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A tower of theta correspondences for \(G_ 2\). (English) Zbl 0881.11051
This paper represents a further development of the ideas of a previous paper [D. Ginzburg, S. Rallis and D. Soudry, Isr. J. Math. 100, 61-116 (1997; Zbl 0881.11050)]. In that paper the authors constructed new and “small” automorphic representations of simple, split groups of type \(D_m\) and \(E_m\) over a number field \(k\). These representations are analogous to the classical theta representations and so lead one to expect that they also lead to (global) correspondences between representations of dual reductive pairs. Indeed they have already studied two analogous examples inside the three-fold cover of a group of type \(G_2\) [Am. J. Math. 119, 251-335 (1997; Zbl 0877.11031)].
Here the authors consider examples where \(G_2\) appears as one factor of a dual reductive pair in groups of type \(D_m\) \((m=4,5,6)\) and \(E_n\) \((n=6,7,8)\). The first of these were already established by other methods (using the classical theta correspondence) by S. Rallis and G. Schiffmann [Am. J. Math. 111, 801-849 (1989; Zbl 0723.11026)]. They are all related inside a ‘tower’ and so interact with one another. The authors establish a number of remarkable properties of the correspondence. Unfortunately even a moderately precise statement of the major results would be too extensive for this review.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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