3.2 Distributions on groups
Let \(G\) be a totally disconnected, locally compact, Hausdorff topological group.
3.2.1 Translations of functions
For \(\varphi : G \to \mathbb {C}\) and \(g \in G\) we define
Right and left translations preserve local constancy and compact support.
Translations of test functions are defined and satisfy the expected identities.
3.2.2 Compactly supported distributions and convolution
A compactly supported distribution is a distribution whose support is compact.
Given \(D\) and \(\varphi \), the function \(x \mapsto D(R_x\varphi )\) is locally constant and compactly supported, and hence defines an element of \(C_c^{\infty }(G)\).
The convolution of \(F\) and \(D\) is defined by
Convolution of compactly supported distributions is associative.
3.2.3 Translation actions on distributions
Left and right translation actions are defined by dualizing the actions on test functions.
3.2.4 Delta distributions and a normalized Haar distribution
The delta distribution at \(g\) evaluates test functions at \(g\).
The indicator function of a compact clopen subset is a compactly supported locally constant function.
We define a compactly supported distribution \(e_H\) attached to a compact open subgroup \(H\) as the average of a test function over \(H\) with respect to the normalized Haar measure on \(H\).
3.2.5 Convolution identities for delta and normalized Haar distributions
For a compact open subgroup \(H\), the normalized Haar distribution is idempotent:
For \(g \in G\) and test function \(\varphi \):
If \(g \in H\), then \(\delta _g * e_H = e_H * \delta _g\).
3.2.6 Invariance of the normalized Haar distribution
For a compact open subgroup \(H\), \(e_H\) is left \(H\)-invariant, right \(H\)-invariant, and hence bi-\(H\)-invariant.
3.2.7 Finite decomposition for bi-invariant distributions
If \(F\) is compactly supported and bi-\(H\)-invariant, then there exist finitely many elements \(g_i \in G\) and coefficients \(a_i \in \mathbb {C}\) such that
for all test functions \(\varphi \).
3.2.8 Local constancy (smoothness)
We say a distribution is left, right, or bi-invariant under an open subgroup if it is fixed by the corresponding translation action.
Let \(D\) be a compactly supported distribution on \(G\). The following statements are equivalent:
\(D\) is left-invariant under some compact open subgroup of \(G\);
\(D\) is right-invariant under some compact open subgroup of \(G\);
\(D\) is bi-invariant under some compact open group of \(G\).
A compactly supported distribution satisfying the above condition is said to be locally constant.