Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:16:52Z http://mathoverflow.net/feeds/question/94535 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94535/representation-of-double-cover-of-un-on-eigenspaces-of-harmonic-oscillator Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator Eric O. Korman 2012-04-19T14:44:00Z 2012-04-22T20:27:12Z <p>Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction of the metaplectic representation to $\tilde U(n)$ commutes with the Hamiltonian of the harmonic oscillator: $H = \sum_i (x_i^2 - \frac{\partial^2}{\partial x_i^2})$ and so decomposes as a direct sum of finite dimensional representations of $\tilde U(n)$ on the eigenspaces of $H$.</p> <p>I am looking for a reference that discusses these representations of $\tilde U(n)$. Specific things I would like to know are if each of these representations is irreducible and if any of them descend to representations of $U(n)$.</p> http://mathoverflow.net/questions/94535/representation-of-double-cover-of-un-on-eigenspaces-of-harmonic-oscillator/94578#94578 Answer by paul garrett for Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator paul garrett 2012-04-19T23:20:59Z 2012-04-22T20:27:12Z <p>As a place-holder answer: while, unfortunately I do not know a good reference offhand, I believe (based on a memory of having done the computation at least twice myself) that direct computation shows that the restriction to meta-U(n) does descend to U(n). </p> <p>Conceivably various expository papers of Steve Kudla include either-or-both discussion of the various pairs inside metaplectic groups (as your meta-version of U(1)xU(n)) and the sort of splitting property of interest.</p> <p>Edit: Oops! Indeed, as commented and amplified in another answer, the descent is just to SU(n).</p> http://mathoverflow.net/questions/94535/representation-of-double-cover-of-un-on-eigenspaces-of-harmonic-oscillator/94602#94602 Answer by Jon Yard for Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator Jon Yard 2012-04-20T06:29:22Z 2012-04-20T06:29:22Z <p>I believe the double cover you are looking for is just $\tilde{U}(n) = \tilde{U}(1) \times SU(n)$, where $$\tilde{U}(1) = \lbrace\exp(\phi T) : 0 \leq \phi &lt; 4\pi\rbrace$$ is the double cover of $U(1)$ and $T$ is a formal generator of its Lie algebra. This cover factors through the usual cover $U(1)\times SU(n) \to U(n)$ as $$(\exp(\phi T),U) \mapsto (e^{i\phi/2},U) \mapsto e^{i\phi/2} U.$$ So the problem reduces to classifying the $\tilde{U}(1)$ representations and the $SU(n)$ ones, and the answer is straightforward. First decompose<br> $$L^2(\mathbb{R}^n) \simeq \bigoplus_{N=0}^\infty V_N,$$ where $V_N$ is the eigenspace of $H$ with eigenvalue $2N+1$. Then $\tilde{U}(1)\times SU(n)$ should act irreducibly on each $V_N$ as $$(\exp(\phi T),U) \mapsto e^{iN\phi/2} \cdot \mathrm{Sym}^N U.$$</p> <p>It is easy to see this. The $\tilde{U}(1)$ action is the same as with the usual $L^2(\mathbb{R})$ case. You can easily derive the $SU(n)$ representations using the Fock space isomorphism of $L^2(\mathbb{R}^n)$ with a Hilbert space of analytic functions: $$L^2(\mathbb{R}^n) \simeq L^2_\mathrm{hol}(\mathbb{C}^n,\pi^{-n}e^{-||z||^2/2}dz).$$ Under this isomorphism, $V_N$ maps to the space of homogeneous polynomials of degree $N$, and $SU(n)$ acts on polynomials in the usual way as $(Uf)(z) = f(Uz)$.</p> <p>This is probably all contained in <a href="http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160140303/abstract" rel="nofollow">Bargmann's classic paper.</a></p> http://mathoverflow.net/questions/94535/representation-of-double-cover-of-un-on-eigenspaces-of-harmonic-oscillator/94851#94851 Answer by Eric O. Korman for Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator Eric O. Korman 2012-04-22T15:48:56Z 2012-04-22T15:48:56Z <p>I just found that Borel and Wallach's * Continuous cohomology, discrete subgroups, and representations of reductive groups* does a detailed analysis of this representation (which they call the oscillatory representation). It's on google books: <a href="http://books.google.com/books?id=_EZY9LhAxosC&amp;lpg=PA257&amp;ots=NIQffrLQCA&amp;dq=borel%20wallach%20cohomology%20lie&amp;pg=PA151#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.com/books?id=_EZY9LhAxosC&amp;lpg=PA257&amp;ots=NIQffrLQCA&amp;dq=borel%20wallach%20cohomology%20lie&amp;pg=PA151#v=onepage&amp;q&amp;f=false</a></p>