The Fourfold Way in Real Analysis

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.