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 G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let ar G be the wonderful compactification of G. For a fixed pair (B,B  ) of opposite Borel subgroups of G, we look at intersections of Lusztigs G-stable pieces and the B  B-orbits in ar G , as well as intersections of BB-orbits and B  B  -orbits in ar G . We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of .

We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (the Density theorem) and that the image is the space of good forms (the trace PaleyWiener theorem).

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 .

In this paper, we study the <i></i>-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that <i></i>-ordinary locus is a union of certain maximal EkedahlKottwitzOortRapoport strata introduced in [12] and we give criteria on the density of the <i></i>-ordinary locus.