March 29

Let W be a finite Coxeter group with generators S. Recall that for J ⊂ S, we denote W_{J} = ⟨J⟩ ≤ W.

Proposition 1 Let I, J ⊂ S. Then W_{I} ∩ W_{J} = W_{I∩J}.

Proposition Definition 1 Let I, J ⊂ S. Then every double coset W_{I}wW_{J} contains a unique element x of minimal length. Moreover, every element of W_{I}xW_{J} has an expression uxv with u ∈ W_{I}, v ∈ W_{J} and

ℓ (uxv) = ℓ (u) + ℓ (x) + ℓ (v) .

x is called the distinguished double coset representative of W_{I}xW_{J}. The set of all DDCRs of W_{I}\W/W_{J} is denoted D_{I,J}.

Observation 1 To consider intersections of parabolic subgroups, that is, intersections of the form ^{y}W_{I} ∩ ^{z}W_{J}, it suffices to consider intersecions of the form W_{I} ∩ ^{w}W_{J} since

yW_{I} ∩ ^{z}W_{J} = ^{y} (W_{I} ∩ ^{y^{−1}z}W_{J}) .

Moreover, if x is a DDCR for W_{I} ∩ ^{w}W_{J}, then we have w = uxv for u ∈ W_{I} and v ∈ W_{J} so that

W_{I} ∩ ^{w}W_{J} = W_{I} ∩ ^{uxv}W_{J} = ^{u} (W_{I} ∩ ^{x}W_{J}) .

Thus, it suffices to look at W_{I} ∩ ^{x}W_{J} for x ∈ D_{I,J}.

Theorem 1 (Kilmoyer, Solomon) Let x ∈ D_{I,J} be fixed for fixed subsets I, J ⊂ S. Let K = I ∩ ^{x}J. Then

W_{I} ∩ ^{x}W_{J} = W_{K}.

Proposition 2 Let J ⊂ S. For all subsets J′ ⊂ J, P_{J′} ∩ L_{J} is a standard parabolic subgroup of L_{J} with Levi decomposition

P_{J′} ∩ L_{J} = L_{J′} (V_{J′} ∩ L_{J}) .

The map P_{J′} → P_{J′} ∩ L_{J} gives a bijection from the set of standard parabolic subgroups {P_{J′} : J′ ⊂ J} to the set of all standard parabolic subgroups of L_{J}.