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

Theorem 4.4.1. Uǫ is a maximal order.
Proof. See [DCK90].
4. General Theory 47
Note. As we have seen in section 3.5, since Uǫ is a maximal order, Zǫ is
integrally closed and Uǫ ∈ Cm for some m ∈ N.
Following example 2.2.4, we can make the following construction: denote
by Qǫ := Q(Zǫ) the field of fraction of Zǫ, we have that Q(Uǫ) := Uǫ ⊗Zǫ Qǫ
is a division algebra, finite dimensional over its center Qǫ. Denote by F the
maximal commutative subfield of Q(Uǫ), we have that
(i) F is a finite extension of Qǫ of degree m,
(ii) Q(Uǫ) has dimension m 2 over Qǫ,
(iii) Q(Uǫ) ⊗Qǫ F ∼ = Mm(F).
So,
Proposition 4.4.2. There is a non empty closed proper subvariety D of
Spec(Zǫ) such that
(i) If χ ∈ Spec(Zǫ) \ D, then U χ ǫ is isomorphic to Mm(C), and (hence)
there is, up to an isomorphism, exactly one irreducible Uǫ module Vχ
with character χ. One has dimVχ = m.
(ii) If χ ∈ D, then dim U χ ǫ ≥ m 2 , but the dimension of every irreducible
U χ ǫ module is strictly less than m.
Proof. Apply theorem 2.2.7.
Note that, from the above discussion, we have:
dim Q(Z0) Qǫ = deg τ
dimQǫ Q(Uǫ) = m 2
dim Q(Z0) Q(Uǫ) = l 2N+n
where, the first equality is a definition, the second has been pointed out
above and the third follows from the P.B.W theorem. Finally, we have
l 2N+n = m 2 deg τ.
Recall that, from proposition 4.2.12, we have deg τ = l n , hence
deg Uǫ(g) = m = l N .