{"product_id":"quantization-and-arithmetic-9783764387907","title":"Quantization and Arithmetic","description":"\u003cp\u003e • Author(s): André Unterberger\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Birkhauser\u003cbr\u003e • BISAC: Group Theory\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003e(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\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45281664336023,"sku":"9783764387907","price":3639.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783764387907.webp?v=1769299881","url":"https:\/\/atlanticbooks.com\/products\/quantization-and-arithmetic-9783764387907","provider":"Atlantic Books","version":"1.0","type":"link"}