  Let $G$ be a reductive group over a $p$-adic field $F$ of characteristic
zero, with $p \gg 0$. In [Kim04], J.-L. Kim studied an equivalence relation
called weak associativity on the set of unrefined minimal $K$-types for $G$ in
the sense of A. Moy and G. Prasad. Following [Kim04], we attach to the set
\(\overline{\mathfrak s}\) of good \(K\)-types in a weak associate class of
positive-depth unrefined minimal $K$-types a $G(F)$-invariant open and closed
subset $\mathfrak g(F)_{\overline{\mathfrak s}}$ of the Lie algebra $\mathfrak
g(F)$ of $G(F)$, and a subset $\tilde G_{\overline{\mathfrak s}}$ of the
admissible dual \(\tilde G\) of \(G(F)\) consisting of those representations
containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak
s}$. Then \(\tilde G_{\overline{\mathfrak s}}\) is the union of finitely many
Bernstein components for $G$, so that we can consider the Bernstein projector
$E_{\overline{\mathfrak s}}$ that it determines. We show that
$E_{\overline{\mathfrak s}}$ vanishes outside the Moy--Prasad $G(F)$-domain
$G(F)_r \subset G(F)$, and reformulate a result of Kim as saying that the
restriction of $E_{\overline{\mathfrak s}}$ to $G(F)_r$, pushed forward via the
logarithm to the Moy--Prasad $G(F)$-domain $\mathfrak g(F)_r \subset \mathfrak
g(F)$, agrees on $\mathfrak g(F)_r$ with the inverse Fourier transform of the
characteristic function of $\mathfrak g(F)_{\overline{\mathfrak s}}$. This is a
variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan and Y.
Varshavsky in arXiv:1504.01353 for the depth-$r$ Bernstein projector.
