LEADER 05970nam 22006613 450 001 9910965481803321 005 20231110223243.0 010 $a9781470471668 010 $a1470471663 035 $a(MiAaPQ)EBC29379019 035 $a(Au-PeEL)EBL29379019 035 $a(CKB)24267686200041 035 $a(OCoLC)1336953688 035 $a(EXLCZ)9924267686200041 100 $a20220721d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aArchimedean Zeta Integrals for CL(3) x GL(2) /$fMiki Hirano, Taku Ishii, Tadashi Miyazaki 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (136 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.278 311 08$aPrint version: Hirano, Miki Archimedean Zeta Integrals for Providence : American Mathematical Society,c2022 9781470452773 327 $aCover -- Title page -- Introduction -- Acknowledgments -- Part 1. Whittaker functions -- Chapter 1. Basic objects -- 1.1. Notation -- 1.2. Groups and algebras -- 1.3. Whittaker functions -- 1.4. Capelli elements -- 1.5. The gamma function and the Bessel functions -- 1.6. Special functions of two variables -- Chapter 2. Preliminaries for ( ,\bR) -- 2.1. Generalized principal series representations -- 2.2. The elements of \g_{\bC} and (\g_{\bC}) -- 2.3. The eigenvalues of generators of (\g_{\bC}) -- Chapter 3. Whittaker functions on (2,\bR) -- 3.1. Representations of (2) -- 3.2. Principal series representations -- 3.3. Principal series Whittaker functions -- 3.4. Essentially discrete series Whittaker functions -- Chapter 4. Whittaker functions on (3,\bR) -- 4.1. Representations of (3) -- 4.2. Principal series representations -- 4.3. Principal series Whittaker functions at scalar -types -- 4.4. Principal series Whittaker functions at 3 dimensional -types -- 4.5. Generalized principal series representations -- 4.6. Generalized principal series Whittaker functions -- Chapter 5. Preliminaries for ( ,\bC) -- 5.1. Principal series representations -- 5.2. The elements of \g_{\bC} and (\g_{\bC}) -- 5.3. The eigenvalues of generators of (\g_{\bC}) -- Chapter 6. Whittaker functions on (2,\bC) -- 6.1. Representations of (2) -- 6.2. Principal series representations -- 6.3. Principal series Whittaker functions -- Chapter 7. Whittaker functions on (3,\bC) -- 7.1. Representations of (3) -- 7.2. Principal series representations -- 7.3. Principal series Whittaker functions -- Part 2. Archimedean zeta integrals for (3)× (2) -- Chapter 8. Preliminaries -- 8.1. The aim of Part 2 -- 8.2. Some formulas for the calculation -- Chapter 9. The local zeta integrals for (3,\bR)× (2,\bR) -- 9.1. The local Langlands correspondence for ( ,\bR). 327 $a9.2. Preparations for (2)-modules -- 9.3. Whittaker functions on (2,\bR) -- 9.4. Whittaker functions on (3,\bR) -- 9.5. The local zeta integrals for (3,\bR)× (2,\bR) -- 9.6. The calculation for '= _{( ?', ?')}? _{( ?', ?')} -- 9.7. The calculation for '= _{( ?',1)}? _{( ?',0)} -- 9.8. The calculation for '= _{( ', ')} -- Chapter 10. The local zeta integrals for (3,\bC)× (2,\bC) -- 10.1. The local Langlands correspondence for ( ,\bC) -- 10.2. Preparations for (2)-modules -- 10.3. Whittaker functions on (2,\bC) -- 10.4. Whittaker functions on (3,\bC) -- 10.5. The local zeta integrals for (3,\bC)× (2,\bC) -- 10.6. The calculation in the case ?> -- - ?' -- 10.7. The calculation in the case - ?'> -- ? -- 10.8. The calculation in the case - ?'? ??- ?' -- Appendix A. Archimedean zeta integrals for (2)× ( ) ( =1,2) -- A.1. The local zeta integrals for (2,\bR)× (1,\bR) -- A.2. The local zeta integrals for (2,\bR)× (2,\bR) -- A.3. The local zeta integrals for (2,\bC)× (1,\bC) -- A.4. The local zeta integrals for (2,\bC)× (2,\bC) -- Bibliography -- Back Cover. 330 $a"In this article, we give explicit formulas of archimedean Whittaker functions on GL(3) and GL(2). Moreover, we apply those to the calculation of archimedean zeta integrals for GL(3) x GL(2), and show that the zeta integral for appropriate Whittaker functions is equal to the associated L-factors"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aCoulomb functions 606 $aRiemann integral 606 $aFunctions, Zeta 606 $aAutomorphic forms 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields$2msc 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Fourier coefficients of automorphic forms$2msc 606 $aTopological groups, Lie groups -- Lie groups -- Semisimple Lie groups and their representations$2msc 615 0$aCoulomb functions. 615 0$aRiemann integral. 615 0$aFunctions, Zeta. 615 0$aAutomorphic forms. 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields. 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Fourier coefficients of automorphic forms. 615 7$aTopological groups, Lie groups -- Lie groups -- Semisimple Lie groups and their representations. 676 $a515/.55 676 $a515.55 686 $a11F70$a11F30$a22E46$2msc 700 $aHirano$b Miki$01801825 701 $aIshii$b Taku$01801826 701 $aMiyazaki$b Tadashi$01801827 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910965481803321 996 $aArchimedean Zeta Integrals for CL(3) x GL(2)$94347225 997 $aUNINA