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.

Let <i>G</i> be a semisimple adjoint group over <b>C</b> and <i></i> be the De ConciniProcesi completion of <i>G</i>. In this paper, we define a Lagrangian subvariety of the cotangent bundle of <i></i> such that the singular support of any character sheaf on <i></i> is contained in .

We give a new description of the set {m Adm}(\mu) of admissible alcoves as an intersection of certain``obtuse cones''of alcoves, and we show this description may be given by imposing conditions vertexwise. We use this to prove the vertexwise admissibility conjecture of Pappas-Rapoport-Smithling. The same idea gives simple proofs of two ingredients used in the proof of the Kottwitz-Rapoport conjecture on existence of crystals with additional structure.

We study a class of perverse sheaves on some spherical varieties which include the strata of the De ConciniProcesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.

In this paper, we study some relation between the cocenter H(G) of the Hecke algebra H (G) of a connected reductive group G over a nonarchimedean local field and the cocenter H(M) of its Levi subgroups M. Given any Newton component of H(G), we construct the induction map i from the corresponding Newton component of H(M) to it. We show that this map is an isomorphism. This leads to the BernsteinLusztig type presentation of the cocenter H(G), which generalizes the work [11] on the affine Hecke algebras. We also show that the map i we constructed is adjoint to the Jacquet functor.