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

4. General Theory 45
Let G be the connected, simply connected Lie group with Lie algebra g,
let H ⊂ G be the maximal torus of G where LieH = h, and let N ± be the
unipotent subgroups of G with Lie algebra n ± . Note that H is canonically
identified with Spec(Z0 0 ) by the paring
(exp(η), ki) = exp 2π √ −1αi(η)
for η ∈ h, the Lie algebra of H. The product G 0 = N − HN + is well known
to be a dense open subset of G (in the complex topology), called the big cell.
We define maps
and by multiplication a map
E : Spec(Z + 0 ) → N+
(4.16)
F : Spec(Z − −
0 ) → N (4.17)
K : Spec(Z 0 0) → H (4.18)
π : EKF : Spec(Z0) = Spec(Z + 0 ) × Spec(Z0 0) × Spec(Z − 0 ) → G
as follows. Fix a reduced expression of the longest element of W, ω0 =
si1 . . .siN , and let Eβ1 , . . .,EβN be the corresponding negative root of g.
Let
fβk =
. . .Tik−1 (fik ) ∈ Z0
ǫik
− ǫ−1
ik
Ti1
which we regard as a complex valued function on Spec(Z0). Then we define
maps E, F and K to be the products
F = exp fβNF
βN . . .exp fβ1F
β1
E = exp Tω0 (fβN )Tω0 (F βN ) . . .exp Tω0 (fβ1 )Tω0 (F β1 )
K(h) = h 2
where h ∈ H
Proposition 4.2.15. The product map π = FKE : Spec(Z0) → G is independent
of the choice of reduced decomposition of ω0, and is a covering of
degree 2 n .
Proof. See [DCP93], §16.
Theorem 4.2.16. (i) Consider the Poisson structure on G defined in example
1.2.14, then we have an identification of Spec(Z0) with a Poisson
dual to G. In particular LieSpec(Z0) = s, where
s = {(x, y) ∈ b+ ⊕ b− : xh + yh = 0} ,
(ii) The symplectic leaves of Spec(Z0) coincide with the preimages of the
conjugacy classes in G under π.
(iii) If C ⊂ G is a conjugacy class and dimC > 0 then π −1 (C) is connected.
Proof. see [DCKP92] or [DCP93] §16.