersion.
2) If yb0α is positive then the mapb
iαb : U−×Q γ∈S(bα)Uγ −→ Gby0sαbB/ bb B (u−, (uγ)γ) 7−→ sβ(u− s Y i=1 uγi)sβyb0sαbB/ bb B is an open immersion.
Our proof of Lemma 10 depends on the following well-known lemma. Lemma 11. 1. The product induces an open immersion
U−× U × T −→ G
(u−, u, t) 7−→ u−ut.
2. Let H be a T -stable closed subgroup of U . Number γ1, . . . , γs the positive roots of G that are
not roots of H. Then the map
Uγ1 × · · · × Uγs× H −→ U
((ui), u) 7−→ u1. . . usu
is an isomorphism.
Proof. The first assertion is a part of the Bruhat decomposition (see e.g. [Hum75, Section 28.5]). The second assertion is the content of [Hum75, Section 28.1].
Proof. We begin by the negative case. Notice that Lemma 7 implies that γ ∈ S(α) if and onlyb if Uγ 6⊆ Uyb0s
b
αB/ bb B. Applying the second assertion of Lemma 11 with H = Uby0sαbB/ bb B, one gets an
isomorphism Y
γ∈S(bα)
Uγ× Uby0s
b
αB/ bb B−→U.
Now, again by Lemma 7, Gyb0s
b
αB/ bb B = T Uyb0sαbB/ bb B. Then the first assertion of Lemma 11 implies
that
U−× Y
γ∈S(bα)
Uγ× Gby0sαbB/ bb B−→G
is an open embedding. Taking the quotient by Gby0s
b
αB/ bb B we conclude.
Proposition 12. Keep the setting as in Lemma 10. 1) If yb0α is negative then the map:b
fαb : U−×Qγ∈S(bα)Uγ× U−bα −→ G/ bb B (u−, (uγ)γ, u−bα) 7−→ (u− s Y i=1 uγi)yb0sαbu−bαBb
is an open immersion. Furthermore the image of fαb is contained in Gyb0B/ bb B ∪ Dαb , and
fαb−1(Dαb) = U−×Qγ∈S(bα)Uγ× {1}.
2) If yb0α is positive then the map:b
fαb : U−×Q γ∈S(bα)Uγ× U−bα −→ G/ bb B (u−, (uγ)γ, u−bα) 7−→ sβ(u− s Y i=1 uγi) sβyb0sαbu−bαBb
is an open immersion. Furthermore the image of fαb is contained in Gyb0B/ bb B ∪ Dαb, and
fαb−1(Dαb) = U−×Qγ∈S(bα)Uγ× {1}.
Proof. We prove part 1). The proof of the positive case is obtained from the following one by replacing every appearance of by0 with sβby0. Identify bG/ bB with the fibered product bG ×Pαb Pαb/ bB.
Via this identification fαb is the map:
fαb : U−×Qγ∈S(bα)Uγ× U−bα −→ G ×b PαbPαb/ bB (u−, (uγ)γ, u−bα) 7−→ (u− s Y i=1 uγiby0sαb : u−bαB/ bb B) .
Since by0sαbB/ bb B and παb(yb0sαbB/ bb B) have the same isotropy in G, we can identify their G-orbits. In
particular we will think about iαb, the open immersion of Lemma 10, as a map to G.by0sαbPαb/Pαb ⊆
b
G/Pαb. Call Vαb ⊆ bG/Pαb the image of iαb. By Lemma 10, Vαb is open in bG/Pαb. Denote the two
components of iαb−1: Vαb−→ U−× Y γ∈S(bα) Uγ, by j1 : Vbα−→ U− and j2 : Vαb −→Qγ∈S(bα)Uγ, respectively.
Observe that U−bα ≃ U−bαB/ bb B = Pαb/ bB \ ( bB/ bB) is open in Pbα/ bB. The image of fαb is
Ωαb:= { [(bg : x)] ∈ bG ×PαbPαb/ bB : [bg] ∈ Vαband sαb−1yb0−1j2([bg]))−1j1([bg]))−1gx ∈ U−bαB/ bb B},
where [bg] = bgPαb/Pαb∈ bG/Pαb. We deduce that Ωαb is open in bG ×PαbPαb/ bB.
Finally we prove that fαb is an isomorphism. If we call φbα : U−bαB/ bb B−→U−bα the inverse of the
natural projection, then it’s easy to see that the map Ωαb−→ U−×
Y
γ∈S(bα)
that sends (bg, x) to:
(iαb−1)1([bg]) , (iαb−1)2([bg]) , φαb sαb−1yb0−1j2([bg])−1j1([bg[)−1gx
is the inverse of fαb.
To conclude, notice that for u−bα = 1, fαb restrict to iαb, hence Ωαb∩ Dαb is an open dense subset of
the divisor.
For u−bα 6= 1, by0sαbu−bαB/ bb B 6=yb0sαbB/ bb B in y0sbαPαb/ bB. Hence yb0sαbu−bαB/ bb B has to be a point of
the open G-orbit, that is the orbit of by0B/ bb B.
We conclude that fαb maps U−×
Y
γ∈S(bα)
Uγ× U−bα\{1}
in the open G-orbit and that fαb−1(Dαb) =
U−×Qγ∈S(bα)Uγ× {1}.
3.3 Conclusion
In the setting of the end of Section 3.1, we are now in position to characterize the image of the embedding:
H0(G/B × bG/ bB, Lν⊗ Lbν)G−→H0(G/B, Lν)(H0)by0 bν. (2)
Fix ϕ ∈ (V∗
ν)(H0)by0 bν and follow it throught the following isomorphisms:
(V∗
ν)(H0)by0 bν ≃ H0(G/B, Lν)(H0)y0 bbν ≃ H0(G/B × G.yb0B/ bb B, Lν⊗ Lbν)G
ϕ 7→ σ 7→ ˜σ.
Fix v0 ∈ Vν(B) and ˜y0 ∈ (Lbν)yb0 − {0}. Explicitly, σ and ˜σ are given by the formulas
∀g ∈ G σ(gB/B) = [g : ϕ(gv0)]
∀g1, g2 ∈ G σ(g˜ 1B/B, g2yb0B/ bb B) = [g1 : ϕ(g2−1g1v0)] ⊗ g2y˜0.
(3) We want to determine when ˜σ extends to a global section. Take bα ∈ b∆ that is a label of some edge descending from the open orbit. Then Dαb := Gyb0sαbB/ bb B is a divisor of bG/ bB along which,
we want to determine the vanishing order of ˜σ. Consider the image Vbα of the map ιαb defined by
Proposition 12.
Proposition 13. 1. Assume that by0α is negative. Then, the section ˜b σ extends to a regular
section on G/B × Vαb if and only if
Xβmϕ = 0 for m > h−by0·bν, (bβ−)∨i.
2. Assume that yb0α is positive. Then, the section ˜b σ extends to a regular section on G/B × Vbα if
and only if
X−βm ϕ = 0 for m > hby0·bν, ( bβ+)∨i.
a group homomorphism φβ : SL2(C)−→G. The same notation is used, in the obvious way, also for
b
G. Now fix an sl2-triple (Xαb, Hαb, X−bα) for α, with Xb αb ∈ gαb.
Suppose first that we are in the negative case, that is by0α = − bb β−, where β = −ρ(by0α). Becauseb
of Proposition 12, ˜σ extends to a section on G/B × Vαb if and only if the map:
G/B × U−×Qγ∈S(bα)Uγ× C∗ −→ Lν⊗ Lbν
(gB/B, u−, (uγ)γ, t) 7−→ ˜σ (gB/B, u−Qsi=1uγiyb0sαbǫ−bα(t) bB/ bB)
extends at t = 0. Notice also that U− and U
γ are subgroups of G and ˜σ is G-invariant, hence the
function above extend at t = 0 if and only if the following map, that with a little abuse will still be called ˜σ, extends at t = 0.
G/B × C∗ −→ Lν⊗ Lbν
(gB/B, t) 7−→ ˜σ (gB/B ,yb0sαbǫ−bα(t) bB/ bB)
. (4)
Now fix sl2-triples for bβ± as in Lemma 8, so that for any t ∈ C∗, ǫ±β(t) = ǫ± bβ+(t)ǫ
± bβ−(t). In SL2, we have 1 t 0 1 0 1 −1 0 = 1 0 t−1 1 −t 0 0 −t−1 1 −t−1 0 1 , for any t ∈ C∗. Hence
ǫαb(t)sαb = ǫ−bα(t−1)αb∨(−t)ǫαb(−t−1). (5)
Where bα can be replaced by any positive root of G or bG for which a corresponding sl2-triple has
been fixed. Now take (gB/B, t) ∈ G/B × C∗, then
(gB/B ,yb0sαbǫ−bα(t) bB/ bB) =(gB/B ,yb0ǫαb(t)sαbB/ bb B)
=(gB/B , yb0ǫ−bα(t−1) bB/ bB)
=(gB/B , ǫβb−(c−1t−1)yb0B/ bb B)
where c is the nonzero constant satisfying by0Xαb = cX− bβ−. But since Uβ+ ⊂yb0Bbyb0−1:
ǫβb−(c−1t−1)yb0B/ bb B = ǫβb−(c−1t−1)ǫβb+(c−1t−1)by0B/ bb B = ǫβ(c−1t−1)yb0B/ bb B.
Now, Formula (3) gives ˜ σ (gB/B , by0sαbǫ−bα(t) bB/ bB) = [g : ϕ(ǫβ(−c−1t−1)gv0)] ⊗ ǫβ(c−1t−1)˜y0. Moreover, ǫβ(c−1t−1)˜y0 = ǫβb−(c−1t−1)ǫβb+(c−1t−1)˜y0 = ǫ − bβ−(ct)(sb β−)−1( bβ−)∨(−ct)ǫ − bβ−(c−1t−1)˜y0 by formula (5) = ǫ − bβ−(ct)(sβb−)−1(−ct)h−by0bν,( bβ − )∨ i since U − bβ−⊂by0Ubyb0−1 and
Rewrite the term ϕ(ǫβ(−c−1t−1)gv0) as ϕ(ǫβ(−c−1t−1)gv0) = hǫβ(c−1t−1)ϕ, gv0i =Pn≥0(c−1n!t−1)nhXβnϕ, gv0i. Finally, we get ˜ σ (gB/B , yb0sαbǫ−bα(t) bB/ bB)= [g : X n≥0 ±(c−1t−1)n+hb y0bν,( bβ−)∨i n! hX n βϕ, gv0i] ⊗ (ǫ− bβ−(ct)(sβb−)−1y˜0).
The term ǫ− bβ−(ct)(sβb−)−1y˜0 is regular on C. Hence ˜σ has no pole along t = 0 if and only if
∀n > h−yb0ν, ( bb β−)∨i =⇒ hXβnϕ, gv0i = 0 ∈ C[G].
But Gv0 spans Vν that is irreducible. As a consequence, hXβnϕ, gv0i = 0 if and only if Xβnϕ = 0.
Suppose now that we are in the positive case, hence by0·α = bb β+ with β = ρ(by0α). The outlineb
of the proof doesn’t change. By the same argument of the previous case, ˜σ extends to a section on G/B × Vαb if and only if the following map, that will still be called ˜σ, extends at t = 0.
G/B × C∗ −→ Lν⊗ Lbν
(gB/B, t) 7−→ ˜σ (gB/B , sβyb0sαbǫ−bα(t) bB/ bB)
(6)
Again fix sl2-triples for bβ± as in Lemma 8. Take (gB/B, t) ∈ G/B × C∗, then
(gB/B , sβyb0sαbǫ−bα(t) bB/ bB) =(gB/B , sβyb0ǫ−bα(t−1) bB/ bB)
=(gB/B , sβǫ−β(c−1t−1)yb0B/ bb B)
where now c is the nonzero constant satisfying by0Xαb = cXβb+. Hence:
˜
σ (gB/B , sβyb0sαbǫ−bα(t) bB/ bB)
= [g : ϕ(ǫ−β(−c−1t−1)(sβ)−1gv0)] ⊗ sβǫ−β(c−1t−1)˜y0.
Similarly to the previous computations: ǫ−β(c−1t−1)˜y0 = ǫ − bβ+(c−1t−1)ǫ − bβ−(c−1t−1)˜y0 = ǫβb+(ct)sβb+ǫβb+(c−1t−1)( bβ+)∨(−c−1t−1)˜y0 = ǫβb+(ct)sβb+(−c−1t−1)h−by0ν,( bb β +)∨ i.
While for the other term
By the same argument of the previous case we conclude that ˜σ has no pole along t = 0 if and only if
∀ n > hyb0bν, (bβ+)∨i =⇒ Xβnϕ = 0.
Remark. Ifyb0α = −βb −, then by0·αb∨ = −( bβ−)∨, hence:
h−yb0·ν, ( bb β−)∨i = hyb0·ν,b yb0·αb∨i = hbν,αb∨i.
Similarly, if by0α = bb β+, then by0·αb∨ = ( bβ+)∨, which implies:
hyb0·bν, (bβ+)∨i = hby0·bν, by0·αb∨i = hν,b αb∨i.
Recall that in the introduction we set D = {bα ∈ bΦ : ρ(by0α) 6∈ Φb 1}, for α ∈ D, let Vb αb be as in
Proposition 13. We prove the theorem of the introduction.
Proof of Theorem 1. The first condition of the theorem is implied by Lemma 9. Then set V := S
b
α∈DG/B × Vαb, which is open in G/B × bG/ bB. The complement of V is of codimension strictly
larger then 1 and G/B × bG/ bB is normal, hence the restriction map H0(G/B × bG/ bB, Lν⊗ Lνb)−→H0(V, Lν⊗ Lbν)
is an isomorphism. Then, from Proposition 13 and from the previous remark we deduce that ϕ ∈ (Vν∗)(H0)y0 bbν is in the image of
H0(G/B × bG/ bB, Lν⊗ Lbν)−→(Vν∗)(H
0) b y0 bν
if and only if the second condition of the theorem holds.
As remarked in the introduction, the conditions of Theorem 1 are in general redundant. This is linked to the fact in Γ( bG/G) we may have different edges between the same vertices. It seems not easy to determine a minimal set of conditions in a uniform way. The following lemma checks that, for the tensor product case, a minimal set of conditions is described by the set:
{(α, 0) ∈ Φ × Φ : α ∈ ∆}.
Lemma 14. For β ∈ Φ+, and σ ∈ Vν∗(yb0·bν), the following are equivalent:
1) Xβm· σ = 0 for m > h−yb0·bν, (bβ−)∨i. 2) X−βm · σ = 0 for m > hyb0·ν, ( bb β+)∨i. Remark. If β ∈ Φ+ and bα 1,αb2∈ b∆ satisfy: b y0·αb1 = − bβ− and by0·αb1= bβ+
then by0sαb1 = sβyb0sαb2, hence their G-orbit is the same, and by the proof we have done we expect
Proof. Call slβ the subalgebra of g spanned by Xβ, Hβ, X−β. Decompose Vν∗ into a direct sum of
slβ irreducible representations: Vν∗ =
L
Vδ. Write σ = Pδσδ accordingly to this decomposition.
Observe that σδ∈ Vδ(by0·bν|CHβ). Since Vδ is an sl2 irreducible representation, if σδ6= 0, then:
Xβm· σδ= 0 ⇐⇒ hby0·bν, β∨i + 2m > hδ, β∨i, and
X−βm · σδ= 0 ⇐⇒ hby0·bν, β∨i − 2m < −hδ, β∨i.
Hence condition 1) is equivalent to:
∀ δ : σδ6= 0, =⇒ hyb0·ν, βb ∨i − 2hyb0·bν, ( bβ−)∨i + 2 > hδ, β∨i.
Similarly condition 2) is equivalent to:
∀ δ : σδ 6= 0, =⇒ hyb0·bν, β∨i − 2hyb0·ν, ( bb β+)∨i − 2 < −hδ, β∨i.
But Lemma 8 implies that hby0·bν, β∨i = hyb0 ·ν, ( bb β−)∨i + hby0·bν, ( bβ+)∨i, then we easily conclude
that 1) and 2) are equivalent.
4
Explicit description on the examples
For each example, we determine a working by0 and a set of simple roots parametrizing the G-stable
divisors of bG/ bB. With the notation of the introduction, we give a subset D0 of D such that the
map bα 7−→ Gby0sαbB/ bb B is a bijection from D0 onto the set of G-stable divisors.
4.1 Tensor product case
Here bG = G × G, bT = T × T and bB = B × B. Moreover Φ1 is empty, X( bT ) = X(T ) × X(T ) and
ρ(λ, µ) = λ + ν. Set yb0 = (e, w0) ∈ cW = W × W . It is clear that Gby0 is open in bG/ bB and that
ℓ(yb0) = dim( bG/ bB) − dim(G/B). By the Bruhat decomposition {G(B/B, w0sαB/B) : α ∈ ∆} is
the collection of codimension one G-orbits in bG/ bB. In particular the set D−0 = {(0, α) : α ∈ ∆}
works. Another possible choice is
D0+= {(α, 0) : α ∈ ∆}.
Note that by0 =yb0−1 and that, for α ∈ ∆, by0(0, α) = (0, w0α) ∈ bΦ−, while by0(α, 0) = (α, 0) ∈ bΦ+.
So, according to our notations, (0, α) = bα− and (α, 0) = bα+. If ν ∈ X(T )+ and λ, µ ∈ X(T ) we
define two sets:
V+(ν, λ, µ) := {v ∈ Vν(λ) : ∀ α ∈ ∆ , Xαmv = 0 for m > hµ, α∨i}
V−(ν, λ, µ) := {v ∈ Vν(λ) : ∀ α ∈ ∆ , X−αm v = 0 for m > hµ, α∨i}
From now on, fix ν, ν1, ν2 ∈ X(T )+ and set bν = (ν1, ν2) ∈ X(T × T )+. We denote by ν∗ :=
−w0ν, so that Vν∗ ≃ V∗
ν. In Theorem 2.1 of [PRRV67] the autors realized, by algebraic methods
isomorphisms:
Mult(ν,bν∗) ≃ V+(ν, ν2− ν1∗, ν1∗) (8)
We explain how this can be recovered from Theorem 1, using D+
0 and D0− as parametrizations
of the G-stable divisors. Notice that, in this case, the conditions 1 of the theorem are empty. Then ρ(by0bν) = (ν1− ν2∗) and forαb+= (α, 0) ∈ D0+, ρ(by0αb+) = α and hν, (b α)b ∨i = hν1, α∨i. Hence, from
Theorem 1, we deduce that
Mult(ν∗,bν) ≃ V−(ν, ν1− ν2∗, ν1).
Since Mult(ν, bν∗) is isomorphic to Mult(ν∗,bν), we recover (7).
Now, ρ(by0(0, −w0α)) = −α, and hν, (0, −wb 0α)∨i = hν2, (−w0α)∨i = hν2∗, α∨i. Then, if we use D−0
to get a minimal number of conditions in the second point of the theorem, we deduce that: Mult(ν∗,ν) ≃ Vb +(ν, ν1− ν2∗, ν2∗).
And since Mult(ν, bν∗) ≃ Mult(ν∗,bν) ≃ Mult(ν∗, (ν2, ν1)), we recover (8).
4.2 Sp2n in SL2n
Fix n ≥ 2. Let V be a 2n-dimensional vector space with fixed basis B = (e1, . . . , e2n). Consider the
following matrices Jn = 1 . .. 1 ; and ω = 0 Jn −Jn 0 . (9)
of size n × n and 2n × 2n respectively. View ω as a symplectic bilinear form of V .
Let G be the associated symplectic group. Set T = {diag(t1, . . . , tn, t−1n , . . . , t−11 ) : ti ∈ C∗}.
Let B be the Borel subgroup of G consisting of upper triangular matrices of G. For i ∈ [1, n], let εi
denote the character of T that maps diag(t1, . . . , tn, t−1n , . . . , t1−1) to ti; then X(T ) = ⊕iZεi. Here
Φ+= {εi± εj : 1 ≤ i < j ≤ n} ∪ {2εi : 1 ≤ i ≤ n},
∆ = {α1 = ε1− ε2, α2= ε2− ε3, . . . , αn−1= εn−1− εn, αn= 2εn}, and
X(T )+ = {Pni=1λiεi : λ1 ≥ · · · ≥ λn≥ 0}.
For i ∈ [1; 2n], set i = 2n + 1 − i. The Weyl group W of G is a subgroup of the Weyl group S2n of
b
G = SL(V ). More precisely
W = {w ∈ S2n : w(i) = w(i) ∀i ∈ [1; 2n]}.
Note that dim G = dim GL2n(C) − dim ∧2V∗ = 2n2+ n and dim bG/ bB = n(2n − 1). Hence we
are looking for by0 ∈ cW such that
dim Gyb
0B/ bb B = 2n.
This is consistent with the fact that ♯Φ1
h= n (Φ1h = {2εi : 1 ≤ i ≤ n}).
An element bw ∈ cW is written as a word [w(1)b w(2) . . .b w(2n)]. Setb b
y0 = [1 ¯1 2 ¯2 . . . ].
Since the ω-orthogonal of he1, e¯1, . . . , e¯ki is hek+1, . . . , e¯ni the stabilizer of yb0B/ bb B is diagonal by
true for any k, this stabilizer consists in block diagonal matrices with blocks of size 2. Moreover, in each block the matrix has to be triangular because of its action on e1, . . . , en. Moreover, these
blocks belong to Sp(2) = SL(2). We just proved that Gby0B/ bb B is contained in a group of dimension 2n. For dimension reasons, we deduce that Gby
0B/ bb B is equal to this subgroup and that G.by0B/ bb B in
open in bG/ bB.
The length of by0 is the number of inversions, that is the set of pairs (i < j) such that j occurs
before i in the word. Fix i ∈ {1, . . . , n}. The number of j > i such that j occurs before i in the word by0 is i − 1. The contribution to these pairs to ℓ(by0) is n(n−1)2 . Similarly, the contribution of
the pairs ¯j > ¯i with ¯i ∈ {n + 1, . . . , 2n} is n(n−1)
2 . Hence ℓ(yb0) = n(n − 1) = dim bG/ bB − dim(G/B)
and y0 has minimal length.
Let bT be the maximal torus of bG consisting in diagonal matrices. Write X( bT ) = ⊕2n
i=1Zεˆi/(ˆε1+
· · · + ˆε2n) with the usual notation. Letαbi = ˆεi− ˆεi+1 be a simple root of bG. If i = 2k is even then
b
y0αbi= ˆε¯k− ˆεk+1 ∈ bΦ1. If i = 2k + 1 is odd then by0αbi= ˆεk+1− ˆεk+1∈ bΦ2. By Lemma 7
D0 = {αb2k : k = 1, . . . , n − 1}
works. Moreover
−ρ(αb2k) = εk+ εk+1 ∀k = 1, . . . , n − 1.
4.3 Spin2n−1 in Spin2n
Let V be a 2n-dimensional vector space endowed with a basis B = (e1, . . . , e2n). Denote by
(x1, . . . , x2n) the dual basis. For i ∈ [1; 2n], set i = 2n + 1 − i. Let bG0 be the orthogonal group
associated to the quadratic form
Q =
n
X
i=1
xixi.
Set bT0 = {diag(t1, . . . , tn, t−1n , . . . , t1−1) : ti ∈ C∗} in bG0. Let bB0 be the Borel subgroup of
b
G0 consisting in upper triangular matrices of bG0. Let ˆεi denote the character of bT0 that maps
diag(t1, . . . , tn, t−1n , . . . , t−11 ) to ti; then X( bT0) = Pn i=1Zεˆi. Here b Φ+= {ˆεi± ˆεj : 1 ≤ i < j ≤ n}, and b ∆ = {αb1 = ˆε1− ˆε2,αb2 = ˆε2− ˆε3, . . . , αbn−1= ˆεn−1− ˆεn,αbn= ˆεn−1+ ˆεn}.
The Weyl group cW of bG0 is a subgroup of the Weyl group S2n of SL(V ). More precisely
c
W = {w ∈ S2n :
w(i) = w(i) ∀i ∈ [1; 2n]
♯w([1; n]) ∩ [n + 1; 2n]) is even }.
For i = 1, . . . , n − 1, sαbi = (i, i + 1)(i + 1 ¯i). Moreover sαbn = (n − 1 ¯n)(n n − 1).
Let H = he1, . . . , en−1,en+e√2n+1, en+1, . . . , e2ni with coordinates (xi)1≤i≤n−1∪ (y) ∪ (x¯i)1≤i≤n−1.
The stabilizer of H in bG0 is G0= SO2n−1(C). Its maximal torus T0 is
{diag(t1, . . . , tn−1, 1, 1, t−1n−1, . . . , t−11 ) : ti∈ C∗}.
The groups G and bG are the universal covers of G0 and bG0 respectively. Here
Φ+= {ε
i± εj : 1 ≤ i < j ≤ n − 1} ∪ {εi : 1 ≤ i ≤ n − 1}, and
∆ = {α1 = ε1− ε2, α2 = ε2− ε3, . . . , αn−2= εn−2− εn−1, αn−1 = εn−1}.
Note that ∆2= {α
n−1} and there is only one G0-stable divisor in bG/ bB. We have
dim bG/ bB = n(n − 1) dim(G/B) = (n − 1)2. Set
b
y0= sbαn−1. . . sαb2sαb1 = (1 n n − 1 . . . 2)(¯1 ¯n . . . ¯2).
We have by0bǫi =bǫi−1 for i = 2, . . . , n and by0bǫ1 =bǫn. In particular
b
y0Φb+ = {bǫn±bǫj : j = 1, . . . , n − 1} ∪ {bǫi±bǫj : 1 ≤ i < j = n − 1}.
One easily deduce that the stabilizer of by0 in G0 is H0.
The only descent is bα1. Then D0= D = {αb1} and −ρ(yb0αb1) = −ρ(ˆεn− ˆε1) = ε1.
4.4 G2 in Spin7
Here G is the group of type G2embedded in Spin7(C) = bG using the first fundamental representation
which is 7-dimensional. We label the simple roots as follows: B3
1 2 3
G2
1 2
The map ρ is characterized by
ρ(αb2) = α2 ρ(αb1) = ρ(αb3) = α1, and satisfies ρ(αb1+αb2) = ρ(αb2+αb3) = α1+ α2 ρ(αb2+ 2αb3) = ρ(αb1+αb2+αb3) = 2α1+ α2 In particular b Φ+∩ bΦ1 = {αb2,αb1+αb2+ 2αb3,αb1+ 2αb2+ 2αb3}.
The working by0 are
b
y0 =bs1bs2bs3 and by0=bs3bs2bs3
Indeed, one can check that ℓ(by0) = 3 and dim(G ∩by0Bbyb0−1) = 3.
Moreover, the only simple roots bα such that dim(G ∩ bw bBwb−1) = 4 where w =b yb0sαb is bα3. The
so obtained divisor is the only G-stable divisor accordingly to ♯∆2= 1.
There are ♯cW
♯W = 4 G-orbits in bG/ bB. The closed orbit has dimension 6 and the open one 9: hence
6 7 8 9 b α1 αb3 b α2 b α3 Here D0 = D = {αb3} and − ρ(yb0αb3) = α1+ α2. 4.5 F4 in E6
Here G is the group of type F4 embedded in bG of type E6. We label the simple roots as follows:
E6
1 3 4 5 6
2
F4
1 2 3 4
The root system E6lies in R8with basis (bǫi)1≤i≤8. More precisely it spans the space x6 = x7 = −x8.
See [Bou02]. The roots are
±bǫi±bǫj 1 ≤ i < j ≤ 5 and ± 1 2(bǫ8−bǫ7−bǫ6+ 5 X i=1 (−1)ν(i)bǫi X ν(i) even. Set ˜ǫ6=bǫ8−bǫ6−bǫ7. Then, the simple roots are
b α1 = 1 2 bǫ1+bǫ8− (bǫ2+bǫ3+bǫ4+bǫ5+bǫ6+bǫ7) = 1 2 bǫ1+ ˜ǫ8− (bǫ2+bǫ3+bǫ4+bǫ5) and b α2 =bǫ1+bǫ2 αb3=bǫ2−bǫ1 αb4=bǫ3−bǫ2 αb5 =bǫ4−bǫ3 αb6 =bǫ5−bǫ4.
Root system F4. The roots are
1
2(±ǫ1± ǫ2± ǫ3± ǫ4) and ± ǫi ± ǫi± ǫj 1 ≤ i < j ≤ 4. The simple roots are
α1 = ǫ2− ǫ3 α2 = ǫ3− ǫ4 α3 = ǫ4 α4= 1
2(ǫ1− ǫ2− ǫ3− ǫ4). The map ρ is characterized by
It is the quotient by ǫ2+ ǫ3− ǫ1− ǫ4 12 ǫ1+ ˜ǫ6+ ǫ4− (ǫ2+ ǫ3+ 3ǫ5) Moreover ♯bΦ+= 36 ♯cW = 51 840 ♯∆2 = 2 ♯W = 1 152 ♯Φ+= 24 ♯cW /W = 45. We have 12 short positive roots in F4:
ǫi 1 2(ǫ1± ǫ2± ǫ3± ǫ4) A working by0 is b y0=bs1bs5bs3bs4bs2bs3bs4bs5bs4bs3bs1bs6.
Indeed, one can check that ℓ(by0) = 12 and dim(G ∩by0Bbyb0−1) = 16.
Moreover, the only simple roots bα such that dim(G ∩ bw bBwb−1) = 17 wherew =b by0s b
αare bα1 and
b
α6. The so obtained divisors are distinct since ♯∆2= 2. Set
D0= {αb1, αb6}.
One checks that
−yb0αb1 =αb1+αb2+ 2αb3+ 2αb4+αb5 7−→ αρ 1+ α2+ 3α3+ α4;
−yb0αb6 =αb1+αb2+αb3+ 2αb4+ 2αb5+αb6 7−→ αρ 1+ 2α2+ 3α3+ 2α4.
References
[Bou02] Nicolas Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley.
[Bri99] M. Brion. Rational smoothness and fixed points of torus actions. Transform. Groups, 4(2-3):127–156, 1999. Dedicated to the memory of Claude Chevalley.
[Bri01] Michel Brion. On orbit closures of spherical subgroups in flag varieties. Comment. Math. Helv., 76(2):263–299, 2001.
[FH91] William Fulton and Joe Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics. [Hum75] James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York, 1975.
Grad-uate Texts in Mathematics, No. 21.
[Kno95] Friedrich Knop. On the set of orbits for a Borel subgroup. Comment. Math. Helv., 70(2):285–309, 1995.
[McG90] William M. McGovern. A branching law for Spin(7, C) → G2 and its applications to
[NS05] Satoshi Naito and Daisuke Sagaki. An approach to the branching rule from sl2n(C) to
sp2n(C) via Littelmann’s path model. J. Algebra, 286(1):187–212, 2005.
[OV90] A. L. Onishchik and È. B. Vinberg. Lie groups and algebraic groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.
[PRRV67] K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan. Representations of complex semi-simple Lie groups and Lie algebras. Ann. of Math. (2), 85:383–429, 1967.
[Res04] Nicolas Ressayre. Sur les orbites d’un sous-groupe sphérique dans la variété des drapeaux. Bull. de la SMF, 132:543–567, 2004.
[Res10] Nicolas Ressayre. Spherical homogeneous spaces of minimal rank. Adv. Math., 224(5):1784–1800, 2010.
[Sun90] Sheila Sundaram. Tableaux in the representation theory of the classical Lie groups. In Invariant theory and tableaux (Minneapolis, MN, 1988), volume 19 of IMA Vol. Math. Appl., pages 191–225. Springer, New York, 1990.
[Ž73] D. P. Želobenko. Compact Lie groups and their representations. American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 40.
