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

5. Quantum universal enveloping algebras for parabolic Lie algebras 66
Proof. First, we observe that T1 is surjective, since M and Ml are projections
over V − and V + respectively, by lemma 4.5.5. Since the n vectors vω are
part of a basis and, the kernel of T1 is a direct summand of rank n, by
surjectivity. It is enough to show that vω is in the kernel of T1. We have
T1(vω) = A l
⎛
⎝
⎞
⎠ − t B l (−ω − w0(ω))
t∈Iω(w l 0 )
ut
+ t ⎛
B (−ω − w0(ω)) − A ⎝
t∈Iω(w0)
= M l (vω) − t B l
w l
0(ω) − w0(ω)
So from lemma 4.5.4 and lemma 4.5.5, we have:
T1(vw) = − t B l
w l
0(ω) − w0(ω)
u ′ t
⎞
⎠
− M(vω).
Let w0 = wl 0w, since w runs through the fundamental weights, we have two
cases:
1. w(ω) = w, therefore w l 0 (ω) − w0(ω) = 0 and T1(vω) = 0.
2. w(ω) = ω, therefore w l 0 (ω) = ω and wl 0 (ω) − w0(ω) = ω − w0(ω) ∈
ker t B l , by definition of t B l , so T1(vω) = 0.
Since T is the direct sum of T1 and N. Its kernel is the intersection of the
2 kernels of these operators. We have computed the kernel of T1 in 5.4.20.
Thus the kernel of T equals the kernel of N restricted to the submodule
spanned by the vω.
Lemma 5.4.21.
N(vω) =
t∈Iω(w l 0 )
Proof. Note that B(ut) = βt, then
N(vw) =
β 1 t −
t∈Iω(w l 0 )
t∈Iω(w0)
β 1 t −
Finally, the claim follows using lemma 4.5.4.
βt = w0(ω) − w l 0(ω).
t∈Iω(w0)
Thus, we can identify N we the map w0 − w l 0
need the following fact
βt. (5.20)
: Λ → Q. At this point we