Special unipotent representations of real classical groups: counting and reduction
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917881579569152 |
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| author | Barbasch, Dan Ma, Jia-Jun Sun, Binyong Zhu, Chen-Bo |
| author_facet | Barbasch, Dan Ma, Jia-Jun Sun, Binyong Zhu, Chen-Bo |
| contents | Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a given bound of the complex associated variety. When $G$ is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of $G$ attached to $\check{\mathcal O}$, in the sense of Arthur and Barbasch-Vogan. Here $\check{\mathcal O}$ is a nilpotent adjoint orbit in the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We give a precise count for the number of special unipotent representations of $G$ attached to $\check{ \mathcal O}$. We also reduce the problem of constructing special unipotent representations attached to $\check{\mathcal O}$ to the case when $\check{\mathcal O}$ is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_05266 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Special unipotent representations of real classical groups: counting and reduction Barbasch, Dan Ma, Jia-Jun Sun, Binyong Zhu, Chen-Bo Representation Theory 22E46, 22E47 Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a given bound of the complex associated variety. When $G$ is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of $G$ attached to $\check{\mathcal O}$, in the sense of Arthur and Barbasch-Vogan. Here $\check{\mathcal O}$ is a nilpotent adjoint orbit in the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We give a precise count for the number of special unipotent representations of $G$ attached to $\check{ \mathcal O}$. We also reduce the problem of constructing special unipotent representations attached to $\check{\mathcal O}$ to the case when $\check{\mathcal O}$ is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups. |
| title | Special unipotent representations of real classical groups: counting and reduction |
| topic | Representation Theory 22E46, 22E47 |
| url | https://arxiv.org/abs/2205.05266 |