REPRESENTATIONS OF AFFINE HECKE ALGEBRAS

J.M. DOUGLASS

From finite to affine: Representations of p-adic groups and the Deligne-Langlands-Lusztig conjecture.

• Affine Weyl groups and Hecke algebras.

• Affine Hecke algebras as endomorphism rings.

• The Deligne-Langlands-Lusztig conjecture.

Applications of Kazhdan-Lusztig polynomials for finite and affine Weyl groups.

• Construction of Kazhdan-Lusztig polynomials.

• Kazhdan-Lusztig polynomials for finite Weyl groups.

– The Kazhdan-Lusztig conjecture.

– Singularities of Schubert varieties.

• Kazhdan-Lusztig polynomials for affine Weyl groups.

– Kazhdan-Lusztig polynomials and weight multiplicities.

The convolution construction of the generic, affine Hecke algebra. • The Steinberg variety of a reductive group.

• Introduction to equivariant K-theory.

• Equivariant K-theory of the Steinberg variety and the generic, affine Hecke algebra.

Geometric constructions of representations of the generic, affine Hecke algebra.

• Irreducible representations – parametrization and construction.

• Induced representations and equivariant K-groups of fixed point varieties.