Let \(F\) be a non-Archimedean local field or a finite field; \(n\) a positive integer, \(k\) equals 1 or 2. Set \(G:=\operatorname{GL}_{n+k}(F)\), \(M=\operatorname{GL}_n(F) \times\operatorname{GL}_n(F)\) embedded diagonally in \(G, U\) and \(P = M U\) the upper-triangular unipotent and parabolic subgroup with Levi factor \(M\). Denote by \( j = j^G_M\) the functor of co-invariants with respect to \(U\) from the category \(C(G)\) of smooth G-modules to \(C(M )\). For irreducible representations \(\pi\) of \(G\) and \(\rho\) of \(M\), Theorem A asserts that \(\dim \operatorname{Hom}_M (j(\pi), \rho) \leq 1\). It is equivalent to Theorem B: let \(G \times M\) act on \(G/U\) by \((g, m)([g\prime ]) = [gg\prime m^{-1} ]\) (this action is well-defined as \(M\) normalizes \(U\)); the space \(C^\infty_c (G/U )\) of compactly supported measures which are locally constant with respect to the \(G\)-action is then a representation of \(G\times M\). This representation is multiplicity free, i.e., for any irreducible representation \(\pi\times \rho\) of \(G \times M\), the dimension of \(\operatorname{Hom}_{G\times M} (C^\infty_c (G/U ), \pi \times \rho)\) is \(0\) or \(1\). By Frobenius reciprocity this is equivalent to Theorem C: embed \(P\) in \(G \times M\) by \(p \mapsto (p, pr_M (p))\), where \(pr : P \rightarrow M\) is the natural projection, then \((G \times M, P )\) is a Gelfand pair, i.e., \(\dim \operatorname{Hom}_P (\pi \times \rho, \mathbb C) \leq 1\). Theorem A with \(k = 1\) implies Theorem D: let \(\tau\) be an irreducible representation of \(H =\operatorname{GL}_n(F)\), \(H\) embedded at the top-left corner of \(G\), then \(\dim \operatorname{Hom}_H (j(\pi)|H, \tau) \leq 1\).
Theorem B is proven using the Gelfand-Kazhdan method from a suitable geometric statement. This interesting and well-written work discusses additional open problems.