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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 (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 $h=1$?

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  • $\begingroup$ What do you mean by $h=1$? Unless you are assuming that $\beta$ is trivial and $\alpha = | \cdot |$, I do not see how the constant function on $G$ with value $1$ could belong to $Ind^G_B \left( \beta \otimes \alpha | \cdot |^{-1} \right)^{sm}$. $\endgroup$
    – user94041
    Commented Jul 18, 2017 at 14:02
  • $\begingroup$ I think that the character $\chi=\beta \otimes \alpha|\cdot|^{-1}$ is the tensor product of two unramified characters (see equation 24 and equation 25 in page 38 of the pdf "math.u-psud.fr/~breuil/PUBLICATIONS/Hangzhou.pdf". By Iwasawa decomposition $Ind_{B(\mathbb{Q}_p)}^{G(\mathbb{Q}_p)}(\chi)\cong Ind_{B(\mathbb{Z}_p)}^{G(\mathbb{Z}_p)}(\chi)$ and since $\chi $ is the tensor product of two unramified characters $\chi$ is trivial on $B(\mathbb{Z}_p)$ which implies (as a vector space) $Ind_{B(\mathbb{Q}_p)}^{G(\mathbb{Q}_p)}(\chi) \cong Ind_{B(\mathbb{Z}_p)}^{G(\mathbb{Z}_p)}(1)$ $\endgroup$
    – MathStudent
    Commented Jul 18, 2017 at 14:54

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

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