We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and q-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

We study categories associated to a decorated marked surface S_△, which is obtained from an unpunctured marked surface S by adding a set of decorating points. For any triangulation T of S_△, let Γ_T be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in S_△ and the reachable rigid indecomposables in the perfect derived category perΓ_T. This is the dual of the bijection, between simple closed arcs in S_△ and reachable spherical objects in the 3-Calabi-Yau category D_{fd}(ΓT), constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient perΓ_T/D_{fd}(ΓT) that defines the generalized cluster categories corresponds to the forgetful map S_△→S (forgetting the decorating points) in a suitable sense.

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we prove a local trace formula for the Ginzburg-Rallis model. By applying this trace formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel work of [Wan15] in which we proved the geometric side of the trace formula.

We study the cluster combinatorics of d-cluster tilting objects in d-cluster categories. Using mutations of maximal rigid objects in d-cluster categories, which are defined in a similar way to mutations for d-cluster tilting objects, we prove the equivalences between d-cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of d+1 triangles of d-cluster tilting objects in [O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (1) (2008) 117–168], we prove that any almost complete d-cluster tilting object has exactly d+1 complements, compute the extension groups between these complements, and study the middle terms of these d+1 triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established for d-cluster categories in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572–618]. They are applied to the Fomin–Reading generalized cluster complexes of finite root systems defined and studied in [S. Fomin, N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005) 2709–2757; H. Thomas, Defining an m-cluster category, J. Algebra 318 (2007) 37–46; K. Baur, R. Marsh, A geometric description of m-cluster categories, Trans. Amer. Math. Soc. 360 (2008) 5789–5803; K. Baur, R. Marsh, A geometric description of the m-cluster categories of type D_n, preprint, arXiv:math.RT/0610512; see also Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm011], and to that of infinite root systems [B. Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008) 25–54].

A decorated surface S is an oriented surface, with or without boundary, and a finite set {s 1,..., s n} of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over [special characters omitted].