Conformally invariant differential operators on Heisenberg groups and minimal representations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916219792130048 |
|---|---|
| author | Frahm, Jan |
| author_facet | Frahm, Jan |
| contents | For a simple real Lie group $G$ with Heisenberg parabolic subgroup $P$, we study the corresponding degenerate principal series representations. For a certain induction parameter the kernel of the conformally invariant system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation which turns out to be the minimal representation. To study this subrepresentation, we take the Heisenberg group Fourier transform in the non-compact picture and show that it yields a new realization of the minimal representation on a space of $L^2$-functions. The Lie algebra action is given by differential operators of order $\leq3$ and we find explicit formulas for the functions constituting the lowest $K$-type.
These $L^2$-models were previously known for the groups $\operatorname{SO}(n,n)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$ by Kazhdan and Savin, for the group $G_{2(2)}$ by Gelfand, and for the group $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, using different methods. Our new approach provides a uniform and systematic treatment of these cases and also constructs new $L^2$-models for $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$ for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups $\widetilde{\operatorname{SO}}(p,q)$ with either $p\geq q=3$ or $p,q\geq4$ and $p+q$ even.
As a byproduct of our construction, we find an explicit formula for the group action of a non-trivial Weyl group element that, together with the simple action of a parabolic subgroup, generates $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_05952 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Conformally invariant differential operators on Heisenberg groups and minimal representations Frahm, Jan Representation Theory Primary 22E45, Secondary 22E46, 35R03, 43A30 For a simple real Lie group $G$ with Heisenberg parabolic subgroup $P$, we study the corresponding degenerate principal series representations. For a certain induction parameter the kernel of the conformally invariant system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation which turns out to be the minimal representation. To study this subrepresentation, we take the Heisenberg group Fourier transform in the non-compact picture and show that it yields a new realization of the minimal representation on a space of $L^2$-functions. The Lie algebra action is given by differential operators of order $\leq3$ and we find explicit formulas for the functions constituting the lowest $K$-type. These $L^2$-models were previously known for the groups $\operatorname{SO}(n,n)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$ by Kazhdan and Savin, for the group $G_{2(2)}$ by Gelfand, and for the group $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ by Torasso, using different methods. Our new approach provides a uniform and systematic treatment of these cases and also constructs new $L^2$-models for $E_{6(2)}$, $E_{7(-5)}$ and $E_{8(-24)}$ for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups $\widetilde{\operatorname{SO}}(p,q)$ with either $p\geq q=3$ or $p,q\geq4$ and $p+q$ even. As a byproduct of our construction, we find an explicit formula for the group action of a non-trivial Weyl group element that, together with the simple action of a parabolic subgroup, generates $G$. |
| title | Conformally invariant differential operators on Heisenberg groups and minimal representations |
| topic | Representation Theory Primary 22E45, Secondary 22E46, 35R03, 43A30 |
| url | https://arxiv.org/abs/2012.05952 |