In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.

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 paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.

Let G ⊆ \tilde{G} be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of G should be the restriction of those of \tilde{G}. Motivated by this, we hope to construct the L-packets of \tilde{G} from those of G. The primary example in our mind is when G = Sp(2n), whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and \tilde{G} = GSp(2n). As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of \tilde{G} from those of G.

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category D(Γ_S) associated to the surface and the corresponding Seidel–Thomas braid group of D(Γ_S) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).