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Parahoric Ext-algebras of a p-adic special linear group Stockton, Jacob
Abstract
Let F be a finite extension of ℚₚ, k the algebraic closure of 𝔽ₚr for some r, and G = SL₂(F). In the context of the mod-p Local Langlands correspondence, it is natural to study the pro-p Iwahori Hecke algebra H := End(k[G/I],k[G/I]) attached to the pro-p Iwahori subgroup I ≤ G. One reason is that when F = ℚₚ, there is an equivalence between the category of H-modules and the category of smooth k-representations of G generated by their I-fixed vectors. Unfortunately, this equivalence fails when F is a proper extension of ℚₚ. We overcome this obstacle somewhat by passing to the derived setting. When F is a proper extension and I is a torsion free group, it was shown by Schneider that we can obtain an equivalence between the derived category of smooth representations and a certain derived category associated to H. Relatively little is known about this equivalence. In understanding more, we can consider the cohomology algebra E* := Ext*(k[G/I],k[G/I]). The goal of this thesis is to study the related algebra EP* := Ext*(k[G/P],k[G/P]) when P is taken to be either the Iwahori subgroup J of G or the maximal compact subgroup K = SL₂(ℤₚ), both of which contain I. We are able to give explicit descriptions of these algebras, including the full product. Surprisingly, we deduce that EK* is commutative (not graded commutative) and that it is isomorphic to the centre of EJ*.
Item Metadata
Title |
Parahoric Ext-algebras of a p-adic special linear group
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Creator | |
Supervisor | |
Publisher |
University of British Columbia
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Date Issued |
2022
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Description |
Let F be a finite extension of ℚₚ, k the algebraic closure of 𝔽ₚr for some r, and G = SL₂(F). In the context of the mod-p Local Langlands correspondence, it is natural to study the pro-p Iwahori Hecke algebra H := End(k[G/I],k[G/I]) attached to the pro-p Iwahori subgroup I ≤ G. One reason is that when F = ℚₚ, there is an equivalence between the category of H-modules and the category of smooth k-representations of G generated by their I-fixed vectors. Unfortunately, this equivalence fails when F is a proper extension of ℚₚ.
We overcome this obstacle somewhat by passing to the derived setting. When F is a proper extension and I is a torsion free group, it was shown by Schneider that we can obtain an equivalence between the derived category of smooth representations and a certain derived category associated to H. Relatively little is known about this equivalence. In understanding more, we can consider the cohomology algebra E* := Ext*(k[G/I],k[G/I]). The goal of this thesis is to study the related algebra EP* := Ext*(k[G/P],k[G/P]) when P is taken to be either the Iwahori subgroup J of G or the maximal compact subgroup K = SL₂(ℤₚ), both of which contain I. We are able to give explicit descriptions of these algebras, including the full product. Surprisingly, we deduce that EK* is commutative (not graded commutative) and that it is isomorphic to the centre of EJ*.
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Genre | |
Type | |
Language |
eng
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Date Available |
2022-04-22
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0412992
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2022-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International