Virtual transfer factors

The Langlands-Shelstad transfer factor is a function defined on some reductive groups over a $p$-adic field. Near the origin of the group, it may be viewed as a function on the Lie algebra. For classical groups, its values have the form $q^c\,{\bf\text{sign}}$, where ${\bf\text{sign}}\in\{-1,0,1\}$, $q$ is the cardinality of the residue field, and $c$ is a rational number. The ${\bf\text{sign}}$ function partitions the Lie algebra into three subsets. This article shows that this partition into three subsets is independent of the $p$-adic field in the following sense. We define three universal objects (virtual sets in the sense of Quine) such that for any $p$-adic field $F$ of sufficiently large residue characteristic, the $F$-points of these three virtual sets form the partition.

The theory of arithmetic motivic integration associates a virtual Chow motive with each of the three virtual sets. The construction in this article achieves the first step in a long program to determine the (still conjectural) virtual Chow motives that control the behavior of orbital integrals.