Compatibility of local and global Langlands correspondences

Abstract

We prove the compatibility of local and global Langlands correspondences for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which was proved up to semisimplification in M. Harris and R. Taylor, <italic>The Geometry and Cohomology of Some Simple Shimura Varieties</italic> , Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript l Baseline left-parenthesis normal upper Pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_l(\Pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Galois group of an imaginary CM-field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> attached to a conjugate self-dual regular algebraic cuspidal automorphic representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi"> <mml:semantics> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:annotation encoding="application/x-tex">\Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n Baseline left-parenthesis double-struck upper A Subscript upper L Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_n(\mathbb A_L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript l Baseline left-parenthesis normal upper Pi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_l(\Pi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the decomposition group of a place <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not dividing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponds to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript v"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Pi _v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the local Langlands correspondence. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript v"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Π </mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Pi _v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is square integrable for some finite place <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v solidus vertical-bar l">