6 1. PRELIMINARIES
uv = ¯¯ u v whenever l(uv) = l(u)+l(v). In what follows we will slightly abuse notation
by denoting an element of the Weyl group and its representative in NormG H with
the same letter.
We will also need a notion of a reduced word for an ordered pair (u, v) of
elements in W . It is defined as follows: if (i1,...,il(u)) is a reduced word for u and
(i1,...,il(v)) is a reduced word for v, then any shuffle of sequences (i1,...,il(u))
and (−i1,..., −il(v)) is called a reduced word for (u, v).
Every ξ ∈ g can be uniquely decomposed
(1.10) ξ = ξ− + ξ0 + ξ+,
where ξ+ ∈ n+, ξ− ∈ n− and ξ0 ∈ h. Consequently, for every x in an open Zarisky
of G there exists a unique Gauss factorization
x = x−x0x+, x+ ∈ N+, x− ∈ N−, x0 ∈ H.
For any x ∈ G0 and a fundamental weight ωi define
Δi(x) = x0
this function can be extended to a regular function on the whole G. In particular,
for G = SLr+1, Δi(x) is just the principal i × i minor of a matrix x. For any pair
u, v ∈ W , the corresponding generalized minor is a regular function on G given by
(1.11) Δuωi,vωi (x) =
These functions depend only on the weights uωi and vωi, and do not depend on
the particular choice of u and v.
1.2.4. The Bruhat decompositions of G with respect to B+ and B− are defined,
G = ∪u∈W B+uB+, G = ∪v∈W B−vB−.
The sets B+uB+ (resp. B−vB−) are disjoint. They are called Bruhat cells (resp.
opposite Bruhat cells). The generalized flag variety is defined as a homogeneous
G/B+ = ∪u∈W B+u.
Clearly, this is a generalization of the complete flag variety in a finite dimensional
Finally, for any u, v ∈ W , the double Bruhat cell is defined as
= B+uB+ ∩ B−vB−,
and the reduced double Bruhat cell, as
= N+uN+ ∩ B−vB−.
Double Bruhat cells and reduced double Bruhat cells will be the subject of our
discussion in the next chapter. Here we will only mention that the variety
biregularly isomorphic to a Zariski open subset of
where r = rank G.
A corresponding birational map from
can be constructed quite
explicitly, though not in a unique way.
Namely, let j = (j1,...,jl(u)+l(v)+r) be a shuffle of a reduced word for (u, v)
and any re-arrangement of numbers i, . . . , ir with
= −1; we define θ(l) = + if