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the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a representation.

It is known that, if $n=3$, then $\phi$ is unitarily equivalent to one of the following:

  • $(det)^p\otimes \Phi_N$

  • $(det)^p\otimes \Phi_N^c$

for some $p\in\mathbb Z$ and $N$ natural number, where $\Phi_N$ is the representation on the space of homogeneous complex polynomials of degree $N$ in $3$ many variables given by $(\Phi_N(u)P)\vec z=P(u^{-1}\vec z)$ and $\Phi_N^c$ is the contragradient i.e., $\Phi_N^c(u)=\Phi_N(u^{-1})^t$, $t$ be the transpose operation.

see http://www.math.utah.edu/~ptrapa/Knapp-Trapa.pdf Lecture 4 Exercise 2

Is such a characterization true for $n>3$? Is there a reference for the proof of this? Or, otherwise, the only known characterization is via highest-weight?

Thanks,

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    $\begingroup$ This is absolutely not true already for $n=3$, already for the adjoint representation. The true statement is that the irrep with highest weight ${a,b,c}$ is the largest component of $\Phi_{a-b}\otimes \Phi_{b-c}^c \otimes (det)^c$. That does generalize, with the general factor being $Sym^N(Alt^k \mathbb C^n)$. For $k=n-1$ one has $Alt^k(\mathbb C^n) \cong det \otimes (\mathbb C^n)^*$, which allowed you to use $\Phi^c_N$ instead. $\endgroup$ Apr 5, 2015 at 18:39
  • $\begingroup$ See: "Unitary representations of real reductive groups" by Jeffrey Adams, Marc van Leeuwen, Peter Trapa and David A. Vogan Jr, arXiv:1212.2192. $\endgroup$ Apr 5, 2015 at 18:40
  • $\begingroup$ While the question may be basic for people who work with Lie group representations, I don't quite understand all the votes to close. (Disclaimer: I have met and corresponded with the OP.) $\endgroup$
    – Yemon Choi
    Apr 6, 2015 at 1:07
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    $\begingroup$ Or has MO got to the stage where specialists in one area can no longer ask questions which they know would be easy for specialists in some other area? $\endgroup$
    – Yemon Choi
    Apr 6, 2015 at 1:09
  • $\begingroup$ Thank you, Yemon. I have to admit I am not at all an expert on Lie gps... Allen, I did not understand your answer. You are saying that any irreducible rep is unitarily equivalent to something of the form $\Phi_{a-b}\otimes\Phi_{b-d}^c\otimes(det)^d$, in case $n=3$, where $a,b,d$ is the highest weight. But when you talk about "the general factor", which part do you mean? Can you please also define Alt groups? In any case, I think that maybe the highest weight characterization is the one that better suits the context, at the end. $\endgroup$ Apr 6, 2015 at 16:55

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