Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

5. Quantum universal enveloping algebras for parabolic Lie algebras 69
Corollary 5.5.6. The set
Ω0 = a ∈ Spec(Z0(p)) : τ −1 (a) ⊂ Ω
is a Zariski dense open subset of Spec(Z0).
We know by the theory developed in chapter 3, that Sǫ(p) ∈ Cm0 , with
m0 = l l(w0)+l(wl 0 )+rank(w0−wl 0) . As we see in proposition 5.4.9, Sǫ(p) is a finite
module over C0, then Cǫ, the center of Sǫ(p) is finite over C0. The inclusion
C0 ֒→ Cǫ induces a projection υ : Spec(Cǫ) → Spec(C0). As before, we have:
Lemma 5.5.7. (i)
Ω ′ = {a ∈ Spec(Cǫ),such that the corresponding semisimple
representation is irreducible}
is a Zariski open set. This exactly the part of Spec(Cǫ) over which
S χ ǫ (p) is an Azumaya algebra of degree m0.
(ii) The set
Ω ′ 0 = a ∈ Spec(Z0(p)) : υ −1 (a) ⊂ Ω ′
is a Zariski dense open subset of Spec(Z0).
Proof. Apply theorem 2.2.7 at Sǫ(p).
Note. Since Spec(Z0) is irreducible, we have that Ω0 ∩ Ω ′ 0 is non empty.
We can state the main theorem of this section
Theorem 5.5.8.
deg Uǫ(p) = l 1
2(l(w0)+l(w l 0 )+rank(w0−w l 0))
Proof. For χ ∈ Ω0 ∩ Ω ′ 0 , we have, using theorem 5.5.5 and lemma 5.5.7,
and
deg Uǫ(p) = m = deg U χ ǫ (p),
deg S χ ǫ (p) = deg Sǫ(p).
But for all t = 0, we have that U t,χ
ǫ is isomorphic to U χ ǫ (p) as algebra. Hence
is well known that the isomorphism class of semisimple algebras are closed
(see [Pro98] or [Pie82]), we have that S χ ǫ (p) = U 0,χ
ǫ is isomorphic to U χ ǫ (p).
Then
deg Uǫ(p) = m = deg U χ ǫ (p) = deg S χ ǫ (p) = deg Sǫ(p).
And by theorem 5.4.19 the claim follows.