$p$-adic Banach space representations of $SL_2({\mathbb Q}_p)$
Let $F/{\mathbb Q}_p$ be a finite extension. This paper is about continuous admissible $p$-adic Banach space representations $\Pi$ of $G = GL_n(F)$ and their restriction to $H = SL_n(F)$. We first show that the restriction of any such absolutely irreducible $G$-representation decomposes as a finite direct sum of irreducible $H$-representations. Then we consider the restriction to $SL_2({\mathbb Q}_p)$ of certain $p$-adic unitary Banach space representations of $GL_2({\mathbb Q}_p)$, relying on the work of Colmez and Colmez-Dospinescu-Paskūnas. It is shown that if $\Pi$ is associated to an irreducible trianguline de Rham representation with distinct Hodge-Tate weights, then $\Pi|_H$ decomposes non-trivially if and only if this is true for the subrepresentation $(\Pi^{\rm lalg})|_H$ of locally algebraic vectors.