Schwartz–Bruhat space over a local field

Let $K$ be a locally compact nondiscrete field. The Schwartz–Bruhat space $\mathcal S(K)$ is defined according to the type of $K$ . If $K\cong\mathbb R$ or $\mathbb C$ , then $\mathcal S(K)$ is the usual Schwartz space: its smooth functions and all their derivatives decay faster than every polynomial. If $K$ is nonarchimedean, then

\mathcal S(K)=C_c^\infty(K),

the complex-valued, locally constant, compactly supported functions on the additive group of $K$ . Equivalently, each such function is constant on cosets of some open fractional ideal and vanishes outside a compact set. This is the local-field specialization of the Schwartz–Bruhat space on a locally compact abelian group.

Fourier stability

Choose a nontrivial continuous additive character $\psi:K\to\mathbb T$ and a Haar measure on the additive group of $K$ . The Fourier transform

\widehat f(y)=\int_K f(x)\overline{\psi(xy)}\,dx

maps $\mathcal S(K)$ onto itself. With the self-dual normalization of Haar measure it satisfies the usual Fourier inversion formula. This common formalism lets archimedean and nonarchimedean local factors be treated in parallel.

Basic nonarchimedean examples

If $\mathcal O_K$ is the valuation ring, then its indicator $1_{\mathcal O_K}$ lies in $\mathcal S(K)$ : the ring is compact and open, so its indicator is both compactly supported and locally constant. A continuous compactly supported function need not belong to $\mathcal S(K)$ ; local constancy is a defining requirement.

Conventions and scope

The notation $C_c^\infty(K)$ in the nonarchimedean case does not refer to derivatives. “Smooth” there means locally constant, equivalently smooth for the action of the totally disconnected additive group. For $K=\mathbb C$ , rapid decay is measured on the underlying real vector space $\mathbb R^2$ .

References

André Weil, Basic Number Theory, 3rd ed., Springer, 1974. DOI record. Relevant: “Lattices and duality over local fields,” pp. 24–42.
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. DOI record. Relevant: pp. 60–61 on the general Schwartz–Bruhat construction.