Modular Symbols with Values in Beilinson-Kato Distributions

For each integer $n\ge 1$, we construct a ${\mathrm{GL}}_n(\mathbb{Q})$-invariant modular symbol ${\mathit{\xi }}_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\mathit{\mu }$ on ${\mathbb{Q}}^2$, with values in the modular function field, serves as the building block for ${\mathit{\xi }}_n$; we define ${\mathit{\xi }}_n$ essentially by taking the $n$-Steinberg product of $\mathit{\mu }$. The most non-trivial part of this construction is the cocycle property of ${\mathit{\xi }}_n$; we prove it by using an induction on $n$ based on the first two cases ${\mathit{\xi }}_1$ and ${\mathit{\xi }}_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations.