What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?

Question

As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into it a while back while trying to answer some questions about the Fourier algebras of compact Lie groups.) The solution could just be to point me to the right bit of the right literature.

In more detail: using R to denotes the reals, regard the special orthogonal group SO(n,R) as an affine variety in M_n(R)=R^{n^2}; and let A be the coordinate ring of this variety, regarded as an R-algebra (I think this is the correct terminology; I just mean the algebra of R-valued functions on the group that is generated by the coordinate functions). I would like to know the largest d for which H_d^alt(A,A) is non-zero, where H_d is Hochschild homology of A with coefficients in itself, and "alt" is the part generated by alternating a.k.a. antisymmetric cycles.

The vague idea I had was that said d should be equal to the dimension of the tangent space (as defined via maximal ideals in A) at a point of the original variety, and hence to the dimension of SO(n,R) as a real manifold. I've tried to do some hunting in the literature, but much of what I come across is for general affine/projective varieties, and so is overly general and not exactly self-contained.

Is there a general picture telling us that the Hochschild homology of A can be calculated in terms of the Lie algebra cohomology of so(n,R)? If so, does this work for compact forms of other semisimple Lie algebras?

Edit: it's been pointed out that, if we know A is a smooth ring in the sense of commutative algebra, then one can wheel in the Hochschild-Kostant-Rosenberg theorem. I had avoided mentioning this, because although I'm sure A should be smooth in this sense, one of my problems is trying to find an explicit reference to this putative fact. It also feels (since the variety SO(n,R) has such good global symmetry) as if there should be a proof which doesn't rely on proving smoothness of A.

Final edit (for now): Having gone and done some reading in a few textbooks, it look s like the simplest - if not the most elementary solution is to "observe" or cite the fact that SO(n,R) is a smooth variety over R, use localization to reduce the computation to Hochschild homology of the local ring at a point, and then use HKR or something similar. This still feels like overkill to me, but at least it looks to give a well-defined proof. (If you can see a better route, please feel free to add a new comment!)

Answer 1

I am very far from an expert on the subject, but I think the Hochschild homology of the coordinate ring should be the algebraic de Rham complex of your variety SO(n, R)--not the cohomology of the complex, just the groups in the complex (with 0 differential if you like). This is the Hochschild-Kostant-Rosenberg theorem and should just require smoothness, not the fact that you have a Lie group.

I don't know what HH^{alt} is, but maybe you can figure it out from this description. Unless I am very confused about something, HH_k is nonzero exactly for 0 <= k <= the dimension of SO(n, R), so your guess about d sounds plausible.

Answer 2

First some general references on homology of lie groups:

• Cohomology Theory of Lie Groups and Lie Algebras Claude Chevalley and Samuel Eilenberg Transactions of the American Mathematical Society, Vol. 63, No. 1 (Jan., 1948), pp. 85-124
• Borel, Armand Homology and cohomology of compact connected Lie groups. Proc. Nat. Acad. Sci. U. S. A. 39, (1953). 1142--1146.
• Bott, R. On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups. Advances in Math. 11 (1973), 289--303.
• Dwyer, W. G.; Wilkerson, C. W. Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. of Math. (2) 139 (1994), no. 2, 395--442.

As far as I remember in the SO(n,R) case the easiest thing to is to write the spectral sequence of the fiber space:

SO(n-1)->SO(n,R)->S^n