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Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ the corresponding induced representation of $G$. When does there exist a (continuous in a proper sense) involutive homomorphism of algebras $\varphi:B(X)\to B(Y)$ such that $$ \varphi\circ\pi=\rho\circ\iota \quad ? $$ This seems to be true at least when $H$ is finite and $\pi$ is irreducible. Is it possible that this is always true?

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    $\begingroup$ I tried to input a diagram, but they seem to do not work here. Is it possible to use diagrams in MO? $\endgroup$ – Sergei Akbarov Sep 15 '14 at 18:00
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This is false in many cases where $G$ is finite. Let $\rho\circ \iota$ and $\pi$ denote the corresponding maps of group algebras $\mathbb{C}[G]$. The equation above can only hold if any element killed by $\pi$ must also be killed by $\rho\circ \iota$, that is $\mathrm{ker}(\pi)\subset \mathrm{ker}(\rho\circ \iota)$. If $\pi$ is irreducible, then $\mathrm{ker}(\pi)$ is the sum of all the other matrix algebra summands in the group algebra, so $\mathrm{ker}(\rho\circ \iota)$ can only contain $\mathrm{ker}(\pi)$ if $\rho\circ \iota$ is just a bunch of copies of the same irrep. Of course, this could happen, but it doesn't have to; for example, let $\pi$ be trivial, and $H$ any non-normal subgroup. The restriction of the permutation rep of $G/H$ to $H$ is non-trivial, so it can't factor through the trivial rep.

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  • $\begingroup$ Ben, can this be true when $H$ is normal? $\endgroup$ – Sergei Akbarov Sep 18 '14 at 16:44
  • $\begingroup$ When $H$ is normal, then $\rho\circ \iota$ is a bunch of copies of $\pi$, so there the desired factoring is true. $\endgroup$ – Ben Webster Sep 19 '14 at 2:34
  • $\begingroup$ My $H$ is normal. Ben, I need a reference. I must refer to a textbook or to a paper in my text. $\endgroup$ – Sergei Akbarov Sep 19 '14 at 13:21
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In addition to @BenWebster's points, my own reaction would be to see that the hoped-for property is fighting against any/all of several natural characterizations of "induction", e.g., as right (or left) adjoint to restriction... That is, some "type" violation is hoped-for, which, especially in light of counter-examples, is hard to see the reason to believe-in.

Thus, I'd ask if there's a context in which something of this sort arises, so that perhaps what might really suffice/work/solve-the-problem is different enough so as to avoid the "un-naturality"/type-violation in the first place, and avoid the tangible counter-example schema mentioned by Ben W... ?

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