L-modules are mixed
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908949140209664 |
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| author | Saper, Leslie |
| author_facet | Saper, Leslie |
| contents | Let X be the locally symmetric space associated to a reductive $\mathbb Q$-group G and an arithmetic subgroup $Γ$. An L-module M is a combinatorial model of a constructible complex of sheaves on $\widehat X$, the reductive Borel-Serre compactification of X whose strata $X_P$ are indexed by $Γ$-conjugacy classes of parabolic $\mathbb Q$-subgroups P of G. We show that any L-module M is "mixed" in the sense it is an iterated mapping cone of maps to or from shifted weighted cohomology L-modules on strata $X_P$ of $\widehat X$ with coefficients in V, an irreducible regular $L_P$-module. These weighted cohomology "building blocks" are indexed (up to multiplicity) by V in the weak micro-support of M which is a computable local invariant. As an application we prove that the intersection cohomology of $\widehat X$ is isomorphic to the weighted cohomology of $\widehat X$, at least excluding $\mathbb Q$-types D, E, and F. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | L-modules are mixed Saper, Leslie Representation Theory Number Theory 11F75 (Primary) 22E40, 32S60 (Secondary) Let X be the locally symmetric space associated to a reductive $\mathbb Q$-group G and an arithmetic subgroup $Γ$. An L-module M is a combinatorial model of a constructible complex of sheaves on $\widehat X$, the reductive Borel-Serre compactification of X whose strata $X_P$ are indexed by $Γ$-conjugacy classes of parabolic $\mathbb Q$-subgroups P of G. We show that any L-module M is "mixed" in the sense it is an iterated mapping cone of maps to or from shifted weighted cohomology L-modules on strata $X_P$ of $\widehat X$ with coefficients in V, an irreducible regular $L_P$-module. These weighted cohomology "building blocks" are indexed (up to multiplicity) by V in the weak micro-support of M which is a computable local invariant. As an application we prove that the intersection cohomology of $\widehat X$ is isomorphic to the weighted cohomology of $\widehat X$, at least excluding $\mathbb Q$-types D, E, and F. |
| title | L-modules are mixed |
| topic | Representation Theory Number Theory 11F75 (Primary) 22E40, 32S60 (Secondary) |
| url | https://arxiv.org/abs/2604.07719 |