{"product_id":"9781470469078-harmonic-analysis-and-gamma-functio","title":"Harmonic Analysis and Gamma Functions on Symplectic Groups","description":"\u003cmeta content=\"text\/html; charset=utf-8\" http-equiv=\"Content-Type\"\u003e\u003cp\u003e\u003cspan\u003eThe method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard \u003ci\u003eL\u003c\/i\u003e-function of GL\u003csub\u003en\u003c\/sub\u003e and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.\u003cbr\u003eOver a p-adic local field F of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group G = Gm × Sp2n. It is associated to the Langlands ?-functions attached to any irreducible admissible representations ? ? ? of G(F) and the standard representation ? of the dual group G?(C), and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal (Braverman and Kazhdan, 2000) for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on GL1(F), which is associated to a ?-function ??(?s) (a product of n + 1 certain abelian ?-functions). Our work on GL1(F) plays an indispensable role in the development of our work on G(F). These two types of harmonic analyses both specialize to the well-known local theory developed in Tate's thesis (Tate, 1950) when n = 0. The approach is to use the compactification of Sp2n in the Grassmannian variety of Sp4n, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis (1986) and many other works) on the doubling local zeta integrals for the standard L-functions of Sp2n.\u003cbr\u003e\u003cbr\u003eThe method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard L-function of GLn and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.\u003cbr\u003e\u003cbr\u003e\u003c\/span\u003e\u003c\/p\u003e","brand":"Rarewaves","offers":[{"title":"Default Title","offer_id":54845460644214,"sku":"9781470469078","price":68.38,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0092\/7504\/8033\/files\/orig_36019651.jpg?v=1757557608","url":"https:\/\/www.rarewaves.com\/products\/9781470469078-harmonic-analysis-and-gamma-functio","provider":"Rarewaves.com","version":"1.0","type":"link"}