Higher localization and higher branching laws
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929547988959232 |
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| author | Li, Wen-Wei |
| author_facet | Li, Wen-Wei |
| contents | For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\mathrm{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08994 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher localization and higher branching laws Li, Wen-Wei Representation Theory 32C38 (Primary) 22E46 (Secondary) For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\mathrm{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included. |
| title | Higher localization and higher branching laws |
| topic | Representation Theory 32C38 (Primary) 22E46 (Secondary) |
| url | https://arxiv.org/abs/2207.08994 |