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

dense subset

G0

= N−HN+

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

ωi

;

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) =

Δi(u−1xv).

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,

resp., by

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

space

G/B+ = ∪u∈W B+u.

Clearly, this is a generalization of the complete flag variety in a finite dimensional

vector space.

Finally, for any u, v ∈ W , the double Bruhat cell is defined as

Gu,v

= B+uB+ ∩ B−vB−,

and the reduced double Bruhat cell, as

Lu,v

= 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

Gu,v

is

biregularly isomorphic to a Zariski open subset of

Cr+l(u)+l(v),

where r = rank G.

A corresponding birational map from

Cr+l(u)+l(v)

to

Gu,v

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

i2

= −1; we define θ(l) = + if