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

4. General Theory 46
4.3 Parametrization of irreducible representation of Uǫ
As usual we assume that ǫ is a primitive l th root of the unity with l odd and
l > di for all i. All representation are on complex vector space.
We know that Zǫ acts by scalar operators on V (cf [CP95]), so there exist
an homomorphism χV : Zǫ ↦→ C, the central character of V, such that
z · v = χV (z)v
for all z ∈ Zǫ and v ∈ V . Note that isomorphic representations have the
same central character, so assigning to a Uǫ module its central character give
a well define map
Ξ : Rap (Uǫ) → Spec (Zǫ),
where Rap (Uǫ) is the set of isomorphism classes of irreducible Uǫ modules,
and Spec (Zǫ) is the set of algebraic homomorphisms Zǫ ↦→ C.
To see that Ξ is surjective, let I χ , for χ ∈ Spec (Zǫ), be the ideal in Uǫ
generated by
kerχ = {z − χ(z) · 1 : z ∈ Zǫ} .
To construct V ∈ Ξ −1 (χ) is the same as to construct an irreducible representation
of the algebra U χ ǫ = Uǫ/I χ . Note that U χ ǫ is finite dimensional and
non zero. Thus, we may take V , for example, has any irreducible subrepresentation
of the regular representation of U χ ǫ .
Composing with the surjective map Spec(Zǫ) → Spec(Z0), we obtain a
surjective map
Φ : Rap(Uǫ) → Spec(Z0).
A priori in order to study representations one should study the representation
theory of the algebra U χ ǫ , for all χ ∈ Spec(Z0). However by [DCKP92] (or
[DCP93] §16), we have that
Theorem 4.3.1. Let χ1 and χ2 ∈ Spec(Z0) such that χ1 and χ2 live in the
same symplectic leaf then U χ1
ǫ = U χ2
ǫ .
4.4 Degree of Uǫ.
Summarizing, if ǫ is a primitive l th root of 1 with l odd and l > di for all i,
we have proved the following facts on Uǫ:
• Uǫ is a domain (cf theorem 4.1.9),
• Uǫ is a finite module over Z0(cf proposition 4.2.10).
Since the L.S. relations holds Uǫ (cf proposition 4.2.4), we can apply the
theory developed in section 4.1.2, and we obtain that Gr Uǫ is a twisted
polynomial algebra, with some elements inverted. Hence all conditions of
theorem 3.5.3 are verified, so