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.

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 will use the combinatorics of the G-stable pieces to describe the closure relation of the partition of partial flag varieties in [L3, section 4].

We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group G, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on G as the category of quasi-coherent sheaves on an explicitly defined derived stack G^.

Let $G$ be a simple simply connected algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. G. Lusztig and Z. Yun, in a recent series of articles, studied the classification of simple $G_0$-equivariant perverse sheaves on the nilpotent cone of $\mathfrak{g}_i$ for $i\in \mathbf{Z}/m$, where $G_0$ is the exponentiation of the degree zero piece $\mathfrak{g}_0$. They proved a decomposition of the equivariant derived category of $\ell$-adic sheaves on the nilpotent cone of $\mathfrak{g}_i$ into blocks, each generated by a certain cuspidal local system via the procedure of {\itshape spiral inductions}. We prove a conjecture of them, stating that there is a bijection between 1) the set of simple perverse sheaves in a fixed block and 2) the set of simple modules of a block of a (trigonometric) degenerate double affine Hecke algebra (dDAHA). This is a dDAHA analogue of the Deligne--Langlands correspondence for affine Hecke algebras proven by Kazhdan--Lusztig. Our results generalise a previous work of E. Vasserot, where the perverse sheaves in the principal block were considered.