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Quantization and Arithmetic by André Unterberger

Quantization and Arithmetic

by André Unterberger
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Book cover type: Paperback
  • ISBN13: 9783764387907
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Birkhauser
  • Publication Date:
  • Pages: 147
  • Original Price: EUR 49.99
  • Language: English
  • Edition: 2008
  • Item Weight: 386 grams
  • BISAC Subject(s): Group Theory, Mathematical Analysis, and Number Theory

(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

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