I am wondering about a definition of a direct sum in page $31$ of [this paper](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0010437X11005525) by R. Liu.


I am following the notations in page $31$ of the above paper. 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 $\mathbb{Q}\_p^{\times}$. The (Langland) corresponding representation of $GL\_2(\mathbb{Q}\_p)$ is then of the form $B(V)=B(\alpha)/L(\alpha)$ where $B(\alpha)=(Ind\_{B(\mathbb{Q}\_p)}^{GL\_2(\mathbb{Q}\_p)}\alpha \otimes x^{k-2}\beta |x|^{-1})^{\mathcal{C}^{-val(\alpha(p))}}$ and $L(\alpha)$ is a certain closed subspace of $B(\alpha);$ the subgroup $B(\mathbb{Q}\_p)$ is the upper triangular Borel subgroup of $GL\_2(\mathbb{Q}\_p)$. Denote $\pi(\alpha)$ the locally algebraic representation $(Ind\_{B(\mathbb{Q}\_p)}^{GL\_2(\mathbb{Q}\_p)}\alpha \otimes x^{k-2}\beta |x|^{-1})^{lalg}$ and $A(\alpha)$ the locally analytic principal series $(Ind\_{B(\mathbb{Q}\_p)}^{GL\_2(\mathbb{Q}\_p)}\alpha \otimes x^{k-2}\beta |x|^{-1})^{an}$. Set $\pi(\beta)$ and $A(\beta)$ by replacing $\alpha$ by $\beta$. 


With all the notations as above, the author says that there exists natural continuous $GL(2)$ equivariant map from $A(\alpha) \oplus\_{\pi(\beta)} A(\beta)$ to the locally analytic vectors of $B(V)$.
I would like to know the definition of this direct sum $A(\alpha) \oplus\_{\pi(\beta)} A(\beta)$. Thank you for your explanation.



A definition of a (amalgamated) direct sum
The key point here is that $\pi(\alpha)$ and $\pi(\beta)$ are isomorphic representations: both of them are the algebraic representation $Sym^{k-2}$ tensored with the smooth representation $Ind\_B^G (\alpha \otimes \beta |\cdot|^{-1})^{sm} = Ind\_B^G (\beta \otimes \alpha |\cdot|^{-1})^{sm}$. (This last isomorphism is at the beginning of section 1.3 of the paper you link.)


If that is the case, then $A(\alpha) \oplus\_{\pi(\alpha)} A(\beta)$ surely denotes the quotient $A(\alpha) \oplus A(\beta) / \{x, \iota(x) : x \in \pi(\alpha)\}$ where $\iota$ is the isomorphism between $\pi(\alpha)$ and $\pi(\beta)$. 


(The usual word for this is "amalgamated direct sum"; "amalgamate" means roughly "stick together".)


