Loop group actions on categories and Whittaker invariants
The present paper is divided in three parts. In the first one, we develop the theory of D-modules on ind-schemes of pro-finite type. This allows to define D-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category. In the second part, we construct...
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| Format: | Journal article |
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Elsevier
2017
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| _version_ | 1826267374613954560 |
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| author | Beraldo, D |
| author_facet | Beraldo, D |
| author_sort | Beraldo, D |
| collection | OXFORD |
| description | The present paper is divided in three parts. In the first one, we develop the theory of D-modules on ind-schemes of pro-finite type. This allows to define D-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category. In the second part, we construct the functors of Whittaker invariants and Whittaker coinvariants, which take as input a DG category acted on by G((t)), the loop group of a reductive group G. Roughly speaking, the Whittaker invariant category of C is the full subcategory C N((t)),χ ⊆C consisting of objects that are N((t))-invariant against a fixed non-degenerate character χ:N((t))→G a of conductor zero. (Here N is the maximal unipotent subgroup of G.) The Whittaker coinvariant category C N((t)),χ is defined by a dual construction. In the third part, we construct a functor Θ:C N((t)),χ →C N((t)),χ , which depends on a choice of dimension theory for G((t)). We conjecture this functor to be an equivalence. After developing the Fourier–Deligne transform for Tate vector spaces, we prove this conjecture for G=GL n . We show that both Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme!) of G((t)). |
| first_indexed | 2024-03-06T20:53:13Z |
| format | Journal article |
| id | oxford-uuid:384bbe64-1c91-4597-851f-233ef6216f05 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:53:13Z |
| publishDate | 2017 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:384bbe64-1c91-4597-851f-233ef6216f052022-03-26T13:49:13ZLoop group actions on categories and Whittaker invariantsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:384bbe64-1c91-4597-851f-233ef6216f05Symplectic Elements at OxfordElsevier2017Beraldo, DThe present paper is divided in three parts. In the first one, we develop the theory of D-modules on ind-schemes of pro-finite type. This allows to define D-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category. In the second part, we construct the functors of Whittaker invariants and Whittaker coinvariants, which take as input a DG category acted on by G((t)), the loop group of a reductive group G. Roughly speaking, the Whittaker invariant category of C is the full subcategory C N((t)),χ ⊆C consisting of objects that are N((t))-invariant against a fixed non-degenerate character χ:N((t))→G a of conductor zero. (Here N is the maximal unipotent subgroup of G.) The Whittaker coinvariant category C N((t)),χ is defined by a dual construction. In the third part, we construct a functor Θ:C N((t)),χ →C N((t)),χ , which depends on a choice of dimension theory for G((t)). We conjecture this functor to be an equivalence. After developing the Fourier–Deligne transform for Tate vector spaces, we prove this conjecture for G=GL n . We show that both Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme!) of G((t)). |
| spellingShingle | Beraldo, D Loop group actions on categories and Whittaker invariants |
| title | Loop group actions on categories and Whittaker invariants |
| title_full | Loop group actions on categories and Whittaker invariants |
| title_fullStr | Loop group actions on categories and Whittaker invariants |
| title_full_unstemmed | Loop group actions on categories and Whittaker invariants |
| title_short | Loop group actions on categories and Whittaker invariants |
| title_sort | loop group actions on categories and whittaker invariants |
| work_keys_str_mv | AT beraldod loopgroupactionsoncategoriesandwhittakerinvariants |