Lattice of the dual of an Anderson $t$-motive in terms of a map to a flag variety

Abstract

Let $M$ be an uniformizable Anderson $t$-motive of rank $r$, $L$ its lattice and $l_*:=\{l_1,\dots, l_r\}$ its basis. We define a map $δ$ from the set of these bases to a flag variety (the present text gives the definition of $δ$ only for elements of the maximal Schubert cell, and for few other cases). If $l_*$ belongs to the maximal Schubert cell then $δ(l_*)$ is described as a set of matrices parametrized by integer points of a tetrahedron; they are called the Siegel element of $l_*$. We give explicit formulas for a Siegel element for $M'$ -- the dual of $M$. As a by-product, we get another proof of the theorem that the lattice of $M'$ is the dual of the lattice of $M$, independent of the proof obtained by U. Hartl and A.-K. Juschka. Generalizations of this result to non-maximal Schubert cells, other tensor operations (tensor product and Hom) and t-motives having non-trivial endomorphism rings, are subjects of further research.