Krashen the party

I'm at UGA for the week, in between SWAG and TAAAG.  Today Danny Krashen gave a great talk on this paper of Auel, First, and Williams.  The paper is one of the latest in a long tradition of papers which construct counterexamples by making a topological computation and then approximating the relevant topological spaces by algebraic varieties.  (To my knowledge, this technique began with Totaro, but it has been exploited to great effect by Antieau, Williams, and most recently, fellow Ravi student Arnav Tripathy.)

Auel, First, and Williams construct an example of a variety \(X\) and a class \(\alpha\in \text{Br}(X)[2]\) of index \(4\), i.e. it is in the image of the map $$H^1(X, PGL_4)\to H^1(X, \mathbb{G}_m),$$ such that any \(PGL_n\)-torsor of class \(\alpha\) which lifts to a \(PO_n\)-torsor satisfies \(8\mid n\).  This shows that a result of Knus, Parimala, and Srinivas is sharp.  Their construction is a certain Azumaya algebra on (an algebraic approximation of) \(BSL_4/\mu_2\).

I want to try to explain a point of view on where these sorts of examples come from which is kind of different from the one in the paper (which involves virtuosic group cohomology computations).  I'm not claiming that I can give a rigorous proof of their result, but at least it's plausible that one can make these sorts of arguments rigorous.  And I think they provide some motivation for looking at \(BSL_4/\mu_2\).  I haven't carefully checked several of the statements I'm making below, so please correct me if I say something wrong.

Generalities on \(PO_n\)-torsors

I like to think of \(PGL_n\)-torsors as twisted sheaves "up to tensoring by line bundles"; for a gentle introduction to this language, see for example Section 2 of this paper of mine.  In this language, what does it mean for a \(PGL_n\)-torsor to lift to a \(PO_n\)-torsor?  I claim that if we think of our \(PGL_n\)-torsor as a twisted sheaf \(\mathscr{E}\), this is the same as the existence of an isomorphism $$\phi: \mathscr{E}\to \mathscr{E}^\vee\otimes \mathscr{L}$$ for some line bundle \(\mathscr{L}\), where \(\phi\) satisfies a symmetry condition---namely, $$\phi=\phi^\vee\otimes \mathscr{L}.$$  (Please let me know if I've screwed this up.)  Note that this immediately implies that \(\alpha\) is two-torsion, since it implies \(\mathscr{E}\) and \(\mathscr{E}^\vee\) have the same Brauer class; but these classes are additive inverses to one another.

Twisted Sheaves on \(BSL_4/\mu_2\)

OK, so what are the sheaves/twisted sheaves/Brauer classes, etc. on \(BG\), where \(G\) is a group (and we are working over an algebraically closed field)?  For convenience, let's assume that \(G\) is perfect with (finite) fundamental group \(\pi_1(G)\), and let \(\tilde G\) be its universal central extension, so there is a short exact sequence $$1\to \pi_1(G)\to {\tilde G}\to G\to 1.$$  In the case that \(G=SL_4/\mu_2\), its universal central extension is just \(SL_4\).  Then vector bundles on \(BG\) are just representations of \(G\); twisted sheaves are representations of \(\tilde G\) sending \(\pi_1(G)\) to the center of \(GL_n\).  The Brauer group is $$\text{Br}(BG)=\text{Hom}(\pi_1(G), \mathbb{G}_m),$$ and the Brauer class of a representation \(\rho: \tilde G\to GL_n\) is $$\rho|_{\pi_1(G)}: \pi_1(G)\to \mathbb{G}_m=Z(GL_n).$$

So a representation \(\rho: SL_4\to GL_n\) factors through \(SL_4/\mu_2\) if and only if it kills \(\mu_2\), in which case it gives rise to a vector bundle on \(BSL_4/\mu_2\); if it does not, it gives rise to a twisted sheaf on \(BSL_4/\mu_2\).  By our comments above $$\text{Br}(SL_4/\mu_2)=\text{Hom}(\mu_2, \mathbb{G}_m)=\mathbb{Z}/2\mathbb{Z}.$$  Now \(SL_4\) and \(SL_4/\mu_2\) have no non-trivial characters, so there are no line bundles on the classifying space of either group.  Thus the condition that a twisted vector bundle \(\mathscr{E}\) (resp. a vector bundle) lift to a \(PO_n\)-torsor (resp. \(O_n\)-torsor) is the same as the existence of an orthogonally self-dual isomorphism $$\phi: \mathscr{E}\to \mathscr{E}^\vee.$$  Thus all we need to show to "prove" the Auel-First-Williams result is: \((*)\)that any orthogonally self-dual representation of \(SL_4\) such that the central \(\mu_2\)  acts centrally and non-trivially has dimension divisible by \(8\).  

Before remarking on this, let me just observe that the standard representation \(V\) of \(SL_4\) gives a twisted sheaf on \(BSL_4/\mu_2\) which is not self-dual (its dual is \(\bigwedge^3 V\)).  But the \(8\)-dimensional representation \(V\oplus V^\vee\) is of course self-dual.  

Now I haven't tried to prove \((*)\), but I imagine one can do it via the Weyl dimension formula...  

What's Missing?

The main issues in turning this into a proof of the Auel-First-Williams result is to (1) complete the proof of \((*)\), and (2) relate vector bundles/twisted sheaves on algebraic approximations of classifying spaces to vector bundles/twisted sheaves on the classifying spaces themselves.  It is not obvious to me how to do (2), but maybe one can make K-theoretic versions of the arguments above?  This is too much fun to just give up, right?