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In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] state Zelevinksy's classification theorem for representations of $GL_n$ in terms of affine Hecke algebras, despite the fact the paper of Zelevinksy they cite works purely with representations. I can't seem to find a source for this, and to me it is not obvious.

To be more precise, I'll first fix some notation. Let $F$ be a non-Archimedean local field, $G=GL_n(F)$, with Levi subgroup $M$, and respective Iwahori subgroups $\mathcal{I}_G$ and $\mathcal{I}_M$. We write the "full" Hecke algebras as $\mathcal{H}(G)$ and $\mathcal{H}(M)$, and the subalgebras of $\mathcal{I}_G$ (resp. $\mathcal{I}_M$) bi-invariant functions as $\mathcal{H}(G, \mathcal{I}_G)$ (resp. $\mathcal{H}(M, \mathcal{I}_M)$). Both [1] and [2] offer presentations of affine Hecke algebras $\mathbb{H}_G$ and $\mathbb{H}_M$, and we fix the specific isomorphisms $\mathcal{H}(G, \mathcal{I}_M)\to \mathbb{H}_G$ and $\mathcal{H}(M, \mathcal{I}_M)\to\mathbb{H}_M$, which I believe are originally due to Iwahori and Matsumoto. Recall that there's a standard equivalence between $G$-reps with non-zero $\mathcal{I}_G$-fixed vector, and $\mathcal{H}(G, \mathcal{I}_G)$-modules. Via the isomorphism(s) of Iwahori-Matsumoto, we obtain an equivalence of $\mathbb{H}_G$-modules.

The theorem that seems to be made implicit use of in [1] and [2] is: Let $\sigma$ be a smooth representation of $M$ with non-zero $\mathcal{I}_M$-fixed vector, and let $\bar{\sigma}$ be the $\mathbb{H}_M$-module obtained through the above equivalences. Then $\mathbb{H}_G\otimes_{\mathbb{H}_M}\bar{\sigma}$ is isomorphic as an $\mathbb{H}_G$-module to the $\mathbb{H}_G$-module obtained from the normalized parabolic induction $\iota_{M\subset P}^G(\sigma)$.

While it may not be terribly deep to try and prove this, I fear aspects of it are tedious, and I'm hoping it has been carried out somewhere in detail.

The closest thing I could find is Theorem 1.4 of [3], which compares compact induction from closed subgroups to tensor products over the full Hecke algebras $\mathcal{H}(G)$ and $\mathcal{H}(M)$. Presumably, replacing these algebras with the $\mathcal{I}_G$ and $\mathcal{I}_M$ bi-invariant functions still leaves you with an $\mathcal{H}(M, \mathcal{I}_M)$-action on $\mathcal{H}(G, \mathcal{I}_G)$, and that somehow forces compact induction to become normalized parabolic induction. Then, one would need to transport everything via the isomorphisms with $\mathbb{H}_G$ and $\mathbb{H}_M$, which I think is rather tedious. For example, it is far from clear that the action of $\mathcal{H}(M)$ on $\mathcal{H}(G)$ described in [3], corresponds to the action of $\mathbb{H}_M$ on $\mathbb{H}_G$ (implied) in [1].

[1] D. Barbasch, C. Ciubotaru, Ladder representation of $GL(n, \mathbb{Q}_p)$, Representations of reductive groups, pages 117 -- 138

[2] B. Leclerc, M. Nazarov and J.-Y. Thibon, Induced representations of affine Hecke algebras and canonical bases of quantum groups

[3] P. Cartier, Representations of $\mathfrak{p}$-adic groups: A survey

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The point of course is that equivalences of categories preserve adjoints, but there is one subtlety as "induction" is a right adjoint whereas the tensor product is a left adjoint. The subtlety is that when working with the Iwahori-Hecke algebra, you are really inducing directly from the Levi; the radical plays pretty much no role and so you have your choice of "the" parabolic or the opposite parabolic, and so Bernstein's second adjointness appears.

This is nicely explained in Section 3.1 of Hellmann's paper "On the derived category of the Iwahori-Hecke algebra" (arXiv), especially diagrams 3.5 and 3.6 and the sentence following them.

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  • $\begingroup$ Sure it preserves why adjoints, but why does it commute with these specific functors? What Hellman calls the Iwahori-Hecke algebra is not the same as the thing that term traditionally describes, being the convolution algebra of smooth compactly supported Iwahori-bi-invariant C-valued functions, though it is Morita equivalent. Even for Hellman's algebra, there isn't a proof, he only declares that there exists some embedding (which he doesn't describe) making the diagrams in (3.5) commute. $\endgroup$
    – Kristaps John Balodis
    Commented Jul 4, 2023 at 17:17
  • $\begingroup$ The algebra Hellmann calls $H(G,I)$ is isomorphic, not just Morita equivalent, to the endomorphism algebra of the progenerator—this is true for any type. Diagram 3.6 explains that the equivalences of categories intertwines the Jacquet functor and the functor ``forget." On one hand, parabolic induction for the opposite parabolic is left adjoint to the Jacquet functor in 3.6. On the other hand, the tensor product is left adjoint to the functor "forget." $\endgroup$
    – Stefan Dawydiak
    Commented Jul 4, 2023 at 17:54
  • $\begingroup$ What, explicitly, is the embedding $\mathcal{H}_M\hookrightarrow\mathcal{H}_G$ that is used to define the "forget" functor? $\endgroup$
    – Kristaps John Balodis
    Commented Jul 4, 2023 at 18:18
  • $\begingroup$ See the very explicit treatment in this survey of Sollveld's: arxiv.org/abs/2009.03007 especially section 3.1. $\endgroup$
    – Stefan Dawydiak
    Commented Jul 4, 2023 at 18:43
  • $\begingroup$ Solleveld's algebras, though isomorphic, seem to be given by yet another presentation. Of course, such an embedding in Sollevled's case would induce an embedding $\mathcal{H}_M\hookrightarrow \mathcal{H}_G$, but the embedding, and thus the $\mathcal{H}_M$-module structure on $\mathcal{H}_G$ would depend on the choice of isomorphism(s). So in order to prove Hellman's (3.6) commutes using Solleveld, one would still need to take the "right" isomorphisms of algebras, and then check the action of the induced embedding $\mathcal{H}_M\to\mathcal{H}_G$ commutes in the way Hellman says it does. $\endgroup$
    – Kristaps John Balodis
    Commented Jul 5, 2023 at 0:49
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The answer to my question is essentially the content of Proposition 2.1.2 of "On the Iwahori-Matsumoto involution and applications" by Chris Jantzen: http://www.numdam.org/item/ASENS_1995_4_28_5_527_0/

A proof in the setting of Hellmann's paper was communicated to me through email by professor Hellmann. Perhaps I will return here and write the full details in the future, but in short, one considers diagrams (3.5) and (3.6) mentioned in the question, but with the opposite equivalence of categories. Then, conisder a free resolution of a given $\mathcal{H}_T$-module $M$. Apply the functors in either direction around the square to said resolution. One will end up with two projective resolutions which are isomorphic, hence their 0th cohomology, and thus the original functors are isomorphic.

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