This paper develops a cohomology theory for locally analytic representations of \(p\)-adic Lie groups on non-Archimedean locally convex vector spaces.
Let \(\mathbb Q_p \subseteq L \subseteq K\), where \(L/\mathbb Q_p\) is a finite extension and \(K\) is spherically complete. Let \(H\) be a finite-dimensional locally \(L\)-analytic group. Let \(D(H)=D(H,K)\) denote the \(K\)-algebra of locally analytic \(K\)-valued distributions on \(H\). Denote by \({\mathcal M}_H\) the category of complete Hausdorff locally convex \(K\)-vector spaces with the structure of a separately continuous \(D(H)\)-module. For each pair of objects \(V\) and \(W\) of \({\mathcal M}_H\), the author defines relative extension groups \({\mathcal E}xt^q_H(V,W)\), torsion groups \({\mathcal T}or^q_H(V,W)\), locally analytic cohomology groups \(H^q_{\text{an}}(H,V)\) and locally analytic homology groups \(H_q^{\text{an}}(H,V)\). The author gives versions of Pontrjagin duality, Shapiro’s lemma and a Hochschild-Serre spectral sequence.
The first application is the following definition of a supercuspidal locally analytic representation. Let \(G\) be a connected reductive \(L\)-group. A topologically irreducible locally analytic \(G\)-representation \(V\) is called supercuspidal if for all proper parabolic subgroups \(P\) of \(G\) and all \(q \geq 0\) we have \(H_q^{\text{an}}(N,V)=0\), where \(N\) denotes the unipotent radical of \(P\).
In addition, the author studies extensions between locally analytic principal series representations.