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In mathematics, the term Hecke algebra is the common name for several related types of associative rings in algebra and representation theory. The most familiar of these is the Hecke algebra of a Coxeter group, also known as Iwahori–Hecke algebra, which is a one-parameter deformation of the group algebra of a Coxeter group. Hecke algebras of more general kind are considered in representation theory of reductive groups over local fields and the theory of automorphic forms.

Hecke algebras are intimately connected with Artin braid groups. 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.

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Hecke algebra of a Coxeter group

Suppose that (W,S) is a Coxeter system with the Coxeter matrix M. Fix a ground ring R (most commonly, R is the ring \mathbb{Z} of integers or an algebraically closed field, such as \mathbb{C}). Let q be a formal indeterminate, and let A=R[q,q-1] be the ring of Laurent polynomials over R. Then the Hecke algebra defined by these data is the unital associative algebra over A with generators Ts for all s ∈ S and the relations:

 T_s T_t T_s \ldots = T_t T_s T_t \ldots, where each side has m_{st}<\infty factors and  s\ne t\in S (braid relations)
(Tsq)(Ts + 1) = 0 for all s\in S (quadratic relation).

This ring is also called the generic Hecke algebra, to distinguish it from the ring obtained from H by specializing the indeterminate q to an element of R (for example, a complex number if R=C).

Warning: in recent books and papers, Lusztig has been using a modified form of the quadratic relation that reads (Tsq1 / 2)(Ts + q − 1 / 2) = 0. After extending the scalars to include q^{\pm 1/2}, the resulting Hecke algebra is isomorphic to the previously defined one. While this does not change the general theory, many formulas look different.

Properties

1. Hecke algebra has a basis {Tw} over A indexed by the elements of the Coxeter group W. In particular, H is a free A-module. If w=s1s2...sn is a reduced decomposition of w ∈ W, then T_w=T_{s_1}T_{s_2}\ldots T_{s_n}. This basis of Hecke algebra is sometimes called the natural basis. The neutral element of W corresponds to the identity of H: Te=1.

2. The elements of the natural basis are multiplicative, namely, Tyw=TyTw when l(yw)=l(y)+l(w), where l denotes the length function on the Coxeter group W.

3. Elements of the natural basis are invertible. For example, from the quadratic relation we conclude that T_s^{-1}=q^{-1}T_s+(q^{-1}-1).

4. Suppose that W is a finite group and the ground ring is the field \mathbb{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 finite-dimensional algebra is semisimple and isomorphic to the complex group algebra of W (corresponding to the case q=1).

5. More generally, if W is a finite group and the ground ring R is a field of characteristic zero, then the Hecke algebra is a semisimple associative algebra over A. 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^{\frac12}].

Canonical basis

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.

Consider a Hecke algebra H over the ring A=\mathbb{Z}[q^{\frac12},q^{-\frac12}], as in the property 4 above. This ring has an involution bar that maps q^{\frac12} toq^{-\frac12} 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_s^{-1}. It can further be proved that this automorphism is involutive (has order two) and takes any Tw to T^{-1}_{w^{-1}}.

Theorem (Kazhdan-Lusztig)

For each w ∈W there exists a unique element C'w which is invariant under the involution i and has the property that in the expansion

 C'_w=(q^{-1/2})^{l(w)}\sum_{y\leq w}P_{y,w}T_y

over the elements of the natural basis, one has P_{w,w}=1, P_{y,w}(q) \in \mathbb{Z}[q] has degree \leq\frac{1}{2}(l(w)-l(y)-1) if y < w in the Bruhat order and Py,w = 0 if y\nleq w.

The elements C'w where w varies over W form a basis of the algebra H, which is called the dual canonical basis of the Hecke algebra H. The canonical basis \{C_w:w\in W\} is obtained in a similar way. The polynomials Py,w(q) making appearance in this theorem are the Kazhdan-Lusztig polynomials.

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.

Hecke algebra of a locally compact group

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 locally compact topological group G and its closed subgroup K. Then the space of bi-K-invariant continuous functions of compact support

C[K\G/K]

can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted

H(G//K)

and called the Hecke ring of the pair (G,K). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. In particular, this holds when

G=SLn(Qp) and K=SLn(Zp)

and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald.

On the other hand, in the case

G = SL2(Q) and K = SL2(Z)

we arrive at the abstract ring behind Hecke operators in the theory of modular forms, which gave the name to Hecke algebras in general.

The case leading to the Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring

H(G//K)

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):

I think it would be most appropriate to call it the Iwahori algebra, but the name Hecke ring (or algebra) given by Iwahori himself has been in use for almost 20 years and it is probably too late to change it now.

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 GLn 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 has been conjectured that all Hecke algebras that are ever needed are mild generalizations of affine Hecke algebras.

Representations of Hecke algebras

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). Modular representations of Hecke algebras and representations at roots of unity turned out to be related with the theory of canonical bases in affine quantum groups and very interesting combinatorics.

Representation theory of affine Hecke algebras was developed by Lusztig with a view towards applying it to description of representations of p-adic groups. It is in many ways quite different in flavor from the finite case. A generalization of affine Hecke algebras, called double affine Hecke algebra, was used by Ivan Cherednik in his proof of the Macdonald conjectures.

References

  • David Goldschmidt Group Characters, Symmetric Functions, and the Hecke Algebra ISBN 0-8218-3220-4
  • Iwahori, Nagayoshi; Matsumoto, Hideya On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Publications Mathématiques de l'IHÉS, 25 (1965), p. 5-48
  • Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge tracts in mathematics, vol. 163. Cambridge University Press, 2005. ISBN 0 521 83703 0
  • George Lusztig, Hecke algebras with unequal parameters, CRM monograph series, vol.18, American Mathematical Society, 2003. ISBN 0-8218-3356-1
  • Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol.15, American Mathematical Society, 1999. ISBN 0-8218-1926-7
  • Lusztig, George, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490--498. doi:10.1016/0021-8693(81)90188-5
  • Colin Bushnell and Philip Kutzko, The admissible dual of GL(n) via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, 1993. ISBN 0-691-02114-7

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