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Let   ${\text G}$   be a reductive group over a   $p$  -adic field   $F$   of characteristic zero, with   $p \gg 0$  , and let   $G={\text G}(F)$  . In [  15 ], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal   $K$  -types for   ${\text G}$   in the sense of A. Moy and G. Prasad. Following [  15 ], 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   ${G}$  -invariant open and closed subset   $\mathfrak{g}_{\overline{\mathfrak{s}}}$   of the Lie algebra   $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$  , and a subset   $\tilde{{G}}_{\overline{\mathfrak{s}}}$   of the admissible dual   $\tilde{{G}}$   of   ${G}$   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 of   ${G}$  , 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   ${G}$  -domain   ${G}_r \subset{G}$  , and reformulate a result of Kim as saying that the restriction of   $E_{\overline{\mathfrak{s}}}$   to   ${G}_r\,$  , pushed forward via the logarithm to the Moy–Prasad   ${G}$  -domain   $\mathfrak{g}_r \subset \mathfrak{g}$  , agrees on   $\mathfrak{g}_r$   with the inverse Fourier transform of the characteristic function of   $\mathfrak{g}_{\overline{\mathfrak{s}}}$  . This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [ 8 ] for the depth-  $r$   Bernstein projector.

The Bernstein Projector Determined by a Weak Associate Class of Good Cosets