Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909620422836224 |
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| author | Zhang, Genkai |
| author_facet | Zhang, Genkai |
| contents | Let $G/K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation $π_ν=\text{Ind}_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K/L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K/L_0\to K/L$ and a circle bundle $K/L_0\to K/L_1$ over a compact Hermitian symmetric space $K/L_1$. We compute the Lie algebra $\mathfrak g$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups Zhang, Genkai Representation Theory Let $G/K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation $π_ν=\text{Ind}_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K/L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K/L_0\to K/L$ and a circle bundle $K/L_0\to K/L_1$ over a compact Hermitian symmetric space $K/L_1$. We compute the Lie algebra $\mathfrak g$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations. |
| title | Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2309.05980 |