Justice Helen McEntee estimates that more than 80 percent of people applying for asylum in Ireland are coming from Britain
Police shot the man dead after elite forces stormed the complex, the Mannheimer Morgen newspaper reported, citing the interior minister of Hesse state on Thursday.
The Darmstaedter Echo said 20 to 50 people had been wounded.
Several media reported that the man had entered the cinema complex at around 3pm local time (1500 GMT).
The Frankfurter Allgemeine said it was unclear if there was a political or ideological motive or whether it was a robbery that went wrong.