Continuous Hecke algebras

Details Continuous Hecke algebras Continuous Hecke algebras

2005-01-13

arxiv.org/abs/math/0501192v3

Mathematics

Quantum Algebra

28 pages, latex; the exposition and the title of the paper are changed in the present version

Scientific paper

We study a generalization of graded Hecke algebras introduced by Drinfeld in 1986, in which the role of a finite group G is played by a reductive algebraic group. This includes continuous generalizations of symplectic reflection algebras and of rational Cherednik algebras. We prove direct and converse PBW theorems, and also study versions of our algebras in which the group G is replaced with its Lie algebra. We also begin to develop the representation theory of continuous Hecke algebras, which unifies representation theories of real reductive groups, Drinfeld-Lusztig degenerate affine Hecke algebras, and symplectic reflection (in particular, Cherednik) algebras. The paper is dedicated to V. Drinfeld's 50-th birthday.

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