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Dual reductive pairs in characteristic 2. (Paires duales réductives en caractéristique 2.) (French) Zbl 0778.11025
A. Weil [Acta Math. 111, 143–211 (1964; Zbl 0203.03305)] gave a very general construction of metaplectic groups based on the representation theory of the Heisenberg group associated with a bilinear form. When the characteristic of the underlying field is 2 the theory of quadratic forms is much more delicate. Here the author uses Dieudonné’s theory of quadratic forms over such a field. First of all one constructs the pseudosymplectic group which is an extension of an orthogonal group by the group of additive quadratic forms. Then the metaplectic group is a central extension of this group by \(\mathbb C^ \times\).
In this paper the author investigates this construction in detail and in particular she constructs and classifies all the reductive pairs in the pseudosymplectic group, that is pairs of subgroups \((G,G')\) which are each others centralizers. She also shows that the metaplectic extension splits over each dual reductive pair.

MSC:
11F27 Theta series; Weil representation; theta correspondences
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
20G25 Linear algebraic groups over local fields and their integers
20G40 Linear algebraic groups over finite fields
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