3. Hochschild-Serre spectral sequence for SL(3, ℝ)

The first big chart here is the E_{1} term of the Hochschild-Serre spectral sequence for computing the 𝔫-cohomology of a Harish-Chandra module X for SL(3, ℝ). Each column shows the contribution of one representation of K = SO(3); the numbers are the weights of the representation of T_{1} = SO(2). For example, the column “5-diml” says that each copy of the 5-dimensional representation of SO(3) occurring in X contributes a three-dimensional piece to the degree one part of E_{1}; the corresponding weights of T_{1} are 1, 0, and −3.

degree  1-diml  3-diml  5-diml  7-diml  9-diml  11-diml
0  0  1  2  3  4  5
1  −1, −2  −1  0, −1  −2  1, 0  −3  2, 1  −4  3, 2  −5  4, 3  −6
2  −3  −2, −3 −2 −3, −4 −1 −4, −5 0 −5, −6 1 −6, −7 2 −7, −8
3  −4  −5  −6  −7  −8  −9

Suppose we concentrate for a moment on the −3 weight space (which has one dimension from each copy of the 5-dimensional representation of K). This weight space might fail to survive to E_{∞} in two ways. First, it could be in the image of a differential from degree 0. But we see from the table (extrapolated infinitely to the right) that the weight −3 does not appear in degree 0; so this is impossible. Second, our weight space could have a non-zero image in the degree 2. By inspection of the degree 2 row of the table, we see that the weight −3 appears there only if X has a copy of the SO(3) representation of dimension 1 or the SO(3) representation of dimension 3.

So inspection of this spectral sequence leads to the following conclusion.

Proposition 1 Suppose X is a Harish-Chandra module for SL(3, ℝ), and that the SO(3) representations of dimensions 1 and 3 do not occur in X. Then the T_{1} weight −3 appears in H^{•}(𝔫, X) only in degree 1, where it has multiplicity equal to the multiplicity of the 5-dimensional representation of SO(3) in X.

An identical argument proves

Proposition 2 Suppose that 2m+1 ≥ 7, and that the SO(3) representations of dimensions 2m−1, 2m−3, and 2m − 5 do not occur in X. Then the T_{1} weight −m − 1 appears in H^{•}(𝔫, X) only in degree 1, where it has multiplicity equal to the multiplicity of the 2m + 1-dimensional representation of SO(3) in X.

How was this spectral sequence constructed? Recall that the E_{1} term in general is built from pieces ⋀^{p}(𝔫∩𝔰)^{∗}⊗H^{q}(𝔫_{𝔨}, X), and that the degree is p+q. The top entries in the two columns for each representation of K are the weights in H^{0}(𝔫_{𝔨}, •) and H^{1}(𝔫_{𝔨}, •) for the indicated representation of K. (Those are provided by the Bott-Kostant theorem.) The rest of each column comes from adding the weights of ⋀^{•}(𝔫 ∩ 𝔰)^{∗}, which are tabulated below.

\begin{matrix}degree & ⋀^{•}(𝔫 ∩ 𝔰)^{∗} \\ 0 & 0 \\ 1 & −1, −2 \\ 2 & −3 \end{matrix}