1.2. SIMPLE LIE ALGEBRAS AND GROUPS 7

jl 0, θ(l) = − if jl 0 and θ(l) = 0 if jl ∈ iZ. Besides, for i ∈ [1,r] and t ∈ C,

put

xi

0(t)

= ρi(diag(t,

t−1)).

Then one defines a map xj: Cl(u)+l(v)+r → Gu,v by

(1.12) xj(t) =

l(u)+l(v)+r

l=1

x|j(ll|)(tl),θ

where elements xi

±(t)

are defined as in (1.8). This map is a biregular isomor-

phism between

(C∗)l(u)+l(v)+r

and a Zariski open subset of

Gu,v.

Parameters

t1,...,tl(u)+l(v)+r constituting t are called factorization parameters. Note that set-

ting tl = 1 in the definition of xj for all l such that θ(l) = 0, one gets a parametriza-

tion for a Zariski open set in

Lu,v,

and so

Lu,v

is biregularly isomorphic to a Zariski

open subset in

Cl(u)+l(v).

In this case, we replace the factorization map above by

(1.13) xj(t) =

l(u)+l(v)

l=1

x|j(ll|)(tl),θ

where now j is a reduced word for (u, v) and t = (t1,...,tl(u)+l(v)). We retain no-

tations j and t, since it will be always clear from context which of the factorizations

(1.12), (1.13) is being used.

Remark 1.1. It will be sometimes convenient to interpret θ as taking values

±1 instead of ±. The proper interpretation will be clear from the context.

It is possible to find explicit formulae for the inverse of the map (1.12) in terms

of the so-called twisted generalized minors. We will not review them here, but

instead conclude the subsection with an example.

Example 1.2. Let G = SLn and (u, v) = (w0,w0), where w0 is the longest

element of the Weyl group Sn. Then

Gw0,w0

is open and dense in SLn and consists

of elements such that all minors formed by several first rows and last columns as well

as all minors formed by several last rows and first columns are nonzero. A generic

element of

Gw0,w0

has nonzero leading principal minors, and by right multiplying

by an invertible diagonal matrix

D−1

can be reduced to a matrix A ∈

Gw0,w0

with

all leading principal minors equal to 1. Subtracting a multiple of the (n − 1)st row

of A from its nth row we can ensure that the (n, 1)-entry of the resulting matrix

is 0, while (generically) all other entries are not. Applying similar elementary

transformations to rows n − 2 and n − 1, then n − 3 and n − 2, and so on up to

1 and 2, we obtain a matrix with all off-diagonal entries in the first column equal

to 0. Starting from the bottom row again and proceeding in a similar manner, we

will eventually reduce A to a unipotent upper triangular matrix B. Applying (in

the same order) the corresponding column transformations to B, we will eventually

reduce it to the identity matrix. If one now translates operations above into the

matrix multiplication, they can be summarized as a factorization of A into

A = (1 + t1en,n−1) · · · (1 + tn−1e2,1)(1 + tnen,n−1) · · · (1 + t2n−3e3,2) · · ·

(1 + t

n(n−1)

2

en,n−1)(1 + t

n(n−1)

2

+1

en−1,n) · · · (1 + tn(n−1)en−1,n).

Furthermore, denote m = n(n − 1) and factor D as

D =

diag(tm+1,tm1 −

+1

, 1,..., 1) · · · diag(1,...,

1,tm+n−1,tm1 −

+n−1

).