Schwartz–Bruhat space over a local field - Imported Knowlpedia Content

Definition

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 Weil, “Lattices and duality over local fields”.

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.