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Kostant cohomology for the trivial sl3 module

Example

Assume the Axiom of Choice, inherited from Kostant's theorem (The Axiom of Choice). Let g=sl3(C), λ=0 and V=C the trivial module. Here W≅S3 has elements 1,s1,s2,s1s2,s2s1,w0 of lengths 0,1,1,2,2,3, and the dot action is w⋅0=wρ−ρ. Kostant's theorem therefore gives Hk(n+,C)=⨁ℓ(w)=kCw⋅0, of dimensions 1,2,2,1 in degrees k=0,1,2,3 with pairwise distinct weights 0,−α1,−α2,−2α1−α2,−α1−2α2,−2α1−2α2; in particular dim⁡H1(n+,C)=2, one class for each simple reflection, and the total dimension is ∣W∣=6. This is the first rank-two check of the multiplicity-free formula, and it shows that a trivial coefficient module does not force all cohomology weights to vanish.

Verification

Given: The Axiom of Choice; g=sl3(C) with its diagonal Cartan subalgebra, positive system Φ+={α1,α2,α1+α2}, simple reflections s1,s2, longest element w0, and the trivial module C.

[L1] The root system of sl3 is type A2, with three positive roots and base {α1,α2}; the Weyl group is generated by s1,s2 subject to s12=s22=1 and s1s2s1=s2s1s2, so W={1,s1,s2,s1s2,s2s1,w0} with lengths 0,1,1,2,2,3 and w0=s1s2s1 (Rank-two root-system classification, Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, Root reflections and the Weyl group action, Length and longest Weyl-group element, Weyl length equals inversion number).

[L2] ρ=12(α1+α2+(α1+α2))=α1+α2, and λ+ρ=ρ is regular; the dot action is w⋅0=wρ−ρ (The Weyl vector rho for a chosen positive system, Integral, dominant, and strictly dominant weights).

[L3] Hk(n+,L(0))=⨁ℓ(w)=kCw⋅0 for every k, and the weights w⋅0 for distinct w are distinct (Kostant's nilradical cohomology theorem, Weyl length equals inversion number).

1.1L1L2

With s1ρ=ρ−α1 and s2ρ=ρ−α2, direct evaluation gives s1⋅0=−α1, s2⋅0=−α2, s1s2⋅0=s1(ρ−α2)−ρ=s1(α1)−ρ=−2α1−α2, s2s1⋅0=−2α2−α1, and w0⋅0=−ρ−ρ=−2ρ=−2α1−2α2, while 1⋅0=0.

2.1L1L3step 1.1

By [L3] the k-th cohomology is the sum of one line for each Weyl element of length k, so its dimension is the number of such elements and its weights are precisely those computed in step 1.1 for the length-k elements. For k>3 the sum is empty, so Hk(n+,C)=0.

3.1L1step 2.1∎

Counting the elements listed in [L1] gives dimensions 1,2,2,1 in degrees 0,1,2,3 and total dimension 6=∣W∣; in degree one the two classes have the distinct weights −α1,−α2 attached to the two simple roots, and both are nonzero because α1,α2 are linearly independent.

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