In [W], Witten considered the indices of elliptic operators on the free loop space CM of a spin manifold M. In particular the index of the formal signature operator on the loop space is exactly the elliptic genus of Landweber-Stong-Ochanine [LS],[O]. Motivated by physics, Witten made the conjectures about the rigidity of these elliptic operators which say that their 51-equivariant indices on M are independent of g G Sl. See [L] for the early history of the subject.

Let $\Lambda$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $\Lambda$ as algebra automorphisms. Under the assumption that $\Lambda$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that the skew group algebra $\Lambda G$ and $\Lambda$ have the same finitistic dimension and the same strong global dimension if the action of $S$ on $E$ is free. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce that $\Lambda G$ is piecewise hereditary if and only if $S$ acts freely on $E$ and $\Lambda$ is piecewise hereditary as well.

We define an analogue of the Casimir element for a graded affine Hecke algebra $$ \mathbb{H} $$ , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology$H$^{$D$}($X$) of an $$ \mathbb{H} $$ -module$X$, and show that$H$^{$D$}($X$) carries a representation of a canonical double cover of the Weyl group $$ \widetilde{W} $$ . Our main result shows that the $$ \widetilde{W} $$ -structure on the Dirac cohomology of an irreducible $$ \mathbb{H} $$ -module$X$determines the central character of$X$in a precise way. This can be interpreted as$p$-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of $$ \mathbb{H} $$ .

Topological cyclic homology is a refinement of Connes–Tsygan’s cyclic homology which was introduced by Bökstedt–Hsiang–Madsen in 1993 as an approximation to algebraic K-theory. There is a trace map from algebraic K-theory to topological cyclic homology, and a theorem of Dundas–Goodwillie–McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing K-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the ∞-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum X with S1-action (in the most naive sense) together with S1-equivariant maps φp:X→XtCp for all primes p. Here, XtCp=cofib(Nm:XhCp→XhCp) is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps φp:X→XtCp in the example of topological Hochschild homology, we introduce and study Tate-diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular, we prove a version of the Segal conjecture for the Tate-diagonals and relate these Frobenius homomorphisms to power operations.

We give two homological proofs of the Noetherianity of the category FI, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.