In this paper we use a homological approach to obtain upper bounds for a few homological invariants of FI_G-modules V. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of V under actions of non-invertible morphisms in the category respectively.

In this paper we describe several characterizations of basic finite-dimensional k-algebras A stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra property. We also discuss whether for a given preorder ≼, F(≼Δ), the category of A-modules with ≼Δ-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.

We give a generalization of the classical tilting theorem of Brenner and Butler. We show that for a 2-term silting complex P in the bounded homotopy category K^b(proj A) of finitely generated projective modules of a finite dimensional algebra A, the algebra B=End_{K^b(proj A)}(P) admits a 2-term silting complex Q with the following properties: (i) The endomorphism algebra of Q in K^b(proj B) is a factor algebra of A, and (ii) there are induced torsion pairs in mod A and mod B, such that we obtain natural equivalences induced by Hom- and Ext-functors. Moreover, we show how the Auslander–Reiten theory of mod B can be described in terms of the Auslander–Reiten theory of mod A.

In this article we propose a geometric description of Arthur packets for p-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note, we investigate the subcategory of C(G) consisting of those modules whose composition factors all have highest weights linked to the Steinberg weight. This subcategory is denoted ST and called the Steinberg component. We give an explicit equivalence between ST and C(G) and we derive some consequences. In particular, our result allows us to relate the Frobenius contracting functor to the projection functor from C(G) onto ST.