Artin Symmetric Functions (2409.09643v3)

Abstract: In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric functions. This ring is built from a family of symmetric functions rings indexed by prime ideals of the base field. We prove many necessary basic results for the ring of arithmetic symmetric functions as well as introduce the analogues of some standard number-theoretic objects in this setting. We prove that the Artin symmetric functions satisfy the same algebraic properties that the Artin L-functions do with respect to induction, inflation, and direct summation of representations. The expansion coefficients of these symmetric functions in different natural bases are shown to be character values of representations of a compact group related to the original Galois group. In the most interesting case, the expansion coefficients into a specialized Hall-Littlewood basis come from new representations built from the original Galois representation using polynomial functors corresponding to modified Hall-Littlewood polynomials. Using a special case of the Satake isomorphism in type GL, as formulated by Macdonald, we show that the Artin symmetric functions yield families of functions in the (finite) global spherical Hecke algebras in type GL which exhibit natural stability properties. We compute the Mellin transforms of these functions and relate them to infinite products of shifted Artin L-functions. We then prove some analytic properties of these Dirichlet series and give an explicit expansion of these series using the Hall-Littlewood polynomial functors.