# Unitary representation of finite-dimensional unitary group

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,

• 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. Apr 5, 2015 at 18:39
• See: "Unitary representations of real reductive groups" by Jeffrey Adams, Marc van Leeuwen, Peter Trapa and David A. Vogan Jr, arXiv:1212.2192. Apr 5, 2015 at 18:40
• 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.) Apr 6, 2015 at 1:07
• 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? Apr 6, 2015 at 1:09
• 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. Apr 6, 2015 at 16:55