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Register a new userDerived Langlands IV:Notes on ${mathcal M}_{c}(G)$-induced representations
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Victor Snaith
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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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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.
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.
We introduce a novel co-learning paradigm for manifolds naturally equipped with a group action, motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common base manifold, which provides multiple views for the geometry of the underlying manifold. The consistency across these fibre bundles provide a common base for performing unsupervised manifold co-learning through the redundancy created artificially across irreducible representations of the transformation group. We demonstrate the efficacy of the proposed algorithmic paradigm through drastically improved robust nearest neighbor search and community detection on rotation-invariant cryo-electron microscopy image analysis.
In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $widehat{mathfrak{gl}_{m|n}}(mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $mu$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $mu$ is nonzero.
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