Timeline for Do all unitary representations weakly converge to zero at infinity?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2019 at 18:54 | answer | added | Giuseppe Negro | timeline score: 0 | |
| Dec 9, 2019 at 14:20 | comment | added | Abdelmalek Abdesselam | @Asaf: It might be worth expanding on your comment and post an answer. Nick gave a counterexample for the wanted property in the most general setting, whereas what you metioned gives positive results in the direction of what the OP seems to be after. | |
| Dec 5, 2019 at 23:25 | comment | added | Asaf | @AbdelmalekAbdesselam, mixing can be stated in terms of vanishing of matrix coefficients defined by the Coopman-von-Nuemann representation of the dynamical system (which is a fancy way of considering $L^{2}(X)$ with the action induced from the measure-preserving transformation). The Howe-Moore theorem implies that for say simple Lie group actions, the action is mixing (and further refinements by works of Harish-Chandra, Kazhdan and Oh quantify this decay). A suitable notion to consider here is joinings (by Furstenberg) and Kronecker factors, as those are the obstacles for mixing. | |
| Dec 5, 2019 at 19:29 | comment | added | Abdelmalek Abdesselam | The most studied situation from the ergodic theory point of view is when you restrict to a one-parameter subgroup in $G$ (notion of mixing for a flow or continuous time dynamical system) or when you look at $\rho_{g^n}$ for $n\in\mathbb{Z}$ (notion of mixing for a transformation or discrete time dynamical system). As I am not an expert in the area, I don't know if there are results for more complicated group actions with $G$ other than $\mathbb{R}$ or $\mathbb{Z}$. | |
| Dec 4, 2019 at 14:44 | comment | added | Giuseppe Negro | @AbdelmalekAbdesselam: Thank you, that is the most important piece of advice that I obtained from asking this question. I must admit that I had no idea that ergodic theory had something to do with these things. | |
| Dec 4, 2019 at 13:58 | comment | added | Abdelmalek Abdesselam | @GiuseppeNegro: It seems from your examples you are interested in representations sending $f$ to $f(\phi_g(x))$ times eventually some weight to have unitarity. Here $\phi_g$ is a transformation of the space $X$. In this case, your question belongs to ergodic theory. Look up, in particular, the notion of mixing. | |
| Dec 4, 2019 at 9:44 | vote | accept | Giuseppe Negro | ||
| Dec 4, 2019 at 9:43 | history | edited | Giuseppe Negro | CC BY-SA 4.0 |
added reference to Howe-Moore's theorem + ergodic theory
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| Dec 3, 2019 at 23:21 | history | edited | Giuseppe Negro | CC BY-SA 4.0 |
corrected misspellings, added SL2 case
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| Dec 3, 2019 at 23:00 | comment | added | Giuseppe Negro | @Asaf: Of course, the $SU(1, 1)$ example is exactly the action of $SL_2(\mathbb R)$ on the hyperbolic plane, up to a Cayley transform, like you say. Thank you for pointing me to the Howe-Moore theorem. | |
| Dec 3, 2019 at 22:28 | history | edited | YCor |
edited tags
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| Dec 3, 2019 at 21:59 | answer | added | Nik Weaver | timeline score: 4 | |
| Dec 3, 2019 at 21:05 | comment | added | Asaf | @NikWeaver, one may take easy modifications, say semisimple with compact factors etc. | |
| Dec 3, 2019 at 20:58 | comment | added | Nik Weaver | @Asaf he said that $G$ is noncompact. | |
| Dec 3, 2019 at 20:57 | comment | added | Nik Weaver | What about $G = \mathbb{R}$, $X= \mathbb{C}$, and $\rho_g f(z) = f(e^{2\pi i g}z)$, i.e., $\rho$ rotates by an angle of $g$? Then $\rho_g = {\rm id}$ whenever $g$ is an integer, so it can't go to zero at infinity. | |
| Dec 3, 2019 at 20:54 | comment | added | Asaf | One needs to obviously exclude the trivial representation. At any case, for simple Lie groups (and semisimple with appropriate additions) you have the famous Howe-Moore theorem (your example is essentially SL2) For compact groups, the answer is negative, representations are finite dimensional in this case and tend to have almost periodic nature. This can be easily seen in the representations of $SO(2)$ on itself, and taking say a trig. polynomial. | |
| Dec 3, 2019 at 20:37 | history | asked | Giuseppe Negro | CC BY-SA 4.0 |