Related papers: The G-Signature Theorem Revisited

Let G be a reductive algebraic group over a field k and let B be a Borel subgroup in G. We demonstrate how a number of results on the cohomology of line bundles on the flag manifold G/B have had interesting consequences in the…

The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections…

For a connected simply connected semisimple algebraic group $G$ we prove existence of invariant tensors in tensor powers of rational $G$-modules and establish relations between existence of such invariant tensors and stability of diagonal…

We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de…

In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined…

We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our…

We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface…

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori.…

We describe the image, under the local Langlands correspondence for tori, of the characters of a torus which are trivial on its Iwahori subgroup. Let $k$ be a non-archimedian local field. Let $\boldsymbol{G}$ be a connected reductive group…

We study the torus equivariant K-homology ring of the affine Grassmannian $\mathrm{Gr}_G$ where $G$ is a connected reductive linear algebraic group. In type $A$, we introduce equivariantly deformed symmetric functions called the K-theoretic…

In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed…

Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component…

Let M be a riemannian manifold. The existence of a spin structure on M, enables to study the topology of M. The obstruction to the existence of the spin structure is given by the second Stiefel-Whitney class. This class is the classifying…

Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an…

Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of…

We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for…

For any rigid analytic group variety $G$ over a non-archimedean field $K$ over $\mathbb Q_p$, we study $G$-torsors on adic spaces over $K$ in the $v$-topology. Our main result is that on perfectoid spaces, $G$-torsors in the \'etale and…

Let G be a compact Lie group. We build a tower of G-spectra over the suspension spectrum of the space of linear isometries from one G-representation to another. The stable cofibres of the maps running down the tower are certain interesting…

Twisting by a G-torsor an object endowed with an action of a group G is a classical tool. For instance one finds in the paragraph 5.3 of the book "cohomologie galoisienne" by Serre, the description of the "op\'eration de torsion" in a…

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has non-positive sectional curvature. As an example, we can take G…