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

moreover, the action of Q ∗ 2
4. General Theory 42
can be obviously extended to an action of the
semidirect product W ⋉ Q∗ 2 .
Let W the subgroup generated by all conjugates σWσ −1 of W by elements
σ ∈ Q∗ 2 .
Theorem 4.2.6. The homomorphism γ ◦ h : Z ↦→ U 0 is injective and its
image is precisely the set U 0W of fixed points of the action of W on U 0 .
Proof. See [DCP93] or [CP95].
Example 4.2.7. Let g = sl2(C). Then W = W and U0W consist of the
. Thus
Laurent polynomials in K1 which are invariant under K1 ↦→ K −1
1
U 0W is generated as an algebra over C(q) by
φ = K1 + K −1
−1
(q − q−1 .
) 2
It easy to check that the quantum Casimir element
Ω = qK1 + q−1K −1
−1
(q − q−1 + EF
) 2
lies in Z and γ −1 (h(Ω)) = φ. It follows from theorem 4.2.6 that Ω generates
Z as C(q)-algebra.
We see now the quantum analogue of the Harish Chandra’s theorem on
the central characters of the classical universal enveloping algebra. Let λ ∈ Λ
and define a homomorphism λ : U 0 ↦→ C(q) by Ki ↦→ q (αi|λ) . Let
χq,λ = λ ◦ γ −1 ◦ h : Z ↦→ C(q)
Theorem 4.2.8. Let λ, µ ∈ Λ. Then χλ = χµ if and only if µ = ω(λ) for
some ω ∈ W.
Proof. See [CP95]
Suppose now that ǫ ∈ C is a l th root of the unity such that l is odd
and l > di for all i. Our aim is to describe the center Zǫ of Uǫ. We shall
assume a different approach from that used in the generic case. Let Z1 =
(Z ∩ A) /(q − ǫ), we call it the Harish Chandra part of the center. This is
not the full center, in fact we have
Proposition 4.2.9. The elements El α, F l α for α ∈ R + , and Kl i for i =
1, . . .,n lie in Zǫ.
Proof. See [DCP93], §21.