Arthur classified the discrete automorphic representations of symplectic and orthogonal groups over a number field by that of the general linear groups. In this classification, those that are not from endoscopic lifting correspond to pairs (ϕ,b), where ϕ is an irreducible unitary cuspidal automorphic representation of some general linear group and b is an integer. In this paper, we study the local components of these automorphic representations at a nonarchimedean place, and we give a complete description of them in terms of their Langlands parameters.

We develop a general procedure to study the combinatorial structure of Arthur packets for p-adic quasisplit Sp(N) and O(N) following the works of Mœglin. This allows us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.

We classify the Ext-quivers of hearts in the bounded derived category D(A_n) and the finite-dimensional derived category D(\Gamma_N A_n) of the Calabi-Yau-N Ginzburg algebra D(\Gamma_N A_n). This provides the classification for Buan-Thomas' colored quiver for higher clusters of A-type. We also give explicit combinatorial constructions from a binary tree with n+2 leaves to a torsion pair in mod k\overrightarrow{A_n} and a cluster tilting set in the corresponding cluster category, for the straight oriented A-type quiver \overrightarrow{A_n}. As an application, we show that the orientation of the n-dimensional ssociahedron induced by poset structure of binary trees coincides with the orientation induced by poset structure of torsion pairs in mod k\overrightarrow{A_n} (under the correspondence above).