Arthur truncation - Knowlpedia

Core idea

Let $G$ be a connected reductive group over a global field, choose a minimal parabolic subgroup, and let $T$ be a sufficiently regular point in the associated real chamber. Arthur's truncation operator $\Lambda^T$ replaces an automorphic function by an alternating sum of its parabolic constant terms, cut off by chamber characteristic functions.

Schematically,

(\Lambda^T\phi)(g)= \sum_P(-1)^{\dim(A_P/A_G)} \sum_{\delta\in P(F)\backslash G(F)} \widehat\tau_P(H_P(\delta g)-T)\,\phi_P(\delta g),

where $P$ ranges over standard parabolic subgroups, $H_P$ is the height map, and $\widehat\tau_P$ selects a positive cone.

Purpose

Automorphic quotients are generally noncompact, so the kernel of a convolution operator need not be integrable on the diagonal. Truncation cancels its asymptotic constant terms in the cusps. The integral of the truncated kernel is then defined and depends polynomial-exponentially on $T$ ; a distinguished constant term yields the trace-formula distribution.

For a cuspidal automorphic form every proper-parabolic constant term vanishes, so truncation leaves it unchanged in the sufficiently regular region.

From truncation to weights

Unfolding the truncated kernel creates the combinatorial weight functions in weighted orbital integrals on the geometric side and weighted characters on the spectral side.

References

James Arthur, “A truncation process for reductive groups,” Bulletin of the American Mathematical Society 83 (1977), 748–750. Project Euclid.
James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings 4, 2005, §§13–14. Clay.