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The Fourfold Way in Real Analysis - An Alternative to the Metaplectic Representation (Hardcover, 2006 ed.)
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The Fourfold Way in Real Analysis - An Alternative to the Metaplectic Representation (Hardcover, 2006 ed.)
Series: Progress in Mathematics, 250
Expected to ship within 10 - 15 working days
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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.
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