The group S L 2

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 29 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

Generators

The Weyl group is

W 0 = S 2 = { 1 , s 1 } with s 1 2 = 1 , and 𝔥 ℤ = ℤ α ∨ , with s 1 α ∨ = – α ∨ .

The Chevalley group G ( 𝔽 ) = S L 2 ( 𝔽 ) is generated by the elements

x α ( f ) = ( 1 f 0 1 ) , x –α ( f ) = ( 1 0 f 1 ) , and if h α∨ ( g ) = ( g 0 0 g–1 ) , n α = ( 0 1 –1 0 ) ,
then x α ( g ) x – α ( – g – 1 ) x α ( g ) = h α ∨ ( g ) n α . The loop group G = S L 2 ( ℂ ( ( t ) ) ) contains
x α + j δ ( c ) = ( 1 c t j 0 1 ) , and h r α ∨ = h r α ∨ ( t – 1 ) = ( t – r 0 0 t r )
and we take
x 1 ( c ) = x α ( c ) = ( 1 c 0 1 ) , n 1 = ( 0 1 – 1 0 ) , x 0 ( c ) = x – α + δ ( c ) = ( 1 0 c t 1 ) , n 0 = ( 0 – t – 1 t 0 )
and
K = S L 2 ( ℂ [ [ t ] ] ) and I = { ( f 1 1 f 1 2 t f 2 1 f 2 2 ) ∈ S L 2 ( ℂ [ [ t ] ] ) ∣ f 12 , f 21 ∈ ℂ [ [ t ] ] , f 11 , f 22 ∈ ℂ [ [ t ] ] × } .

The affine Weyl group

W = { t r α ∨ s 1 ∣ r ∈ ℤ } = { s 0 , s 1 ∣ s 0 s 0 2 = 1 , s 1 2 = 1 }
with
t r α ∨ = ( s 0 s 1 ) r , s 0 = t α ∨ s 1 , s 1 t r α ∨ s 1 = t – r α ∨ , for  r ∈ ℤ .
The bijection between W and alcoves is given by the picture
PICTURE
Using
n 1 – 1 x β ( c ) n 1 = x s 1 β ( – c ) and n 0 – 1 x β ( c ) n 0 = x s 0 β ( – c ) , for   β = ± α + r δ ,
analyze the point which is pictorially represented by
PICTURE
x 1 ( c 1 ) n 1 – 1 x 0 ( c 2 ) n 0 – 1 x 1 ( c 3 ) n 1 – 1 x 0 ( c 4 ) n 0 – 1 x 1 ( c 5 ) n 1 – 1 x 0 ( c 6 ) n 0 – 1 x 1 ( c 7 ) n 1 – 1 PICTURE = x α 1 ( c 1 ) x α + δ ( – c 2 ) x α + 2 δ ( c 3 ) ⋯ x α + 6 δ ( c 7 ) n 1 – 1 PICTURE = x α ( c 1 – c 2 t + c 3 t 2 – c 4 t 3 + c 5 t 4 – c 6 t 5 + c 7 t 6 ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 .
Note that x α ( c t k ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 = ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 x – α ( – c t k – 6 ) , and x – α ( – c t k – 6 ) ∈ I if k > 6 , so that
x α ( c 1 – c 2 t + c 3 t 2 – c 4 t 3 + c 5 t 4 – c 6 t 5 + c 7 t 6 ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 I PICTURE = x α ( c 1 – c 2 t + c 3 t 2 – c 4 t 3 + c 5 t 4 – c 6 t 5 + c 7 t 6 + t 7 p ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 I
for all p ∈ ℂ [ [ t ] ] .

Now suppose that c 1 = c 2 = 0 and c 3 ≠ 0 . Letting p = c 3 – c 4 t + c 5 t 2 – c 6 t 3 + c 7 t 4 the above expression is

x α ( p t 2 ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 = x α ( p t 2 ) ( n 1 – 1 n 0 – 1 ) 2 n 1 – 1 h α ∨ ( p – 1 ) h α ∨ ( p ) n 0 – 1 n 1 – 1 PICTURE = x α + 2 δ ( p ) x α + 2 δ ( – p ) x – α – 2 δ ( p – 1 ) x α + 2 δ ( – p ) h α ∨ ( p ) n 0 – 1 n 1 – 1 PICTURE = x – α – 2 δ ( p – 1 ) n 0 – 1 n 1 – 1 x α ( – p ) h α ∨ ( p )
where x α ( – p ) h α ∨ ( p ) ∈ I . Also, if p – 1 = c 3 – 1 + c 4 ′ t + c 5 ′ t 2 + ⋯ then
x – α – 2 δ ( p – 1 ) n 0 – 1 n 1 – 1 I = x – α ( c 3 – 1 t – 2 + c 4 ′ t – 1 + c 5 ′ + c 6 ′ t + c 7 ′ t 2 ) n 0 – 1 n 1 – 1 I
as all terms of degree greater than 2 can be absorbed into I . This shows that the point of G / I given by
x α ( p t 2 ) ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 I = n – α – 2 δ ( p – 1 ) n 0 – 1 n 1 – 1 I   ∈ U + ( n 1 – 1 n 0 – 1 ) 3 n 1 – 1 I ∩ U – n 0 – 1 n 1 – 1 I .
In pictures
PICTURE
is the same as the point
PICTURE
where,
if p = c 3 – c 4 t + c 5 t 2 – c 6 t 3 + c 7 t 4 , then c 3 – 1 + c 4 ′ t + c 5 ′ t 2 + c 6 ′ t 3 + c 7 ′ t 4 = p – 1   mod   t 5 ℂ [ [ t ] ] .