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Exceptional \(\Theta\)-correspondence. I. (English) Zbl 0878.22011
This paper takes up ideas developed in a paper by D. A. Kazhdan and G. Savin [Piatetski-Shapiro Festschrift, Part I: Papers on representation theory, Israel Math. Conf. Proc. 2, 209-223 (1990; Zbl 0737.22008)] in which the authors constructed small (minimal) representations of various simple, split groups over a local field \(F\). These are in many ways analogous to the metaplectic representations of the metaplectic groups (double covers of symplectic groups). The authors study in considerable detail “dual reductive pairs” in these groups. More precisely, they consider the pairs \(SO(2n-1) \times SO(3)\) in \(SO(2n+2)\), \(G_2 \times \text{PGL} (3)\) in \(E_6\), \(G_2 \times \text{PGSp}_6\) and \(G_2 \times F_4\) in \(E_8\). They prove a number of correspondences between the certain representations of these groups. The methods are interlinked and depend on fine details of the geometry of the exceptional groups combined with an analysis of the restriction of the minimal representation to certain parabolic subgroups. Although they do not go into the matter they indicate that these results will be of considerable significance for the study of global correspondences.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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