Cocenters of Hecke algebras
H
\mathcal {H}
play an important role in studying mod
ℓ
\ell
or
C
\mathbb C
harmonic analysis on connected
p
p
-adic reductive groups. On the other hand, the depth
r
r
Hecke algebra
H
r
+
\mathcal {H}_{r^+}
is well suited to study depth
r
r
smooth representations. In this paper, we study depth
r
r
rigid cocenters
H
¯
r
+
r
i
g
\overline {\mathcal {H}}^\mathrm {rig}_{r^+}
of a connected reductive
p
p
-adic group over rings of characteristic zero or
ℓ
≠
p
\ell \neq p
. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth
r
r
rigid cocenter, hence find an explicit basis of
H
¯
r
+
r
i
g
\overline {\mathcal {H}}^\mathrm {rig}_{r^+}
.