We prove a conjecture of Miemietz and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case in Varagnolo and Vasserot (in press) [15].

We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and are related to local exterior square L-functions.

Let A be an abelian category and B be the Happel-Reiten-Smalo tilt of A with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between A and B, which is compatible with the inclusion of B into the derived category of A. As applications, we obtain new derived equivalences related to splitting torsion pairs, TTF-triples and two-term silting subcategories, respectively. We prove that for the realization functor of any bounded t-structure, its denseness implies its fully-faithfulness.

Following our previous work \cite{HLY}, we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a rooted cluster algebra are corresponded one-by-one to the regular $\mathcal D$-classes of the semigroup consisting of partial seed endomorphisms of the initial seed. Moreover, for a rooted cluster algebra from a Riemannian surface, they are also corresponded to the isomorphism classes of the so-called paunched surfaces.

We prove that any derived equivalence between triangular algebras is standard, that is, it is isomorphic to the derived tensor functor given by a two-sided tilting complex.