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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.17956 |
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| _version_ | 1866910412933431296 |
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| author | Barchini, Leticia Trapa, Peter E. |
| author_facet | Barchini, Leticia Trapa, Peter E. |
| contents | Fix an integral semisimple element $λ$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $λ$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of $\mathrm{ad}(λ)$ in $\mathfrak{g}$. Under a natural hypothesis (which is always satisfied for classical subgroups of $\mathrm{GL}(n)$), we embed the closure of each $L$ orbit on $\mathfrak{g}(-1)$ into the closure of an orbit of a symmetric subgroup $K$ containing $L$ on a partial flag variety for $G$. We use this to relate the local intersection homology of the later orbit closures to the former orbit closures. This, in turn, relates multiplicity matrices for split real and $p$-adic groups. We also describe relationships between "microlocal packets'' of representations of these groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relating Real and P-adic Kazhdan-Lusztig Polynomials Barchini, Leticia Trapa, Peter E. Representation Theory 22E45 Fix an integral semisimple element $λ$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $λ$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of $\mathrm{ad}(λ)$ in $\mathfrak{g}$. Under a natural hypothesis (which is always satisfied for classical subgroups of $\mathrm{GL}(n)$), we embed the closure of each $L$ orbit on $\mathfrak{g}(-1)$ into the closure of an orbit of a symmetric subgroup $K$ containing $L$ on a partial flag variety for $G$. We use this to relate the local intersection homology of the later orbit closures to the former orbit closures. This, in turn, relates multiplicity matrices for split real and $p$-adic groups. We also describe relationships between "microlocal packets'' of representations of these groups. |
| title | Relating Real and P-adic Kazhdan-Lusztig Polynomials |
| topic | Representation Theory 22E45 |
| url | https://arxiv.org/abs/2402.17956 |