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Let $L$ be a finite extension of $\mathbf{Q}_p$ and $G$ the group of $L$-points of a split connected reductive group $\mathbf{G}$ over $L$, $T$ the $L$-points of a split maximal torus in $\mathbf{G}$, and $B$ the $L$-points of a Borel containing (the underlying torus of) $T$. Let $E$ be another finite extension of $\mathbf{Q}_p$ containing $L$ and all its Galois conjugates.

Question: If $\chi$ is a smooth $E$-valued character of $T$ and $\psi$ is the (restriction to $T$ of the) highest weight of some irreducible algebraic representation of $\mathbf{G}_E$ over $E$, what is the correct definition of the locally algebraic parabolic induction $\mathrm{Ind}_B^G(\chi\psi)_{\mathrm{lalg}}$?

An algebraic function on $G$ should be an element of the space of locally analytic functions on $G$ in the image of the ``restriction map" from the affine ring of $\mathbf{G}_E$ (restriction meaning we restrict to $G\subseteq\mathbf{G}(E)=\mathbf{G}_E(E)$). I guess a locally algebraic function should be an element of the space of locally analytic functions on $G$ in the image of the natural map from the tensor product of the space of algebraic functions and the space of smooth functions (although I can't see that this map is injective if we don't restrict to compactly supported smooth functions) on $G$. So a reasonable definition for the locally algebraic induction would be the set of locally algebraic functions $f:G\rightarrow E$ such that $f(bg)=\chi(b)\psi(b)f(g)$ (where we regard $\psi$ as a character of $B$ via the quotient map to $T$) with $G$ acting by right translation of the argument. Alternatively we could define the locally algebraic induction to be the space of $\mathbf{G}$-locally algebraic vectors in the locally analytic induction of $\chi\psi$. Or one could just tensor the algebraic induction of $\psi$ with the smooth induction of $\chi$. I don't know whether or not these definitions are equivalent in general.

There is a definition for $\mathrm{GL}_{2/\mathbf{Q}_p}$ in Emerton's ``A local-global compatibility conjecture in the $p$-adic Langlands programme for $\mathrm{GL}_{2/\mathbf{Q}}$," where $\chi$ is allowed to be a general locally analytic character. This definition doesn't quite coincide with the definitions given above in general as far as I can tell, although for smooth $\chi$ I think it coincides with the last two (the definition is near the top of page 8 of the paper, which can be found here:

Aside from this, I haven't seen a general definition in the literature on $p$-adic representations of $p$-adic groups.

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