Let $\pi$ be an admissible representation of a locally compact totally disconnected group. I have a technical problem about the proof of


*$\pi$ is irreducible if and only if its contragredient is so*


given in 2.15(c) of the '76 article of Bernstein and Zelevinsky. There $\pi$ is assumed to have a nontrivial proper subrepresentation $E\_1$, and it is asserted that the orthogonal complement of $E\_1$ be a nontrivial proper subrepresentation of the contragredient, whence the result. What I cannot figure out is the nontriviality of this orthogonal complement. We simply have to find a nonzero smooth functional which vanishes on $E\_1$; this shall follow from $E\_1\neq E$ (as properness of the orthogonal complement follows from $E\_1\neq 0$), but how?



A technical problem on the contragredient representation in the context of locally compact totally disconnected groups
This follows from two facts:


1. The complement $E\_1^\perp$ of $E\_1$ in $\tilde{E}$ is isomorphic to the contragredient of $E/E\_1$.
2. If $V$ is admissible and nonzero then $\tilde{V}$ is nonzero (and admissible). For if $\tilde{V}=0$ then $V = \tilde{\tilde{V}} = 0$.


