## Virtual transfer factors

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- by Julia Gordon and Thomas C. Hales
- Represent. Theory
**7**(2003), 81-100 - DOI: https://doi.org/10.1090/S1088-4165-03-00183-3
- Published electronically: March 3, 2003

## Abstract:

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 $q^c \mathrm {sign}$, where $\mathrm {sign}\in \{-1,0,1\}$, $q$ is the cardinality of the residue field, and $c$ is a rational number. The $\mathrm {sign}$ 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.

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## Bibliographic Information

**Julia Gordon**- Affiliation: The Fields Institute, 222 College St., Toronto, Ontario, M5T 3J1, Canada
- Email: julygord@umich.edu
**Thomas C. Hales**- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- Email: hales@pitt.edu
- Received by editor(s): December 6, 2002
- Published electronically: March 3, 2003
- © Copyright 2003 Julia Gordon and Thomas C. Hales
- Journal: Represent. Theory
**7**(2003), 81-100 - MSC (2000): Primary 11F85, 22E50
- DOI: https://doi.org/10.1090/S1088-4165-03-00183-3
- MathSciNet review: 1973368