  We construct local and global metaplectic double covers of odd general spin
groups, using the cover of Matsumoto of spin groups. Following Kazhdan and
Patterson, a local exceptional representation is the unique irreducible
quotient of a principal series representation, induced from a certain
exceptional character. The global exceptional representation is obtained as the
multi-residue of an Eisenstein series, it is an automorphic representation and
decomposes as the restricted tensor product of local exceptional
representations. As in the case of the small representation of SO(2n+1) of
Bump, Friedberg and Ginzburg, exceptional representations enjoy the vanishing
of a large class of twisted Jacquet modules (locally), or Fourier coefficients
(globally). Consequently they are useful in many settings, including lifting
problems and Rankin-Selberg integrals. We describe one application, to a
calculation of a co-period integral.
