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A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli;  Giovanni Felder — 2008

Annales scientifiques de l'École Normale Supérieure

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology H H 2 n ( 𝒟 n , 𝒟 n * ) of the algebra of differential operators on a formal neighbourhood of a...

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder;  A. Veselov — 2005

Journal of the European Mathematical Society

We consider both standard and twisted actions of a (real) Coxeter group G on the complement ℳ G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H * ( ℳ G , 𝒞 ) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S n , the Weyl groups...

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