Archimedean zeta integrals for 𝐺𝐿(3)×𝐺𝐿(2)

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2022-07-01 DOI:10.1090/memo/1366

Miki Hirano, Taku Ishii, Tadashi Miyazaki

{"title":"Archimedean zeta integrals for 𝐺𝐿(3)×𝐺𝐿(2)","authors":"Miki Hirano, Taku Ishii, Tadashi Miyazaki","doi":"10.1090/memo/1366","DOIUrl":null,"url":null,"abstract":"<p>In this article, we give explicit formulas of archimedean Whittaker functions on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 3 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL(3)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL(2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Moreover, we apply those to the calculation of archimedean zeta integrals for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L left-parenthesis 3 right-parenthesis times upper G upper L left-parenthesis 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>×</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GL(3)\\times GL(2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and show that the zeta integral for appropriate Whittaker functions is equal to the associated <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-factors.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1366","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 1

Abstract

In this article, we give explicit formulas of archimedean Whittaker functions on G L ( 3 ) GL(3) and G L ( 2 ) GL(2) . Moreover, we apply those to the calculation of archimedean zeta integrals for G L ( 3 ) × G L ( 2 ) GL(3)\times GL(2) , and show that the zeta integral for appropriate Whittaker functions is equal to the associated L L -factors.