Restricted root theory

We study the restriction of the non-compact roots. For α ∈ C_i, C_ij, hα, γ_ii =

−¹₂hγ_i, γ_ii. Thus, from Lemma 2.26, α + γ_i is a root, which is non-compact and we have that the positive non-compact roots are the union of the following disjoint sets

• Γ = {γ1, . . . , γr},

• Qi = {α ∈ ∆⁺_Q | π(α) = ¹₂γ_i},

• Qij = {α ∈ ∆⁺_Q| π(α) = ¹₂(γ_j + γ_i)}.

The following translations are bijections

C_i −−→ Q^+γi _i Q_i −−→ C^−γi _i (2.26.2) Q_ij −−→ C^+γi _ij C_ij −−→ Q^−γi _ij.

The following theorem will be fundamental in what follows.

Theorem 2.27. The restricted root system Σ(Ad(c)g, it⁻) consists of the roots ±γ_j (1 ≤ j ≤ r) with multiplicity 1, the roots ±¹₂γ_j±¹₂γ_k (j 6= k) with multiplicity a, and possibly the roots ±¹₂γ_j with even multiplicity 2b. The restricted root system is then of type (BC)_r or C_r (see [Hel01], Ch.X) and its Weyl group consists of the signed permutations of the set Γ.

When b = 0 (resp. b 6= 0) the space G/H, the Lie algebra g or the group G is of tube type (resp. of non-tube type). We have seen above that the spaces of tube type have a realization as a domain of tube type. The condition b = 0 is equivalent to (g, h) having a restricted root system of type C_r, which is in turn equivalent to the condition in Proposition 2.22.

We define the positive compact/non-compact roots of tube/non-tube type as

∆⁺_C,nt= C₀∪ [

1≤j≤r

C_j ∆⁺_Q,nt= [

1≤j≤r

Q_j ∆⁺_nt = ∆⁺_C,nt∪ ∆⁺_Q,nt

∆⁺_C,t = [

1≤i<j≤r

C_ij ∆⁺_Q,t = Γ ∪ [

1≤i6=j≤r

Q_ij ∆⁺_t = ∆⁺_C,t∪ ∆⁺_Q,t.

A relevant number in what follows will be the dual Coxeter number N , which is defined as

N = a(r − 1) + b + 2. (2.27.1)

Denoting n = dim_Cm⁺ and n_T = dim_Cm⁺_T, we have n_T = r(r − 1)

2 a + r and n = n_T + rb.

And therefore,

N = 2n_T + (n − n_T)

r = n + n_T

r .

2.3.2 More results on restricted roots

We show now some results not found in the literature which will be needed later. The squared norm of any of the elements of Γ is an important constant that we denote by hγ_i, γ_ii. It does not depend on i as all γ_i have the same length (cf. [Hel08], Ch. V, Th. 4.1., but it can be deduced also from the next lemma).

Lemma 2.28. In the non-tube type case (b 6= 0), we have the following:

1. For any α ∈ Qi, Qij, Ci, Cij, hα, αi = hγi, γii, for any 1 ≤ i 6= j ≤ r.

2. Let α ∈ Q_i and β ∈ Q_j. If β − α is a root, then hβ, αi = ¹₂hγ_i, γ_ii.

3. Let β ∈ Q_i, Q_ij. For any root α 6= β, the string β + nα contains either one or two elements.

If b = 0 and all the roots have the same length, the results (3) is still true for α, β ∈ Q_ij.

Proof. We prove in detail (1), as we will repeat this type of arguments in this section.

Let α ∈ Q_i. The γ_i-string based on α is {α − γ_i, α}, since if α − 2γ_i and α + γ_i were roots, they would project to −³₂γi and ³₂γi respectively, which is not possible. The α-string based on γi is {γi− α, γi}, since if γi− 2α and γi+ α were roots, they would project to 0 and ³₂γ_i. At first glance, γ_i − 2α could be a root, but γ_i − 2α would be non-compact and only compact roots project to 0. The same argument works for α ∈ Q_ij, but the observation about γ_i− 2α is not needed. Once we have that the strings are {α−γ_i, α} and {γ_i−α, γ_i}, we apply Lemma 2.26, and get 2hγ_i, αi = hα, αi and 2hα, γ_ii = hγ_i, γ_ii, i.e., hα, αi = hγ_i, γ_ii. For α ∈ C_i, C_ij, we write α as α⁰+ γ_i for α⁰ ∈ Q_i, Q_ij, and hα, αi = hα⁰ + γ_i, α⁰ + γ_ii = hα⁰, α⁰i + 2hα, γ_ii + hγ_i, γ_ii = hα⁰, α⁰i, which equals ¹₂hγ_i, γ_ii by the just proved.

For the second statement, if β − α is a root, β − α ∈ Q_ji, and then hβ − α, β − αi = hγi, γii. From the first statement we obtain hβ, αi = ¹₂hγi, γii.

The only case in which a string could have more than two elements is that of α, β in the same set Q_i or Q_ij, in such a way that {β − 2α, β − α, β} projects to {−¹₂γ_i, 0,¹₂γ_i} or {−¹₂(γ_i− γ_j), 0, ¹₂(γ_i− γ_j)}. But this will imply 2hα, αi = 2hβ, αi = hγ_i, γ_ii, which is not possible from the first statement.

Since the arguments depend only on the length of the roots, the last statement follows.

Remark 2.29. In the tube type case, we may have that hα, αi 6= hγ_i, γ_ii. Between the irreducible cases, this only happens for sp(2n, R).

Lemma 2.30. Let β ∈ Q_i. Then, for exactly half of the roots α ∈ Q_ij, β − α is a root.

Proof. We use the notation α ± S = {α ± s | s ∈ S}. First, we show that the sets (β + Q_j) ∩ ∆ and [(β − γ_i) + (Q_j− γ_j)] ∩ ∆ = [(β − γ_i) − C_j] ∩ ∆ are disjoint. Note that if β = ¹₂γ_i+ β^⊥, for β^⊥ ∈ (t^C)^∗, then, hβ^⊥, β^⊥i = ³₄hγ_i, γ_ii by (1) in Lemma 2.28.

Let α_p, α_q ∈ Q_j be such that β + α_p = (β − γ_i) + (α_q − γ_j). Write α_p = ¹₂γ_j + α^⊥_p and α_q = ¹₂γ_j + α^⊥_q. By (2) of Lemma 2.28, hβ, α_pi = hβ^⊥, α^⊥_pi = ¹₂hγ_i, γ_ii. From β + αp = (β − γi) + (αq− γj) we have 2β^⊥− α^⊥_p − α^⊥_q = 0. But this cannot be true as h2β^⊥− α^⊥_p − α_q^⊥, β^⊥i = −¹₄hγi, γii. So, the sets are disjoint.

Secondly, we show that C_ij is contained into the union of β_i+ Q_j and (β − γ_i) + (Q_j− γ_j). Given δ ∈ C_ij, we have that hβ − γ_i, δi = hβ, δi −¹₂hγ_i, γ_ii. So, at least one of hβ − γ_i, δi or hβ, δi is not zero.

And finally, it remains to show that exactly one half of C_ij comes from the sum of non-compact roots, and the other half from the sum of compact roots. This comes from hβ, αi = hβ − γ_i, α − γ_ji, which implies that for every root β − α which is sum of non-compact roots, there is a different root (β − γ_i) − (α − γ_j) which is sum of compact ones.

Remark 2.31. From this lemma, if b 6= 0, then between the b sums of βi+ Qj there are

a

2 roots, and b − ^a₂ sums which are not roots. This implies that a is even and 2b ≥ a.

Lemma 2.32. Let β ∈ Q_i. Then we have that

hβ, αi =











hγ_i, γ_ii for α = β

1

2hγ_i, γ_ii for α = β − γ_i

1

2hγ_i, γ_ii for α ∈ Q_i, α 6= β 0 for α ∈ C_i, α 6= β − γ_i.

Proof. The first case follows directly from (1) in Lemma 2.28. For α ∈ Qi\ {β} we have that

hβ, α − γ_ii = hβ, αi − 1

2hγ_i, γ_ii.

If we do α = β we have the case α = β − γ_i. The sum α + β is not a root, as it would be compact and project to γ_i. So, hα, βi ≥ 0, and from Lemma 2.28, (2), we have that hα, βi is 0 or ¹₂hγ_i, γ_ii. From the equation, only one of hβ, αi, hβ, α − γ_ii can be non-zero. Let us see that hβ, α − γ_ii = 0. If hβ, α − γ_ii 6= 0, then one of β ± (α − γ_i) would be a root, but β + (α − γ_i) would be compact projecting to 0, and

β − (α − γ_i) would project to γ_i and therefore it would be γ_i, which is only possible when β = α.

2.3.3 Sums of roots

We take advantage of the decomposition into compact and non-compact roots to state some results about sums of roots which will be relevant when proving facts about the Toledo character to be defined below.

For the sake of simplicity, we use the following notation for any indexed summands f (i), f (i, j) and any sum over a set S:

X

i

f (i) := X

1≤i≤r

f (i) X

i<j

f (i, j) := X

1≤i<j≤r

f (i, j) X

S

α :=X

α∈S

α

An example will clarify this:

X

i<j

X

Qij

α := X

1≤i<j≤r

X

α∈Qij

α.

The following lemma computes the sum of all the positive non-compact roots of tube type, ∆⁺_Q,t = Γ ∪S

1≤i<j≤rQ_ij.

Lemma 2.33. For any 1 ≤ i < j ≤ r, we have that X

α∈Qij

α = a

2(γ_i+ γ_j).

As a consequence,

X

α∈∆⁺_Q,t

α = N − b 2

r

X

j=1

γ_j.

Proof. Consider the bijections (2.26.2). On the one hand,P

α∈Qijα−aγ_i =P

α∈Cijα.

On the other hand, P

α∈Qijα − aγj = −P

α∈Cijα. Adding the two expressions we obtainP

α∈Qij = ^a₂(γi+ γj), and adding for all possible i and j:

X

∆⁺_Q,t

α =X

i<j

X

Qij

α +

r

X

j=1

γ_j = a(r − 1) + 2 2

r

X

j=1

γ_j = N − b 2

r

X

j=1

γ_j,

where the last equality follows from the definition of the dual Coxeter number, (2.27.1).

For the positive non-compact roots of non-tube type, ∆⁺_Q,nt = S

1≤i≤rQ_i we do not have a similar result for its sum. But the following lemma will suffice for our purposes.

Lemma 2.34. For β ∈ Q_ij we have that

* X

∆⁺_Q,nt

α, β +

=

*b 2

r

X

j=1

γ_j, β +

And for β ∈ Qi we have that,

* X

∆⁺_Q,nt

α, β +

=

*N + b 2

r

X

j=1

γ_j, β +

.

Proof. Consider the expression P =

* X

i<j

X

Qij

α,X

k

X

Qk

β +

.

On one hand, by Lemma 2.33, P =

*a(r − 1) 2

r

X

j=1

γj,X

k

X

Qk

β +

= br · a(r − 1) 2

1

2hγi, γii.

On the other hand, for any α ∈ Q_ij we have

* α,X

k

X

Qk

β +

=

*

α, X

Qi∪Q_j

β +

= bh1 2γ_i,1

2γ_ii + bh1 2γ_j,1

2γ_ji = b

2hγ_i, γ_ii.

So we have

P = ar(r − 1) 2

* α,X

k

X

Q_k

β +

. Therefore, for α ∈ Q_ij,

* X

k

X

Qk

β, α +

= 2P

ar(r − 1) b 2

1

2hγ_i, γ_ii =

*b 2

r

X

j=1

γ_j, β +

. For β ∈ Q_i, from Lemmas 2.30 and 2.32 we have that

* X

j

X

Qj

α, β +

=h

(r − 1)a

2 + (b − 1) + 2i1

2hγ_i, γ_ii

= N + b 2 ·1

2hγi, γii =

*N + b 2

r

X

j=1

γj, β +

.

From Lemmas 2.33 and 2.34, we obtain the following proposition.

Proposition 2.35. For β ∈ ∆⁺_Q:

* X

α∈∆⁺_Q

α, β +

= N hγ_i, γ_ii .

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