[NOTE: This behavior has been improved in After Effects CS5.]
When you export a FLV file with transparency/alpha from After Effects, the default setting in the Render Queue's Output Module sets the Alpha so it's premultiplied (matted) with the Composition's background color. However, the FLV format needs the alpha channel to be always premultiplied with black. As a result, the Flash authoring application can't handle the transparency information correctly, if the background color in the After Effects Composition is not black.
To avoid this behavior, set the "Color" pop-up in the Output Module to "Straight (unmatted)" before exporting to the FLV format.
Technically, the FLV encoder will get a straight alpha from After Effects and convert it so it's premultiplied and matted with black.
For more information on alpha channel types, see Alpha channel interpretation: premultiplied or straight in After Effects Help.
There's some more information about this issue in this blog post:
Europe, Middle East and Africa