One of the most quoted and used facts in publications on the representation theory of \(p\)-adic groups is Casselman’s square-integrability criterion from his 1974 unpublished notes. In the paper under review, the authors consider a so-called \(p\)-adic symmetric space; this is a space of the form \(G(F)/H(F)\) where \(H\) is the algebraic subgroup fixed under an involutive \(F\)-automorphism of a connected, reductive group \(G\) over a local field \(F\). In particular, the authors are interested in representations of \(G(F)\) which can be realized on the space of square-integrable functions on \(G(F)/H(F)\). Consider a finitely generated admissible representation \((\pi,V)\) of \(G(F)\) and a linear form \(\lambda \in (V^{\ast})^{H(F)}\). Look at the generalized \(H(F)\)-matrix coefficients \(\varphi_{\lambda,v} : g \mapsto <\lambda,\pi(g^{-1})v>\) for \(v \in V\) and \(g \in G(F)\). Nontrivial realizations of \(\pi\) on the spaces of functions on \(G(F)/H(F)\) is via these generalized matrix coefficients as \(\lambda\) varies over \(H(F)\)-invariant non-zero linear forms as above.
In earlier work, the authors had studied representations whose \(H(F)\)-matrix coefficients are compactly supported modulo \(Z(G(F))H(F)\). In the paper under review, the authors study representations for which the generalized \(H(F)\)-matrix coefficients are square integrable modulo \(Z(G(F))H(F)\) – this property is christened as \(\pi\) being \(H(F)\)-square integrable with respect to \(\lambda\). The main theorem is a criterion for \(H(F)\)-square-integrability which is analogous to Casselman’s criterion for usual square integrability. In fact, Casselman’s criterion is recovered from the main theorem applied to \((G(F) \times G(F))/G(F)\) where the involution is the interchange of the two copies. Interestingly, it is unknown yet whether \(\pi\) being \(H(F)\)-square integrable with respect to some \(\lambda \in (V^*)^{H(F)}\) implies the same with respect to every such \(\lambda\).