  Let V=V_0+V_1 be a real finite dimensional supervector space provided with a
non-degenerate antisymmetric even bilinear form B. Let spo(V) be the Lie
superalgebra of endomorphisms of V which preserve B. We consider spo(V) as a
supermanifold. We show that a choice of an orientation of V_1 and of a square
root i of -1 determines a very interesting generalized function on the
supermanifold spo(V), the superPfaffian.
  When V=V_1, spo(V) is the orthogonal Lie algebra so(V_1) and the
superPfaffian is the usual Pfaffian, a square root of the determinant.
  When V=V_0, spo(V) is the symplectic Lie algebra sp(V_0) and the
superPfaffian is a constant multiple of the Fourier transform of one the two
minimal nilpotent orbits in the dual of the Lie algebra sp(V_0), and is an
analytic square root of the inverse of the determinant in the open subset of
invertible elements of spo(V).
  Our opinion is that the superPfaffians (there are four of them, corresponding
to the two orientations on V_1, and to the two square roots of -1) are
fundamental objects. At least, they occur in the study of equivariant
cohomology of supermanifolds, and in the study of the metaplectic
representation of the metaplectic group with Lie algebra spo(V). In this
article, we present the definition and some basic properties of the
superPfaffians.
