On the computability of some positive-depth supercuspidal characters near the identity

TL;DR

This paper demonstrates that the values of certain supercuspidal characters of p-adic symplectic and orthogonal groups near the identity are algorithmically computable through a geometric parameter space and motivic functions.

Contribution

It introduces a geometric parameter space for minimal depth supercuspidal representations and shows that their character values are computable via specialization of motivic exponential functions.

Findings

Character values near the identity are parameterized by residue-field points of a geometric space.

Character values can be recovered from constructible motivic exponential functions.

A large part of the character table for these groups is algorithmically computable.

Abstract

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depends on the group and a real number $r>0$) for the set of equivalence classes of the representations of minimal depth $r$ satisfying some additional assumptions. This parameter space is essentially a geometric object defined over $\Q$. Given a non-Archimedean local field $\K$ with sufficiently large residual characteristic, the part of the character table near the identity element for $G(\K)$ that comes from our class of representations is parameterized by the residue-field points of $B$. The character values themselves can be recovered by specialization from a constructible motivic exponential function. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group $G(\K)$ is computable.