The n-dimensionalmetaplectic groupSp(n, R) is the twofoldcoverof the
sympl- n n tic group Sp(n, R), which is the group of linear
transformations ofX = R ×R that preserve the bilinear (alternate) form x
y [( ), ( )] =? x, ? + y, ? . (0. 1) ? ? 2 n There is a unitary
representation of Sp(n, R)intheHilbertspace L (R ), called the
metaplectic representation, the image of which is the groupof
transformations generated by the following ones: the linear changes of
variables, the operators of multiplication by exponentials with pure
imaginary quadratic forms in the ex- nent, and the Fourier
transformation; some normalization factor enters the de?- tion of the
operators of the ?rst and third species. The metaplectic representation
was introduced in a great generality in [28] - special cases had been
considered before, mostly in papers of mathematical physics - and it is
of such fundamental importancethat the two concepts (the groupand the
representation)havebecome virtually indistinguishable. This is not going
to be our point of view: indeed, the main point of this work is to show
that a certain ?nite covering of the symplectic group (generally of
degree n) has another interesting representation, which enjoys analogues
of most of the nicer properties of the metaplectic representation. We
shall call it the anaplectic representation - other coinages that may
come to your mind sound too medical - and shall consider ?rst the
one-dimensional case, the main features of which can be described in
quite elementary terms.
