On $L^1$-$L^2$ dichotomy for flat symmetric spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929602357624832 |
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| author | Gupta, Sanjeev Kumar Spronk, Nico |
| author_facet | Gupta, Sanjeev Kumar Spronk, Nico |
| contents | For rank 1 flat symmetric spaces, continuous orbital measures admit absolutely continuous convolution squares, except for Cartan type AI. Hence $L^1$-$L^2$ dichotomy for these spaces holds true in parallel to the compact and non-compact rank 1 symmetric spaces. We also study $L^1$-$L^2$ dichotomy for flat symmetric spaces of ranks $p=2,3$ of type AIII, i.e.\ associated with $SU(p,q)/S(U(p)\times U(q))$ where $q\geq p$. For continuous orbital measures given by regular points $L^1$-$L^2$ dichotomy holds. We study such measures given by certain singular points when $p=2$, and show that $L^1$-$L^2$ dichotomy fails. This is the first time such results are observed for any type of symmetric spaces of rank 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $L^1$-$L^2$ dichotomy for flat symmetric spaces Gupta, Sanjeev Kumar Spronk, Nico Representation Theory 43A90, 43A85, 22E30 For rank 1 flat symmetric spaces, continuous orbital measures admit absolutely continuous convolution squares, except for Cartan type AI. Hence $L^1$-$L^2$ dichotomy for these spaces holds true in parallel to the compact and non-compact rank 1 symmetric spaces. We also study $L^1$-$L^2$ dichotomy for flat symmetric spaces of ranks $p=2,3$ of type AIII, i.e.\ associated with $SU(p,q)/S(U(p)\times U(q))$ where $q\geq p$. For continuous orbital measures given by regular points $L^1$-$L^2$ dichotomy holds. We study such measures given by certain singular points when $p=2$, and show that $L^1$-$L^2$ dichotomy fails. This is the first time such results are observed for any type of symmetric spaces of rank 2. |
| title | On $L^1$-$L^2$ dichotomy for flat symmetric spaces |
| topic | Representation Theory 43A90, 43A85, 22E30 |
| url | https://arxiv.org/abs/2411.15564 |