Coassembly is a homotopy limit map

We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.Comment: Accepted version. Several improvements from the referee, including a more elegant proof of Lemma 3.

Chern Classes and Compatible Power Operations in Inertial K-theory

Let [X/G] be a smooth Deligne-Mumford quotient stack. In a previous paper the authors constructed a class of exotic products called inertial products on K(I[X/G]), the Grothendieck group of vector bundles on the inertia stack I[X/G]. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When G is diagonalizable these give rise to an augmented $\lambda$-ring structure on inertial K-theory. One well-known inertial product is the virtual product. Our results show that for toric Deligne-Mumford stacks there is a $\lambda$-ring structure on inertial K-theory. As an example, we compute the $\lambda$-ring structure on the virtual K-theory of the weighted projective lines P(1,2) and P(1,3). We prove that after tensoring with C, the augmentation completion of this $\lambda$-ring is isomorphic as a $\lambda$-ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles $T^*P(1,2)$ and $T^*P(1,3)$, respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the Hyper-Kaehler Resolution Conjecture.Comment: Many improvements. Special thanks to the referee for helpful suggestions. To appear in Annals of K-Theory. arXiv admin note: text overlap with arXiv:1202.060

Topological K-theory of affine Hecke algebras

Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that $K_* (C*_r (R,q))$ does not depend on the parameter q. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras. Thus, for the computation of these K-groups it suffices to work out the case q=1. These algebras are considerably simpler than for q not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine $K_* (C*_r (R,q))$ for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-algebra of any classical p-adic group. For the computations in the case q=1 we study the more general situation of a finite group \Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product $C(M) \rtimes \Gamma$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.Comment: In the second version, paragraph 1.2 was moved to an appendix. Apart from that, only a few minor correction

The extension problem for graph C∗C^*-algebras

We give a complete $K$-theoretical description of when an extension of two simple graph $C^{*}$-algebras is again a graph $C^{*}$-algebra.Comment: Accepted version, to appear in Annals of K-theor