 of notation. See \cite{akshay}, section 2 or \cite{Ronchetti} for details.
\end{defn}
\begin{rem} The classical Hecke algebra of double $K_p$-cosets is then simply the degree zero subalgebra of $\mathcal H^*_R \left( G_p, K_p \right)$.
\end{rem}
Let now $C^{\bullet}$ be a complex of (smooth) $G_p$-modules with $R$-coefficients. Then the hypercohomology of $K_p$ with coefficients in $C^{\bullet}$ can be computed as the cohomology of the complex $\ensuremath{\mathrm{Hom}}_{G_p} \left( \cInd{G_p}{K_p} R, C^{\bullet} \right)$.
One can equivalently think of this complex as the homomorphisms $\ensuremath{\mathrm{Hom}}_{\mathcal D(G_p)} \left( \cInd{G_p}{K_p} R, C^{\bullet} \right)$ in the derived category of smooth $G_p$-modules with coefficients in $R$.
This complex obviously has an action (by pre-composition) of the endomorphisms of $\cInd{G_p}{K_p} R$ in the derived category and hence this latter description makes it obvious that (by taking cohomology of the respective complexes) we get an action of the derived Hecke algebra $\mathcal H^*_R \left( G_p, K_p \right)$ on $H^* \left( K_p, C^{\bullet} \right)$.
We will call a \emph{derived Hecke operator} any endomorphism of $H^* \left( K_p, C^{\bullet} \right)$ arising in this way.
\begin{rem} \label{derivedidentitycoset} A special case of derived Hecke operators is obtained by classes supported on the `identity coset' - which is to say, cohomology classes in $H^* \left( K_p, R \right) \subset \mathcal H^*_R \left( G_p, K_p \right)$. See for instance the derived diamond operators constructed in the next section \ref{deriveddiamond}.
\end{rem}
This very abstract setup can then be made concrete for plenty of interesting complexes $C^{\bullet}$.
We describe next how to interpret arithmetic cohomology in this context.
For each open compact subgroup $K_p \subset G_p$, we let $C^{\bullet} (K_p)$ be the cochain complex of $Y(K^p \times K_p)$ with $R$-coefficients.
Let \[ C^{\bullet} = \varinjlim_{K_p} C^{\bullet} (K_p) \] be the direct limit over open compact subgroups of $G_p$ - as explained in section 2.6 of \cite{akshay}, this complex knows the cohomology of the tower at $p$, in the sense that one has a quasi-isomorphism \[ \left( C^{\bullet} \right)^{K_p} \sim C^{\bullet} \left( K_p \right) \] and thus taking the hypercohomology of $K_p$ yields the cohomology of the corresponding arithmetic manifold: \[ H^* \left( K_p, C^{\bullet} \right) = H^* \left( Y \left( K^p \times K_p \right), R \right). \]
By the general setup outlined above, we obtain that $H^* \left( Y \left( K^p \times K_p \right), R \right)$ is a graded $\mathcal H^*_R \left( G_p, K_p \right)$-module. We will use this repeatedly.

\subsection{Derived diamond operators} \label{deriveddiamond}
We now introduce the derived diamond operators. These are operators on arithmetic cohomology which increase the cohomological degree, and come (just like the classical diamond operators) from the torus. However, while the classical diamond operators are obtained via group elements in the torus, the derived diamond operators come from classes in the cohomology of this torus.

As defined in section \ref{sectionordinarycohomology}, on the complex $C^{\bullet} \left( Y_{b,c}, \ensuremath{\mathbb Z} / p^n \right)$ and on its cohomology $H^* \left( Y_{b,c}, \ensuremath{\mathbb Z} / p^n \right)$ we have two actions: \begin{enumerate}
    \item There is the classical action of the $U_p$-operators, - which in our setup are the operators $U_p^i = I(b,c) \lambda_i(p) I(b,c)$.
    \item We also have the diamond action of the operators $\langle \alpha \rangle$ for each $\alpha \in I(1,c) / I(b,c)$.
\end{enumerate}
By lemma \ref{Heckeincohomology} these two actions commute, even at the level of complexes.

\begin{lem} The covering $Y_{c,c} \twoheadrightarrow Y_{b,c}$ is Galois, with Galois group $T_{b,c} = \mathrm T \left( \ensuremath{\mathbb Z} / p^b \right) / \mathrm T \left( \ensuremath{\mathbb Z} / p^c \right)$.
\end{lem}
\begin{proof} From lemma 6.1 of \cite{KT}, we have that since $K(c,c)$ is normal in $K(b,c)$, the covering $Y_{c,c} \twoheadrightarrow Y_{b,c}$ is a Galois covering, with Galois group $ K(b,c) / K(c,c) \cdot \left( \mathrm Z_{\mathrm G}(\ensuremath{\mathbb Q}) \cap K(b,c) \right)$.
Since $K(b,c)$ and $K(c,c)$ coincide away from $p$, we have that \[ K(c,c) \cdot \left( \mathrm Z_{\mathrm G}(\ensuremath{\mathbb Q}) \cap K(b,c) \right) = K(c,c) \cdot \left( \mathrm Z_{\mathrm G}(\ensuremath{\mathbb Q}) \cap I(b,c) \right). \]
Now $I(b,c)$ is a pro-$p$-group (being contained in the pro-$p$-Iwahori subgroup $I(1,1)$), and from our assumption that $p > \left| \mu \left( \mathrm Z_{\mathrm G} \right) \right|$ we have that the finite group $\mathrm Z_{\mathrm G}(\ensuremath{\mathbb Q})$ has no $p$-torsion.
Hence, the intersection $\mathrm Z_{\mathrm G}(\ensuremath{\mathbb Q}) \cap I(b,c)$ is trivial, and we obtain that the covering is Galois with Galois group \[ K(b,c) / K(c,c) \cong I(b,c) / I(c,c) \cong \mathrm T \left( \ensuremath{\mathbb Z} / p^b \right) / \mathrm T \left( \ensuremath{\mathbb Z} / p^c \right). \]
\end{proof}
We have an action of the cohomology of $T_{b,c}$ on the cohomology of the base, as explained by Venkatesh: by the universal property of the classifying space, the cover $Y_{c,c} \twoheadrightarrow Y_{b,c}$ corresponds to a map $\pi_{b,c}: Y_{b,c} \ensuremath{\longrightarrow} B T_{b,c}$ - taking cohomology yields a map $\pi_{b,c}^*: H^* \left( T_{b,c}, R \right) \ensuremath{\longrightarrow} H^* \left( Y_{b,c}, R \right)$ and the action on $H^* \left( Y_{b,c}, R \right)$ of a cohomology class in $H^* \left( T_{b,c}, R \right)$ is obtained by applying $\pi_{b,c}^*$ and then taking the cup product.
We call this action a \emph{derived diamond action}.

We can rephrase the same action in a more algebraic way. Let $R$ be a general coefficient ring and $C_{\bullet} (Y_{c,c}, R)$ the complex of singular chains with $R$-coefficients, whose homology computes the homology of $Y_{c,c}$. This complex has a natural action of $T_{b,c}$ via deck transformations.

Then it is standard that \[ H^* \left( Y_{b,c}, R \right) \cong H^* \left( \ensuremath{\mathrm{Hom}}_{R[T_{b,c}]} \left( C_{\bullet} (Y_{c,c}, R), R \right) \right), \] where on the left we have the singular cohomology of $Y_{b,c}$ and on the right the cohomology of the complex of $T_{b,c}$-equivariant homomorphisms.

In particular, the right-hand side has a natural action of $H^* \left( \ensuremath{\mathrm{Hom}}_{\mathcal D(R[T_{b,c}])}