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
).
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