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

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 the 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 $\mathbb{Q}$. Given a non-Archimedean local field $\mathbb{K}$ with sufficiently large residual characteristic, the part of the character table near the identity element for $G(\mathbb{K})$ 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, in the terminology of Cluckers and Loeser in a recent paper. 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 $G(\mathbb{K})$ is computable.