Test-function space on a local group

Definition

Let $F$ be a local field and let $G(F)$ be a local group. Its test-function space is denoted

\mathcal D(G(F))=C_c^\infty(G(F)).

The superscript has two different, category-dependent meanings:

if $F$ is archimedean and $G(F)$ is a Lie group, it consists of smooth complex functions with compact support;
if $F$ is nonarchimedean and $G(F)$ is locally profinite, it consists of locally constant complex functions with compact support.

In both cases $\mathcal D(G(F))$ carries its standard locally convex inductive-limit topology. This topology, not only the underlying vector space, determines what it means for a distribution to be continuous.

Nonarchimedean description

A nonarchimedean test function is fixed by translation by some compact open subgroup on each chosen side. Spaces with fixed compact support and fixed open invariance are finite-dimensional, and their directed union gives $C_c^\infty(G(F))$ . The notation “smooth” therefore means locally constant; it does not refer to derivatives.

For a compact open subgroup $K$ , the bi- $K$ -invariant test functions form the Hecke algebra $\mathcal H(G(F),K)$ . Allowing $K$ to shrink recovers the full test-function space.

Convolution and measures

After choosing a Haar measure, test functions can be convolved. Changing the Haar measure rescales formulas that identify functions with measures, so trace formulas, orbital integrals, and character distributions must state compatible measure normalizations.

Adelic form

For an algebraic group over a global field, the adelic test-function space is a restricted tensor product of the local spaces. At almost every finite place where the group is unramified, one fixes a hyperspecial subgroup $K_v$ and uses the distinguished function $1_{K_v}$ . This is the group analogue of the restricted tensor product of local Schwartz–Bruhat spaces.

Scope warning

The Euclidean test-function space $\mathcal D(\Omega)$ is the archimedean open-set model, not a definition of locally constant $p$ -adic test functions. An arbitrary locally compact group that is neither a Lie group nor locally profinite has no canonical $C_c^\infty$ until an additional smooth structure or test-function category is specified.

References

François Bruhat, “Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $p$ -adiques,” Bulletin de la Société Mathématique de France 89 (1961), 43–75. Numdam.
James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings 4, 2005, §§1–3. Clay.