Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913743835758592 |
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| author | Droschl, Johannes |
| author_facet | Droschl, Johannes |
| contents | In this paper we prove a conjecture of Kudla and Rallis. Let $χ$ be a unitary character, $s\in \mathbb{C}$ and $W$ a symplectic vector space over a non-archimedean field with symmetry group $G(W)$. Denote by $I(χ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(χ,s)$ along the natural embedding $G(W)\times G(W)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(χ,s)$ of $G(W)\times G(W)$. Let $π$ be an irreducible smooth complex representation of $G(W)$. We then prove \[\dim _\mathbb{C}\mathrm{Hom}_{G(W)\times G(W)}(I_{W,W}(χ,s),π\otimes π^\lor)=1.\] We also give analogous statements for $W$ orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local Theta correspondence for symplectic-orthogonal and unitary dual pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11280 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series Droschl, Johannes Representation Theory 22E46, 22E50 In this paper we prove a conjecture of Kudla and Rallis. Let $χ$ be a unitary character, $s\in \mathbb{C}$ and $W$ a symplectic vector space over a non-archimedean field with symmetry group $G(W)$. Denote by $I(χ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(χ,s)$ along the natural embedding $G(W)\times G(W)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(χ,s)$ of $G(W)\times G(W)$. Let $π$ be an irreducible smooth complex representation of $G(W)$. We then prove \[\dim _\mathbb{C}\mathrm{Hom}_{G(W)\times G(W)}(I_{W,W}(χ,s),π\otimes π^\lor)=1.\] We also give analogous statements for $W$ orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local Theta correspondence for symplectic-orthogonal and unitary dual pairs. |
| title | Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series |
| topic | Representation Theory 22E46, 22E50 |
| url | https://arxiv.org/abs/2310.11280 |