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

1. Poisson algebraic Groups 17
Let X = G/B± the flag manifold. By proposition 1.4.8, we have
X =
Xω where Xω ⋍ B±ωB±/B±
ω∈W
Definition 1.4.9. Xω is the so-called Schubert cell of X corresponding to
ω ∈ W.
It is well know that Xω is naturally isomorphic to C l(ω) , where l(ω) is
the length of ω.
Proposition 1.4.10. The following Bruhat decomposition holds:
D(G) = P · (ω1, ω2) · P, (1.9)
(ω1,ω2)∈W×W
where P = HG ∗ and H is the distinguished Cartan subgroup of G associated
to h.
Consider the following sets:
C (ω1,ω2) = (G ∗ · (ω1, ω2) · G ∗ )/G ∗ ,
B (ω1,ω2) = C (ω1,ω2) ∩ ˜ G,
A (ω1,ω2) = p −1
h∈H
hB (ω1,ω2)
Proposition 1.4.11. (i) each symplectic leaf in ˜ G is of the form hB (ω1,ω2)
for some h ∈ H and (ω1, ω2) ∈ W × W.
(ii) each symplectic leaf in G is of the form hA (ω1,ω2) for some h ∈ H and
(ω1, ω2) ∈ W × W.
Proposition 1.4.12. Denote s(ω1, ω2) = codimhker(ω1ω −1
2
C (ω1,ω2) ⋍ H s(ω1,ω2) × C l(ω1)+l(ω2)
− 1). Then
Example 1.4.13. The following is the full list of the symplectic leaves in
SL2(C):
t 0
Tt =
0 t−1
, t = 0,
t b
TtA (e,ω0) =
0 t−1
: b = 0 , t = 0,
t 0
TtA (ω0,e) =
c t−1
: c = 0 , t = 0,
a b
TtA (ω0,ω0) = : b, c = 0,
c d
b
= t2 , t = 0.
c