Journal of the Ramanujan Mathematical Society

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Base Change and (GLn(F), GLn-1(F))-Distinction


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1 Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel
     

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Let F be a nonarchimedean local field and E a finite cyclic extension of F of prime degree d. Further let GLn-1 be embedded into GLn as block matrices in the usual way. It is not true in general that the base change lift of a GLn-1(F)-distinguished representation of GLn(F) is GLn-1(E)-distinguished. We obtain a precise condition for an irreducible GLn-1(F)-distinguished representation π of GLn(F) to be taken to a GLn-1(E)-distinguished representation by the base change map. If π is unitarizable and GLn-1(F)-distinguished, then we show that the base change lift of π is GLn-1(E)-distinguished. We then analyse the fiber of the base change map over a GLn-1(E)-distinguished representation of GLn(E) and determine the number of GLn-1(F)-distinguished representations in the fiber.
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  • Base Change and (GLn(F), GLn-1(F))-Distinction

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Authors

Arnab Mitra
Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel
C. G. Venketasubramanian
Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

Abstract


Let F be a nonarchimedean local field and E a finite cyclic extension of F of prime degree d. Further let GLn-1 be embedded into GLn as block matrices in the usual way. It is not true in general that the base change lift of a GLn-1(F)-distinguished representation of GLn(F) is GLn-1(E)-distinguished. We obtain a precise condition for an irreducible GLn-1(F)-distinguished representation π of GLn(F) to be taken to a GLn-1(E)-distinguished representation by the base change map. If π is unitarizable and GLn-1(F)-distinguished, then we show that the base change lift of π is GLn-1(E)-distinguished. We then analyse the fiber of the base change map over a GLn-1(E)-distinguished representation of GLn(E) and determine the number of GLn-1(F)-distinguished representations in the fiber.