Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard torus of all diagonal matrices) i.e. there is some $q$ such that if $V^{\mu}_{\mathbb{C}}$ is the $\mathbb{C}$-points of the irreducible algebraic representation with highest weight $\mu$, then $$ H^{q}(\mathfrak{gl}_{n}(\mathbb{R}), SO_{n}(\mathbb{R}); \Pi_{\infty} \otimes V^{\mu}_{\mathbb{C}})\otimes\Pi_{f} \neq 0. $$ I've read in a few places that we can extract the Hodge-Tate weights of the associated Galois representations directly from the weight $\mu$ but I haven't been able to find the actual recipe, can anyone enlighten me?
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$\begingroup$ Try Theorem 2.1.1 of Barnet-Lamb/Gee/Geraghty/Taylor 2014, wwwf.imperial.ac.uk/~tsg/Index_files/potentialautomorphy.pdf. (This is assuming $\Pi$ polarized, but actually the same recipe is valid more generally.) $\endgroup$– David LoefflerCommented Feb 28, 2019 at 7:21
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