On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O
Abstract For a complex reflection group W with reflection representation $$\mathfrak {h}$$ h...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131391 |
| _version_ | 1826199814335889408 |
|---|---|
| author | Losev, Ivan Shelley-Abrahamson, Seth |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Losev, Ivan Shelley-Abrahamson, Seth |
| author_sort | Losev, Ivan |
| collection | MIT |
| description | Abstract
For a complex reflection group W with reflection representation
$$\mathfrak {h}$$
h
, we define and study a natural filtration by Serre subcategories of the category
$$\mathcal {O}_c(W, \mathfrak {h})$$
O
c
(
W
,
h
)
of representations of the rational Cherednik algebra
$$H_c(W, \mathfrak {h})$$
H
c
(
W
,
h
)
. This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in
$$\mathcal {O}_c(W, \mathfrak {h})$$
O
c
(
W
,
h
)
of given support. We apply these techniques to count the number of irreducible representations in
$$\mathcal {O}_c(W, \mathfrak {h})$$
O
c
(
W
,
h
)
of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of
$$\mathcal {O}_c(W, \mathfrak {h})$$
O
c
(
W
,
h
)
for exceptional Coxeter groups W in many new cases. |
| first_indexed | 2024-09-23T11:26:11Z |
| format | Article |
| id | mit-1721.1/131391 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:26:11Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1313912024-01-02T19:35:51Z On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O Losev, Ivan Shelley-Abrahamson, Seth Massachusetts Institute of Technology. Department of Mathematics Abstract For a complex reflection group W with reflection representation $$\mathfrak {h}$$ h , we define and study a natural filtration by Serre subcategories of the category $$\mathcal {O}_c(W, \mathfrak {h})$$ O c ( W , h ) of representations of the rational Cherednik algebra $$H_c(W, \mathfrak {h})$$ H c ( W , h ) . This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in $$\mathcal {O}_c(W, \mathfrak {h})$$ O c ( W , h ) of given support. We apply these techniques to count the number of irreducible representations in $$\mathcal {O}_c(W, \mathfrak {h})$$ O c ( W , h ) of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of $$\mathcal {O}_c(W, \mathfrak {h})$$ O c ( W , h ) for exceptional Coxeter groups W in many new cases. 2021-09-20T17:16:53Z 2021-09-20T17:16:53Z 2018-01-23 2020-09-24T21:10:34Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131391 en https://doi.org/10.1007/s00029-018-0390-6 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer International Publishing AG, part of Springer Nature application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Losev, Ivan Shelley-Abrahamson, Seth On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title | On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title_full | On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title_fullStr | On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title_full_unstemmed | On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title_short | On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$ O |
| title_sort | on refined filtration by supports for rational cherednik categories mathcal o o |
| url | https://hdl.handle.net/1721.1/131391 |
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