On the Hecke Module of $\text{GL}_n(k[[z]])\backslash \text{GL}_n(k((z)))/\text{GL}_n(k((z^2)))$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910504148008960 |
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| author | Jin, Yuhui |
| author_facet | Jin, Yuhui |
| contents | Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i
\end{pmatrix} (i>1)\}$ if $char(k) \neq 2$ and $k$ is a finite field. These cosets form a (spherical) Hecke module $\mathcal{H}(G,H,K)$ over the (spherical) Hecke algebra $\mathcal{H}(G,K)$ of double cosets in $K\backslash G/H$, where $K=\text{GL}_m(k[[z]])$ and $H=\text{GL}_m(k((z^2)))$ and $G=\text{GL}_m(k((z)))$. Similarly to Hall polynomial $h_{λ,ν}^μ$ from the Hecke algebra $\mathcal{H}(G,K)$, coefficients $h_{λ,ν}^μ$ arise from the Hecke module. We will provide a closed formula for $h_{λ,ν}^μ$, under some restrictions over ${λ,ν,μ}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19017 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hecke Module of $\text{GL}_n(k[[z]])\backslash \text{GL}_n(k((z)))/\text{GL}_n(k((z^2)))$ Jin, Yuhui Combinatorics Representation Theory Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if $char(k) \neq 2$ and $k$ is a finite field. These cosets form a (spherical) Hecke module $\mathcal{H}(G,H,K)$ over the (spherical) Hecke algebra $\mathcal{H}(G,K)$ of double cosets in $K\backslash G/H$, where $K=\text{GL}_m(k[[z]])$ and $H=\text{GL}_m(k((z^2)))$ and $G=\text{GL}_m(k((z)))$. Similarly to Hall polynomial $h_{λ,ν}^μ$ from the Hecke algebra $\mathcal{H}(G,K)$, coefficients $h_{λ,ν}^μ$ arise from the Hecke module. We will provide a closed formula for $h_{λ,ν}^μ$, under some restrictions over ${λ,ν,μ}$. |
| title | On the Hecke Module of $\text{GL}_n(k[[z]])\backslash \text{GL}_n(k((z)))/\text{GL}_n(k((z^2)))$ |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2406.19017 |