Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}\_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}\_{\geq 2}$.


Then $V$ is uniquely determined by a pair of smooth characters $\alpha,\beta$ of $\mathbb{Q}\_p^{\times}$. Breuil and Berger gives in [their paper](https://www.math.u-psud.fr/%7Ebreuil/PUBLICATIONS/cristallinGL2.pdf) (see page $43$, equations $22$ and $23$) a $GL\_2(\mathbb{Q}\_p)$ equivariant Intertwining operator $I^{sm}$ between the smooth representations (locally constant functions with values in a finite extension $E$ of $\mathbb{Q}\_p$)


$$Ind\_B^G (\beta \otimes \alpha |\cdot|^{-1})^{sm} \rightarrow Ind\_B^G (\alpha \otimes \beta|\cdot|^{-1})^{sm}$$


In terms of locally constant functions on $\mathbb{Q}\_p$, it is given by


$$I^{sm}(h)(z)=\int\_{\mathbb{Q}\_p}(\alpha\beta^{-1})(x)|x|^{-1}h(z+x^{-1})dx$$


I do not understand why the image of $I^{sm}$ consists of smooth (locally constant) functions with values in $E$. What happens when I take $h=1$?



Smooth intertwining operators
As Breuil say in the paper you cite that these are *algebraic* Intertwining and so do not bother much about convergence.


