How is the "special aptitude for in-depth independent scientific work" proven if the professor candidate does not have a doctorate? My PhD is almost finished (writing last paper of the cumulative dissertation) and I have applied for a full W2 professorship at a German university of applied sciences .

I have been invited by the appointment committee to give a lecture. However, I was informed that I would only be shortlisted if I can demonstrate a “special aptitude for in-depth independent scientific work” (German: “besondere Befähigung zu vertiefter selbstständiger wissenschaftlicher Arbeit”). It is required by all German higher education laws (e.g. §25 NHG).
The request was made by HR relayed through the committee as far as I can tell.
(I am a German citizen, so this is not visa-related.)

How do I show this without a doctorate? My list of publications is comparatively extensive for a doctoral candidate.
With your PhD thesis close to handing in, and in particular a cumulative PhD thesis together with "list of publications is comparatively extensive for a doctoral candidate" they may say that you have already shown the qualification, and that they expect you to finish before starting the professorship.
I don't know about FHs (universities of applied science), but at universities the procedure will usually take many months, so I'd say this expectation can be quite realistic. 
More generally speaking, FHs (and technical/applied departments at universities) want to have professors with industry experience. In that context, experience of e.g. many years as a group leader in industrial R&amp;D that did not end up in a formal thesis can easily be at least equivalent to the amount of research required for a PhD thesis or a habilitation. 
Again, in these fields it is maybe more common than in more theoretical fields that the decision of doing a PhD thesis is not made directly afer Diplom/Master but after a professional career is well established and the candidate finds themselve doing &amp; liking research.
Last and hopefully least, there is the rather unpleasant possibility that the application procedure is rigged up to comply with formal procedural requirements but make sure an already known favorite candidate gets the call: in that case, "clearly impossible" candidates may be encouraged to apply rather than telling them that their application is hopeless and they shouldn't waste their time.


That being said, I don't see particular red flags in the described situation: as OP does not yet fulfil the criterion that would automatically put a check-mark to the aptitude for independent scientific work (doctorate) saying they need to show it in some other manner is probably the normal formal way of HR burocracy.

Recommendation: talk to the committee/HR, saying that as you have not yet finished your PhD (but are close to handing in) and seeing that whether you should list your publications? Should say that the PhD certificate will be handed in once you have it? Or something else?  

I don't know for sure, but I almost expect that the committee will have to certify that you have the required aptitude rather than you declaring anything.