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

4. General Theory 41
4.2.1 The center of Uǫ
The center of Uǫ consists of “two parts”, one coming from the center at a
“generic q”, the other from the fact that ǫ is a root of 1. We begin by
describing the center at “q generic”.
Let Z be the center of U. Any element z ∈ Z can be written as a linear
combinations of the elements of the basis of U given by the P.B.W. theorem.
Since z commutes with the Ki for all i, it follows that
z =
η∈Q + r,t∈Par(η)
E r ϕr,tF t
where ϕr,t ∈ U0 and
Par(η) = (m1, . . .,mN) ∈ N N :
miαi = η .
Definition 4.2.5. The map h : Z ↦→ U 0 defined by h(z) = ϕ0,0 is called the
Harish Chandra homomorphism
Property. h is a homomorphism of algebras.
Proof. See [DCP93], §18.
h allows us to describe Z as a ring. Any element φ ∈ U0 may be regarded
as a C(q)-valued function on the weight lattice Λ in an obvious way: if
φ = n i=1 Kti
i , where ti ∈ Z for all i, set ∀ λ ∈ Λ
φ(λ) = qÈi ti(αi|λ)
and extended to U 0 by linearity. Define an automorphism of C(q)-algebras
γ : U 0 ↦→ U 0 by setting γ(Ki) = qiKi. Then
Property. For all φ ∈ U 0
where ρ = 1
2 (α1 + . . . + αn).
γ(φ)(λ) = φ(λ + ρ)
Let Q ∗ 2 = {σ : Q ↦→ Z2}, it is easy to see that Q ∗ 2
automorphisms by
i
acts on U as a group of
σ · Kβ = σ(β)Kβ (4.11)
σ · Fα = σ(α)Fα (4.12)
σ · Eα = Eα (4.13)
for all α ∈ R + , β ∈ Q, and σ ∈ Q ∗ 2 . Note that W acts on Q∗ 2 :
(ω · σ)(β) = σ(ω −1 (β));