ORBITAL INTEGRAL CORRESPONDENCE FOR THE PAIR (G2, Sp(1;R))

The groups Sp(1;R), O(3;4) form a dual pair in the §sense of Howe. This leads to a correspondence of §irreducible unitary representations between the §double connected cover of Sp(1;R) and some §irreducible unitary representations of O(3;4). By §a property of double transitivity, Rallis &§Schiffmann showed that the restriction of the§resulting representation to G2 remains irreducible, §but don't compute the characters of these §representations. Neither do they compute the§lowest term of the expansion of such a character, §which should be the Fourier§transform of an orbital integral corresponding to a §nilpotent orbit. The goal of this work is to make §progress in this direction. We showed that this §theory can be §extended to include the case of G2. Then we interpret §the Jacobson-Rallis-Schiffmann theorem as a statement §that there is an injection from§the regular semisimple orbits of sp(1;R) to those of §g2, via unnormalized maps used in CIT. We attempt to§extend this statement to nilpotent orbits and arrive §at a conjecture, and compute the Cauchy Harish-§Chandra integral for orbits in sp(1,R), §and find they look like the Fourier transforms of §orbital integrals of g2.