I get download results on the first page, same on the second and third, and I also get a tutorial on installing it (defunct), but none mentions AIR_HOME, and I can't infer from any other documentation what was the intent behind that name. If I google for AIR_HOME, I get results related to air conditioning...
But It appears that either Ant doesn't do path expansion properly, or it is some fault in the script itself, because if I call it like this: $ env PLAYERGLOBAL_HOME=~/flex/SDK/4.6/frameworks/libs/player/11.1/playerglobal.swc AIR_HOME=../../4.6/airhome/ ant then it "forgets" the ../../ in front of the 4.6.