Modular symbols with values in Beilinson-Kato distributions

For each integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we construct a <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 Q 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:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_n(\mathbb {Q})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant modular symbol <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\xi }_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with coefficients in a space of distributions that takes values in the Milnor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -group of the modular function field. The Siegel distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic mu"> <mml:semantics> <mml:mi mathvariant="bold-italic"> μ </mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {\mu }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Q}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with values in the modular function field, serves as the building block for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\xi }_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; we define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\xi }_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> essentially by taking 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> -Steinberg product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic mu"> <mml:semantics> <mml:mi mathvariant="bold-italic"> μ </mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {\mu }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The most non-trivial part of this construction is the cocycle property of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\xi }_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; we prove it by using an induction on <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> based on the first two cases <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\xi }_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic xi 2"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> ξ