Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
Equivariant Embeddings of Algebraic Groups
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P ⊃T P • (σ ∩ ∆ P (T )) is a simplicial decomposition <strong>of</strong> Γ(G × T T σ , [e, e]) ⊂ X ∗ (G). We remark<br />
that this is a finite simplicial decomposition, for there are only finitely many parabolic subgroups<br />
<strong>of</strong> G containing T . (The Borel subgroups <strong>of</strong> G that contain T are all conjugate to a fixed one<br />
by elements <strong>of</strong> the Weyl group, W (T, G), which is finite, and the set <strong>of</strong> parabolic subgroups <strong>of</strong> G<br />
containing a given Borel subgroup B ⊃ T are indexed by subsets <strong>of</strong> the set <strong>of</strong> simple roots ∆(T, B)<br />
relative to B.)<br />
For instance, consider the GL 2 -embedding GL 2 × D 2<br />
A 2 . Here σ = {γ m,n (t) = ( t m 0<br />
0 t n )<br />
: m, n ∈<br />
N 0 } ⊂ X ∗ (D 2 ). There are only three parabolic subgroups <strong>of</strong> GL 2 containing D 2 , namely B + =<br />
{ ( )<br />
a b<br />
0 d }, B − = { ( )<br />
a 0<br />
c d } and GL2 itself. Then<br />
∆ B +(D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m ≥ n},<br />
∆ B −(D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m ≤ n},<br />
∆ GL2 (D 2 ) = {γ m,n ∈ X ∗ (D 2 ) : m = n},<br />
so σ ∩ ∆ B +(D 2 ) = {γ m,n : m ≥ n ≥ 0}, σ ∩ ∆ B −(D 2 ) = {γ m,n : n ≥ m ≥ 0} and σ ∩ ∆ GL2 (D 2 ) =<br />
{γ m,n : m = n ≥ 0}. Now<br />
B + • (σ ∩ ∆ B +(D 2 )) = {t ↦→<br />
(<br />
B − • (σ ∩ ∆ B −(D 2 )) = {t ↦→<br />
(<br />
)<br />
t m bt n (1−t m−n )<br />
0 t n<br />
)<br />
t m 0<br />
ct m (1−t n−m ) t n<br />
GL 2 • (σ ∩ ∆ GL2 (D 2 )) = {t ↦→ ( t m 0<br />
0 t n )<br />
: m = n ≥ 0},<br />
: m ≥ n ≥ 0},<br />
: n ≥ m ≥ 0},<br />
with limits<br />
⎧<br />
[( 1 b<br />
0 1<br />
[( 1 b<br />
0 1<br />
)<br />
, (0, 0)<br />
]<br />
)<br />
, (0, 1)<br />
]<br />
m > n > 0<br />
m > n = 0<br />
lim [(p • γ m,n(t)) · [e, (1, 1)]] =<br />
t→0<br />
⎪⎨<br />
⎪⎩<br />
[( 1 c 0<br />
1<br />
) , (0, 0)] n > m > 0<br />
[( 1 c 0<br />
1<br />
) , (1, 0)] n > m = 0<br />
[( 1 0 0<br />
1<br />
) , (0, 0)] m = n > 0<br />
[( 1 0 0<br />
1<br />
) , (1, 1)] m = n = 0<br />
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