The affine flag variety

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

Last updates: 22 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

Intersections U – v I ∩ I w I

Let

U – = ⟨ x – α ( f ) ∣ f ∈ 𝔽 , α ∈ R + ⟩
so that the root subgroup 𝔛 α + k δ ⊆ U – exactly when α + k δ ∈ R – + ℤ δ . The periodic orientation is the orientation of the hyperplanes H α + k δ such that

Then

v ( α + k δ ) ∈ R – + ℤ δ if and only if the orientation of H α + k δ is PICTURE ( v is on the negative side of H α + k δ , v s α+kδ is on the positive side.

We shall use the identity

x α ( f ) n α – 1 = x – α ( f ) x α ( – f – 1 ) h α ∨ ( f ) (main folding law)
to rewrite points of I w I as elements of U – v I . Suppose that
x i 1 ( c 1 ) n i 1 – 1 ⋯ x i ℓ ( c ℓ ) n i ℓ – 1 = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v b .
Then
x i 1 ( c 1 ) n i 1 – 1 ⋯ x i ℓ ( c ℓ ) n i ℓ – 1 x j ( c ) n j – 1 = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v b x j ( c ) n j – 1 x i 1 ( c 1 ) n i 1 – 1 ⋯ x i ℓ ( c ℓ ) n i ℓ – 1 x j ( c ) n j – 1 = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( c ∼ ) n j – 1 b ′ , since   b x j ( c ) n j – 1 ∈ I s j I .
If v α j ∈ R – + ℤ δ PICTURE ( v on negative side of hyperplane labeled j and v sj on positive side) then this is equal to
x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) x v α j ( c ∼ ) n v s j b ′ ∈ U – vI ∩ I w s j I .
If v α j ∉ R – + ℤ δ and c ∼ ≠ 0 , PICTURE ( v on positive side of hyperplane labeled j and v sj on negative side with a label c on the arrow) then
x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( c ∼ ) n j – 1 b ′ = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x – α j ( c ∼ – 1 ) x α j ( – c ∼ ) h α ∨ ( c ∼ ) b ′ x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( c ∼ ) n j – 1 b ′ = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x – α j ( c ∼ – 1 ) b ′ ′ x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( c ∼ ) n j – 1 b ′ = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) x γ ℓ + 1 ( c ∼ – 1 ) n v b ′ ′ ∈ U – v I ∩ I w s j I .
where γ ℓ + 1 = – v α j . So

PICTURE ( v on positive side of hyperplane labeled j and v sj on negative side with a label c on the arrow) becomes PICTURE (fold with ( v on positive side of hyperplane labeled j and a label c –1 on the arrow) .

If v α j ∉ R – + ℤ δ and c ∼ = 0 , PICTURE ( v on positive side of hyperplane labeled j and v sj on negative side with a label 0 on the arrow) then

x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( 0 ) n j – 1 b ′ = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x – α j ( 0 ) n j b ′ x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) n v x j ( c ∼ ) n j – 1 b ′ = x γ 1 ( c 1 ′ ) ⋯ x γ ℓ ( c ℓ ′ ) x γ ℓ + 1 ( 0 ) n v s j b ′ ∈ U – v s j I ∩ I w s j I ,
where γ ℓ + 1 = – v α j . So
PICTURE ( v on positive side of hyperplane labeled j and v sj on negative side with a label 0 on the arrow) becomes PICTURE ( v on positive side of hyperplane labeled j and v sj on negative side with a label 0 on the arrow) .

If w ∈ W ∼ and w → = s i 1 ⋯ s i ℓ is a minimal length walk to w define

𝒫 ( w → ) v = { labeled folded paths   p  of type   w → which end in   v } , for   v ∈ W ∼ .
The folding map gives a bijection
𝒫 ( w → ) v ⟷ U – v I ∩ I w I and G = ⊔ v ∈ W ∼ U – v I .
Let q = Card ( k ) . Then the characteristic functions { X v ∣ v ∈ W ∼ } of the double cosets U – v I are a basis of the right H ∼ - module C ( U – ∖ G / I ) and
X v T s j = { X v s j , if PICTURE   q X v s j + ( q – 1 ) X v , if PICTURE