Virtual Transfer Factors

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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 qᶜ s, where s is -1, 0, or 1, q is the cardinality of the residue field, and c is a rational number. The function s 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.