Distribution on a local group

Definition

Let $F$ be a local field, let $G(F)$ be a local group, and let $\mathcal D(G(F))$ be its test-function space. A distribution on $G(F)$ is a continuous linear functional

D:\mathcal D(G(F))\longrightarrow\mathbb C.

Thus continuity is measured using smooth compactly supported functions in the archimedean case and locally constant compactly supported functions in the nonarchimedean case.

The distribution is conjugation-invariant if

D(f^x)=D(f), \qquad f^x(g)=f(x^{-1}gx),

for every $x\in G(F)$ . Invariant distributions are the natural common language for orbital integrals and representation characters.

Functions and orbital measures

After choosing a Haar measure, a locally integrable function $\Theta$ defines a distribution by

f\longmapsto\int_{G(F)}f(g)\Theta(g)\,dg.

Not every distribution is represented by a function. In particular, an orbital integral is naturally a distribution supported on a conjugacy orbit, while a Harish–Chandra character is first defined distributionally and only then represented by a function on a regular locus.

Stable distributions

For a reductive group, a stable distribution is an invariant distribution satisfying an additional factorization condition through stable orbital-integral data. “Invariant” and “stable” are therefore not synonyms.

Scope warning

This definition is the local-group analogue of a Schwartz distribution on an open subset of Euclidean space. The test-function categories differ, so the Euclidean page should not be used as the direct definition of a distribution on a nonarchimedean group.

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.
Harish-Chandra, “Admissible invariant distributions on reductive $p$ -adic groups,” in Lie Theories and Their Applications, Queen's Papers in Pure and Applied Mathematics 48, 1978.