Progress in Mathematics
2 primary works
Book 209
Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi
by Andre Unterberger
Published 11 December 2002
Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002. The subject of this book is the study of automorphic distributions, by which is meant distributions on R2 invariant under the linear action of SL(2,Z), and of the operators associated with such distributions under the Weyl rule of symbolic calculus. Researchers and postgraduates interested in pseudodifferential analyis, the theory of non-holomorphic modular forms, and symbolic calculi will benefit from the clear exposition and new results and insights.
Book 250
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 xR 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.
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.