The stable cohomology of SL(F_p)

Persiflage 2019-08-20

Back by popular demand: an actual mathematics post!

Today’s problem is the following: compute the cohomology of $\mathrm{SL}(\mathbf{F}_p)$ for a (mod-p) algebraic representation.

Step 0 is to say what this problem actually is. It makes sense to talk about certain algebraic representations of $\mathrm{SL}_n(\mathbf{F}_p)$ as n varies (for example, the standard representation or the adjoint representation, etc.). For such representations, one can prove stability phenomena for the corresponding cohomology groups. But my question is whether one can actually compute these groups concretely.

The simplest case is the representation $\mathbf{L} = \mathbf{F}_p$ and here one has a complete answer: these cohomology groups are all zero in higher degree, a computation first done by Quillen and which is closely related to the fact that the $K_n(\mathbf{F}_p) \otimes \mathbf{F}_p = 0.$

Most of the references I have found for cohomology computations of special linear groups in their natural characteristic consider the case were p is very large compared to n, but let me remind the reader that we are exactly the opposite situation. One of the few references is a paper of Evens and Friedlander from the ’80s which computes some very special cases in order to compute $K_3(\mathbf{Z}/p^2 \mathbf{Z}).$

Note, however, that p should still be thought of as “large” compared to the partition which defines the corresponding stable local system(s).

In order to get started, let us make the following assumptions:

ANZATZ: There exists a space X with a $\mathrm{SL}(\mathbf{Z}_p)$ pro-cover such that:

1. The corresponding completed cohomology groups with $\mathbf{F}_p$ coefficients are $\mathbf{F}_p$ for i = 0 and vanish otherwise. 2. If $\mathbf{L}$ is the mod-p reduction of (an appropriately chosen) lattice in a (added non-trivial irreducible) algebraic representation of $\mathrm{SL}(\mathbf{Z}_p),$ then $H^i(X,\mathbf{L}) = 0$ for i small enough compared to the weight of $\mathbf{L}.$

Some version of this is provable in some situations and it may be generally true, but let us ignore this for now. (One explicit example is given by the locally symmetric space for $\mathbf{SL}(\mathbf{Z}[\sqrt{-2}])$ and taking the cover corresponding to a prime $\mathfrak{p}$ of norm p satisfying certain global conditions.) The point is, this anzatz allows us to start making computations. From the first assumption, one deduces by Lazard that

$H^i(X(p),\mathbf{F}_p) = \wedge^i M,$

where M is the adjoint representation. But now one has a Hochschild-Serre spectral sequence:

$H^i(\mathrm{SL}(\mathbf{F}_p), \mathbf{L} \otimes \wedge^j M) \Rightarrow 0.$

The point is now that one can now start to unwind this (even knowing nothing about the differentials) and make some conclusions, for example:

1. $H^1(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = 0.$ 2. $H^2(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = H^0(\mathrm{SL}(\mathbf{F}_p),\mathbf{L} \otimes M).$

In particular, the first cohomology always vanishes, and the second cohomology is non-zero only for the adjoint representation where it is one dimensional. (One can see the non-trivial class in H^2 in this case coming from the failure of the tautological representation to lift mod p^2.) Note of course I am not claiming that the first cohomology vanishes for all representations, but only the “algebraic” ones, and even then with p large enough (compared to the weight). Note also that one has to be careful about the choice of lattices, but that is somehow built into the stability — for n fixed, the dual of M is given by trace zero matrices in $M_n(\mathbf{F}_p)$ and so (from the cohomology side) “ $\mathbf{L} = M$ ” is the correct object to consider rather than its dual since the dual is not stable even in degree zero. But I think you can secretly imagine that p is big enough and the weight small enough so that you can choose n so that all these representations are actually irreducible).

The first question is whether 1 & 2 are known results — I couldn’t find much literature on these sort of questions (they are certainly consistent with the very special cases considered by Evens and Friedlander).

The second question is what about degrees bigger than 2? For H^3 things start getting a little murkier, but it seems possible that H^3 always vanishes. Beyond that (well even before that) I am just guessing. But one might hope to even come up with a guess the the answer which is consistent with the spectral sequence above.

Added: Some of the discussions in the comments below contain some minor inaccuracies, but in back and forth conversations with Will via email he has formulated a pretty convincing conjectural answer to my questions (and also my secret unasked questions). Hopefully I will come back to this post later when these are all proved!