On certain cohomology groups attached to ${\mathfrak{p}}^{\mathrm{\infty }}$-towers of quaternionic Hilbert modular varieties

For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the ${\mathfrak{p}}^{\mathrm{\infty }}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $PG{L}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $\pi$ of $D^{\ast }(\mathbb{A})$ this yields a natural construction of infinitesimal deformations of the Galois representation attached to $\pi$.