Mathematics

We show that the stable module ∞ -category of a finite group G decomposes in three different ways as a limit of the stable module ∞ -categories of certain subgroups of G . Analogously to Dwyer's terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The key step in the proof is extending the stable module ∞ -category to be defined for any G -space, then showing that this extension only depends on the S -equivariant homotopy type of a G -space. The methods used are not specific to the stable module ∞ -category, so may also be applicable in other settings where an ∞ -category depends functorially on G .

We study the deformation of the Howe dual pairs (U(n),u(1,1)) and (U(n),u(2|1)) to the context of a rational Cherednik algebra H 1,c (G,E) associated with a real reflection group G acting on a real vector space E of even dimension. For the case where E is two-dimensional and G is a dihedral group, we provide complete descriptions for the deformed pair and the relevant joint-decomposition of the standard module or its tensor product with a spinor space.

In this paper we give an alternative construction of a certain class of Deformed Double Current Algebras. These algebras are deformations of U(End( k r )[x,y]) and they were initially defined and studied by N.Guay in his papers. Here we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank.

Let Bu n G (X) be the moduli stack of G -torsors on a smooth projective curve X for a reductive group G . We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on Bu n G (X) . This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on DMod(Bu n G (X)) . We also prove a "second adjointness" result for these enhanced functors.

We give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is syzygy filtration method introduced in \cite{sen2019}. In particular, we give characterization of the finitistic dimension one Nakayama algebras.

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.

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 M c (G) -admissible representations. In my non-expert manner, I prove the analogue of Jacquet's 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.

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.

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth G -representations over k , G a p -adic reductive group and k a field of characteristic p , is equivalent to the derived category of a certain differential graded k -algebra H ∙ G , whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra side, which we construct in this paper. This relies on the theory of six-functor formalisms for differential graded categories developed by O.\ Schnürer. We also discuss the adjoint functors of derived parabolic induction.