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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.
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.