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Register a new userDerived Langlands III: PSH algebras and their numerical invariants
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Authors
Victor Snaith
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This sequel to Derived Langlands II studies some PSH algebras and their numerical invariants, which generalise the epsilon factors of the local Langlands Programme. It also describes a conjectural Hopf algebra structure on the sum of the hyperHecke algebras of products of the general linear groups over a $p$-adic local field or a finite field.
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This is a sequel to the authors book Derived Langlands which introduced an embedding of the category of admissible representations of a locally p-adic group in to the derived category of the monomial category of the group. This article gives a reformulation in terms of the hyperHecke algebra and relates this viewpoint to a number of topics, including the Bernstein centre of the category of admissible representations.
This is the fifth article in the Derived Langlands series which consists of one monograph and four articles. In this article I describe the Hopf algebra and Positive Selfadjoint Hopfalgebra (PSH) aspects to classification of a number of new classes of presentations and admissibility which have appeared earlier in the series. The paper begins with a very estensive. partly hypothetical, of the synthesis of the entire series. Many of the proofs and ideas in this series are intended to be suggestive rather than the finished definitive product for extenuating circumstances explained therein.
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
This is Part IV of a thematic series currently consisting of a monograph and four essays. This essay examines the form of induced representations of locally p-adic Lie groups G which is appropriate for the abelian category of ${mathcal M}_{c}(G)$-admissible representations. In my non-expert manner, I prove the analogue of Jacquets Theorem in this category. The final section consists of observations and questions related to this and other concepts introduced in the course of this series.
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