Hecke Algebras.

Intro -- Table of Contents -- Introduction -- Chapter I: The Abstract Hecke Algebra -- 1. The Group Algebra -- 2. Formal Linear Combinations of Right and Double Cosets -- 3. Hecke Pairs -- 4. The Product -- 5. The Symmetric Group as an Example -- 6. Isomorphisms of Hecke Algebras -- 7. Anti-Homomorphisms -- 8. The Canonical Bilinear Form -- Remarks -- Chapter II: Structure Theory of Hecke Algebras for Finite Groups -- 1. Maschke's Theorem for Hecke Algebras -- 2. The Use of Group Algebras -- 3. Hecke Algebras of Small Dimension -- 4. Quadratic Residues and Non-Residues -- 5. The Canonical Involution on the Hecke Algebra -- Remarks -- Chapter III: Examples of Hecke Algebras for Finite Groups -- 1. The Dihedral Group -- 2. The Integral Orthogonal Group -- 3. The Symmetric Group -- 4. A Borel Subgroup of the General Linear Group over a Finite Field -- 5. A Maximal Parabolic Subgroup -- Remarks -- Chapter IV: The Hecke Algebra Associated with SL(2 -- Z) -- 1. Right Coset Decompositions -- 2. Double Coset Decompositions -- 3. The Hecke Algebra Associated with (Γ,M) -- 4. The Primary Component H[sub(p)] -- 5. An Embedding -- 6. Congruence Subgroups -- 7. The Hecke Algebra Associated with Γ[sub(0)](q) -- Remarks -- Chapter V: The Hecke Algebra Associated with the Unimodular Group -- 1. The Unimodular Group -- 2. The Elementary Divisor Theorem -- 3. Elementary Divisors -- 4. q-Kernels -- 5. Right Coset Decompositions -- 6. The Hecke Algebra Associated with (u[sub(n)],Inv(n -- Z)) -- 7. Particular Double Cosets in the Primary Component -- 8. The Primary Component -- 9. Formal Power Series -- 10. The Hecke Algebra over Q -- 11. An Embedding -- Remarks -- Chapter VI: The Hecke Algebra Associated with the Slegel Modular Group -- 1. Symplectic Similitudes -- 2. The Symplectic Elementary Divisor Theorem -- 3. Right Coset Decompositions.

4. The Hecke Algebra Associated with (Γ[sub(n)],Δ[sub(n)]) -- 5. Particular Double Cosets in the Primary Component -- 6. The Primary Component -- 7. The Commutation Relation with the Siegel Φ-Operator -- 8. A Rationality Theorem for n = 2 -- Remarks -- References.

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