I’m looking for a way to cast a shadow without a visible game object.
This is about a horror game where invisible or partially visible things cast shadows. I’m using lots of point lights, so projectors or blob shadows won’t do it, though I don’t need precise outlines, a blurry shadow would be just as good.
Unfortunately, neither particle systems nor transparent materials cast shadows at all, and cutting the object away using camera culling also removes the object shadow. 
Any ideas?
You could create a script, attached to your main camera, that would have OnPreRender and OnPostRender methods. It has a reference to your invisible shadowcaster, and disable then re-enable the renderer before and after the camera renders it. If I understand the rendering system correctly the lights should still render their shadowmaps outside of this time period, so voila, shadows but no object.
You could probably also write a shader that renders only to shadow maps, but I haven’t the slightest idea how you’d go about that.
A shader sounds like the best approach - I’ll be having a variable number of those invisible shadow-casters.
For a start, though, a particle material that casts shadows would be a great compromise. I’ll as in shaderlab.