Representations of $GL_n(D)$ near the identity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917799102775296 |
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| author | Guy, Henniart Marie-France, Vignéras |
| author_facet | Guy, Henniart Marie-France, Vignéras |
| contents | For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $π$ of $G=GL_n(D)$, we show that for small enough compact open pro-$p$ subgroup $K$ of $G$, the restriction of $π$ to $K$ is the same as that of a virtual representation $\sum c_π(λ) Ind_{P_λ}^G 1$, where the sum is over partitions $λ$ of $n$ and $P_λ$ a parabolic subgroup of $G$ associated to $λ$. When $K$ is a Moy-Prasad subgroup of $G$ we determine from the $c_π(λ)$ a polynomial $P_{π,K}$ of degree $d(π)$ independent of the choice of $K$, such that for large enough integers $j$ the dimension of the points of $π$ fixed under the congruence subgroup $K_j$ of $K$ is $P_{π,K}(q^j)$ where $q$ is the cardinality of the residue field of $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06581 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Representations of $GL_n(D)$ near the identity Guy, Henniart Marie-France, Vignéras Representation Theory Number Theory 20G05, 11F70 For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $π$ of $G=GL_n(D)$, we show that for small enough compact open pro-$p$ subgroup $K$ of $G$, the restriction of $π$ to $K$ is the same as that of a virtual representation $\sum c_π(λ) Ind_{P_λ}^G 1$, where the sum is over partitions $λ$ of $n$ and $P_λ$ a parabolic subgroup of $G$ associated to $λ$. When $K$ is a Moy-Prasad subgroup of $G$ we determine from the $c_π(λ)$ a polynomial $P_{π,K}$ of degree $d(π)$ independent of the choice of $K$, such that for large enough integers $j$ the dimension of the points of $π$ fixed under the congruence subgroup $K_j$ of $K$ is $P_{π,K}(q^j)$ where $q$ is the cardinality of the residue field of $D$. |
| title | Representations of $GL_n(D)$ near the identity |
| topic | Representation Theory Number Theory 20G05, 11F70 |
| url | https://arxiv.org/abs/2305.06581 |