Quantization and arithmetic

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(12) (4) Let ? be the unique even non-trivial Dirichlet character mod 12, and let ? be the unique (odd) non-trivial Dirichlet character mod 4. Consider on the line the distributions m (12) ? d (x)= ? (m)? x? , even 12 m? Z m (4) d (x)= ? (m)? x? . (1.1) odd 2 m? Z 2 i? x UnderaFouriertransformation, orundermultiplicationbythefunctionx ? e , the? rst(resp. second)ofthesedistributionsonlyundergoesmultiplicationbysome 24th (resp. 8th) root of unity. Then, consider the metaplectic representation Met, 2 a unitary representation in L (R) of the metaplectic group G, the twofold cover of the group G = SL(2, R), the de? nition of which will be recalled in Section 2: it extends as a representation in the spaceS (R) of tempered distributions. From what has just been said, if g ˜ is a point of G lying above g? G, andif d = d even g ˜ ?1 or d , the distribution d =Met(g˜ )d only depends on the class of g in the odd homogeneousspace?\G=SL(2, Z)\G, uptomultiplicationbysomephasefactor, by which we mean any complex number of absolute value 1 depending only on g ˜. On the other hand, a function u? S(R) is perfectly characterized by its scalar g ˜ productsagainstthedistributionsd , sinceonehasforsomeappropriateconstants C , C the identities 0 1 g ˜ 2 2 | d , u | dg = C u if u is even, 2 0 even L (R) ?\G