Interpreting the Harish-Chandra—Howe local character expansion via branching rules

Abstract: The Harish-Chandra–Howe local character expansion expresses the character of an admissible representation of a $p$-adic group $G$ as a linear combination of Fourier transforms of nilpotent orbital integrals $\widehat{\mu}_{\mathcal{O}}$ near the identity. We show that for $G=\mathrm{SL}(2,k)$, where the branching rules to maximal compact open subgroups $K$ are known, each of these terms $\widehat{\mu}_{\mathcal{O}}$ can be interpreted as the character $\tau_{\mathcal{O}}$ of an infinite sum of representations of $K$, up to an error term arising from the zero orbit. Moreover, the irreducible components of $\tau_{\mathcal{O}}$ are explicitly constructed from the $K$ -orbits in $\mathcal{O}$. This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.

number theoryrepresentation theory

Audience: researchers in the discipline

The 2020 Paul J. Sally, Jr. Midwest Representation Theory Conference

Series comments: The 44th Midwest Representation Theory Conference will address recent progress in the theory of representations for groups over non-archimedean local fields, and connections of this theory to other areas within mathematics, notably number theory and geometry.

In order to receive information on how to participate (to be sent out closer to the conference), please register by October 14 here: forms.gle/zFAnQBnuPGRnKzMr7