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Some generalizations of the Weil representations of the symplectic and unitary groups. (English) Zbl 0877.20030
The authors give several characterizations of irreducible Weil representations of the symplectic groups \(\text{Sp}_{2n}(q)\) (\(q\) odd) and the special unitary groups \(\text{SU}_n(q)\). The main goal is to complete the classification of complex irreducible representations of classical groups such that the number of distinct eigenvalues of a certain element of order \(p\) is less than \(p\). To this end the authors first develop some machinery that enables to proceed by induction on the rank of the group in question. This machinery includes characterizing the Weil representations in terms of the irreducible constituents of their restriction to related classical subgroups of smaller rank. It is proved that a nontrivial irreducible representation \(\Theta\) of \(\text{Sp}_{2n}(q)\) with \(q\) odd, \(n>1\) (resp. \(\text{SU}_n(q)\) with \(n>2\)) is a Weil representation whenever all the irreducible constituents of the restriction \(\Theta|Sp_{2n-2}(q)\) (resp. \(\text{SU}_{n-1}(q)\)) are either trivial or Weil representations of \(\text{Sp}_{2n-2}(q)\) (resp. \(\text{SU}_{n-1}(q)\)). Next the authors characterize the Weil representations of the groups above in terms of the degree of the minimal polynomial of certain group elements in certain representations.

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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