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Consider the representation correspondence for a reductive dual pair (G 1 , G 2 ) over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of «first occurrence», the representation correspondence takes a cuspidal unipotent representation of G 1 to one of G 2 . This should serve as a fundamental case in studying the correspondence in general over both finite and local fields

Unipotent representations and reductive dual pairs over finite fields