1,721,049 research outputs found
Spin-structures and proper group actions
No full textAbstract not availablePeter Hochs, Varghese Matha
Geometric quantization and families of inner products
No full textAvailable online 24 July 2015Abstract not availablePeter Hochs, Varghese Matha
An equivariant index for proper actions III: the invariant and discrete series indices
No full textAbstract not availablePeter Hochs, Yanli Son
A fixed point formula and Harish-Chandra's character formula
No full textThe main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations.Peter Hochs and Hang Wan
Shelstad's character identity from the point of view of index theory
Full text linkShelstad’s character identity is an equality between sums of characters of tempered representations in corresponding L-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.Peter Hochs and Hang Wan
Formal geometric quantisation for proper actions
Full text linkPublished online: 23 April 2015We define formal geometric quantisation for proper Hamiltonian actions by possibly noncompact groups on possibly noncompact, prequantised symplectic manifolds, generalising work of Weitsman and Paradan. We study the functorial properties of this version of formal geometric quantisation, and relate it to a recent result by the authors via a version of the shifting trick. For (pre)symplectic manifolds of a certain form, quantisation commutes with reduction, in the sense that formal quantisation equals a more direct version of quantisation.Peter Hochs, Varghese Matha
A fixed point theorem on noncompact manifolds
No full textWe generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using KK-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah–Segal–Singer’s result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant K-theory and K-homology, and a nonlocalised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.Peter Hochs and Hang Wan
Quantising proper actions on Spin c-manifolds
No full textParadan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to SpinC-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of K-theory of group C*-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to SpinC-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.Peter Hochs and Varghese Matha
Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions
Full text linkWe prove that a general class of nonlinear, non-autonomous odes in Fréchet spaces are close to odes in a specific normal form, where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ode are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ode in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ode is a finite-dimensional pde. We show that our methods apply to a relevant class of nonlinear, non-autonomous pdes in this way.Peter Hochs and A. J. Robert
An index theorem for higher orbital integrals
No full textPublished online: 17 July 2021Recently, two of the authors of this paper constructed cyclic cocycles on Harish–Chandra’s Schwartz algebra of linear reductive Lie groups that detect all information in the K-theory of the corresponding group C∗-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions.Peter Hochs, Yanli Song and Xiang Tan
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