On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918310790037504 |
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| author | Frahm, Jan van Haastrecht, Robin Weiske, Clemens Zhang, Genkai |
| author_facet | Frahm, Jan van Haastrecht, Robin Weiske, Clemens Zhang, Genkai |
| contents | We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20344 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup Frahm, Jan van Haastrecht, Robin Weiske, Clemens Zhang, Genkai Representation Theory 17B15, 17B60, 22D30, 43A80, 43A85 We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two. |
| title | On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup |
| topic | Representation Theory 17B15, 17B60, 22D30, 43A80, 43A85 |
| url | https://arxiv.org/abs/2601.20344 |