Let $X\_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these 
$$
\sum\_{i=0}^{2\dim X\_0}(-1)^i[IH^i(X)],
$$
regarded as a virtual representation of the Weil group $W(\overline{k}/k)$ of $k$ (i.e. the cyclic group in Gal generated by Frobenius), is independent of $\ell.$ Of course it is also self-dual (up to Tate twist), and (together with Gabber's purity) each summand is independent of $\ell.$ 


My question is: does this alternating sum have any "geometric" interpretation? Or, to what extent (and in which way) this Euler characteristic depends on the singular locus of $X?$


E.g. when $X$ is in addition smooth, this can be interpreted in terms of the number of $k$-rational points. Being independent of $\ell,$ it's not surprising to expect it to have some "motivic" meaning, e.g. in terms of the geometry. Certainly one could also expect something for the individual $IH^i(X),$ but it seems that the Euler characteristic is always easier to relate to geometry. 


Answers/comments in the general setting would be great, and those addressing to Shimura varieties and their Satake compactifications are especially appreciated.



"geometric" interpretation of the alternating sum of intersection cohomology groups 
It's a weighted sum over the $k$-rational points, where each point is weighted by the trace of Frobenius on its local intersection cohomology. This follows instantly from the Grothendieck trace formula. I'm not sure there's really a better description of it beyond that in general. Of course, what you can say about the local intersection cohomologies depends a lot on your setting.


