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

4. General Theory 50
Since T is the direct sum of T1 and N, its kernel is the intersection of
the two kernels of these operators. We have computed the kernel of T1 in
proposition 4.5.5, so the kernel of T is equal to the kernel of N restricted
to the submodule spanned by vectors vω. Thus, we can identify N with the
map 1 − w0 : Λ → Q. At this point we need the following fact
Lemma 4.5.6. Let θ = n i=1 aiαi the highest root of the root system R. Let
Z ′′ = Z[a −1
1 , . . .,a−1 n ], and let Λ ′′ = Λ ⊗Z Z ′′ and Q ′′ = Q ⊗Z Z ′′ . Then the
Z ′′ submodule (1 − w0)Λ ′′ of Q ′′ is a direct summand.
Proof. See [DCKP95] or [DCP93] §10.
We call l > 1 a good integer if l is relative prime to t and to all the ai
Theorem 4.5.7. If l is a good integer, then
deg Uǫ(b) = l 1
2 (l(ω0)+rank(1−ω0))
Proof. From the above discussion we see that deg U_(b) = l s , where s =
(N + n) − (n − rank(1 − w0)) with N = l(w0), which together with theorem
4.5.3 prove the claim.
Note. This method for the calculation of the degree does not use the center
of Uǫ(b).