This connection found a spectacular application in Vaughan Jones' construction of new invariants of knots.
Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo.
Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation.
Start with the following data: The multiparameter Hecke algebra HR(W, S, q) is a unital, associative R-algebra with generators Ts for all s ∈ S that satisfy the following relations: Warning: in later books and papers, Lusztig used a modified form of the quadratic relation that reads
This algebra is universal in the sense that every other multiparameter Hecke algebra can be obtained from it via the (unique) ring homomorphism A → R which maps the indeterminate qs ∈ A to the unit qs ∈ R. This homomorphism turns R into a A-algebra and the scalar extension HA(W, S) ⊗A R is canonically isomorphic to the Hecke algebra HR(W, S, q) as constructed above.
The elements of the natural basis are multiplicative, namely, Tyw = Ty Tw whenever l(yw) = l(y) + l(w), where l denotes the length function on the Coxeter group W. 3.
Suppose that W is a finite group and the ground ring is the field C of complex numbers.
Jacques Tits has proved that if the indeterminate q is specialized to any complex number outside of an explicitly given list (consisting of roots of unity), then the resulting one-parameter Hecke algebra is semisimple and isomorphic to the complex group algebra C[W] (which also corresponds to the specialization q ↦ 1) [citation needed].
Moreover, extending earlier results of Benson and Curtis, George Lusztig provided an explicit isomorphism between the Hecke algebra and the group algebra after the extension of scalars to the quotient field of R[q±1/2] A great discovery of Kazhdan and Lusztig was that a Hecke algebra admits a different basis, which in a way controls representation theory of a variety of related objects.
The generic multiparameter Hecke algebra, HA(W, S, q), has an involution bar that maps q1/2 to q−1/2 and acts as identity on Z.
Then H admits a unique ring automorphism i that is semilinear with respect to the bar involution of A and maps Ts to T−1s.
which is invariant under the involution i and if one writes its expansion in terms of the natural basis: one has the following: The elements
The Kazhdan–Lusztig notions of left, right and two-sided cells in Coxeter groups are defined through the behavior of the canonical basis under the action of H. Iwahori–Hecke algebras first appeared as an important special case of a very general construction in group theory.
Let (G, K) be a pair consisting of a unimodular locally compact topological group G and a closed subgroup K of G. Then the space of K-biinvariant continuous functions of compact support, Cc(K\G/K), can be endowed with a structure of an associative algebra under the operation of convolution.
More generally if (G, K) is a Gelfand pair then the resulting algebra turns out to be commutative.
Iwahori showed that the Hecke ring H(G//B) is obtained from the generic Hecke algebra Hq of the Weyl group W of G by specializing the indeterminate q of the latter algebra to pk, the cardinality of the finite field.
George Lusztig remarked in 1984 (Characters of reductive groups over a finite field, xi, footnote): Iwahori and Matsumoto (1965) considered the case when G is a group of points of a reductive algebraic group over a non-archimedean local field K, such as Qp, and K is what is now called an Iwahori subgroup of G. The resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra, where the indeterminate q has been specialized to the cardinality of the residue field of K. Work of Roger Howe in the 1970s and his papers with Allen Moy on representations of p-adic GL(n) opened a possibility of classifying irreducible admissible representations of reductive groups over local fields in terms of appropriately constructed Hecke algebras.
(Important contributions were also made by Joseph Bernstein and Andrey Zelevinsky.)
These ideas were taken much further in Colin Bushnell and Philip Kutzko's theory of types, allowing them to complete the classification in the general linear case.
Many of the techniques can be extended to other reductive groups, which remains an area of active research.
It follows from Iwahori's work that complex representations of Hecke algebras of finite type are intimately related with the structure of the spherical principal series representations of finite Chevalley groups.
George Lusztig pushed this connection much further and was able to describe most of the characters of finite groups of Lie type in terms of representation theory of Hecke algebras.
This work used a mixture of geometric techniques and various reductions, led to introduction of various objects generalizing Hecke algebras and detailed understanding of their representations (for q not a root of unity).