Timeline for Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2015 at 21:10 | answer | added | LI Jialun | timeline score: 0 | |
| Dec 14, 2013 at 20:09 | comment | added | Asaf | See Akshay's introductory article about $SL_{2}(\mathbb{R})$ here - math.nyu.edu/~venkatesh/research/ml.pdf . The rep. theory of $SL_{2}(\mathbb{C})$ is actually easier because of the absence of the discrete series. For full proofs and formulas - see Knapp's book, I think he does this case by brute force computations, hence you don't need to learn all the related alg. groups and Lie theory. | |
| Dec 14, 2013 at 20:03 | comment | added | Asaf | First of all, it might be better to divide from the left by $\Gamma$, and from the right by $K$, to resolve some difficulties (I've used the second way around, because I'm doing dynamics...). Anyways, decomposing an admissible representation with respect to $K$ will give you various $K$-isotypic types, for which the the one which corresponds to the trivial character will appear in the locally symmetric space (if you happen to have $0$ weight vector in your representation). | |
| Dec 14, 2013 at 19:57 | comment | added | user7894 | How exactly is the spectral decomposition (discrete and continuous) of the laplacian on $L^2(K\G/\Gamma)$ related to the decomposition of the representation ? | |
| Dec 14, 2013 at 19:17 | comment | added | user7894 | Indeed, I assume the dimension of the limit set is greater than $1$ | |
| Dec 14, 2013 at 18:52 | comment | added | Asaf | One way to get it should be by taking universal covers, complexifiying and taking real form... One should probably move to connected components as well... | |
| Dec 14, 2013 at 18:51 | comment | added | Asaf | I think it's one of the exceptional isomorphism (I'm not an expert on alg. grps, but the fact that $SL_{2}(\mathbb{C})/SL_{2}(\mathbb{Z}[i])$ is isomorphic to $SO(3,1)(\mathbb{R})/SO(3,1)(\mathbb{R})$ is a bit exceptional, just like the $SL_2(\mathbb{R})$ vs $SO(2,1)(\mathbb{Z})$ case. | |
| Dec 14, 2013 at 17:42 | comment | added | Marc Palm | @Asaf The isomorphism can also be deduced from the Iwasawa decomposition, right? | |
| Dec 14, 2013 at 17:12 | answer | added | Marc Palm | timeline score: 2 | |
| Dec 14, 2013 at 16:28 | comment | added | Asaf | Do you have some assumptions over the Hausdorff dimension of the limit set? Those thing make dramatic impact on say $\lambda_{0}$. About your second question, the double coset space $K\backslash G / \Gamma$ is isomorphic to $\mathbb{H}^{3}/ \Gamma$ by the isomorphism of $SL_{2}(\mathbb{C})$ to $SO(3,1)(\mathbb{R})$. | |
| Dec 14, 2013 at 10:28 | review | First posts | |||
| Dec 14, 2013 at 10:29 | |||||
| Dec 14, 2013 at 10:10 | history | asked | user7894 | CC BY-SA 3.0 |