View source for Iwahori–Hecke algebra

{{Short description|Deformation of the group algebra of a Coxeter group}}
{{For|other mathematical rings called Hecke algebras|Hecke algebra (disambiguation)}}
{{No footnotes|date=June 2020}}
In [[mathematics]], the '''Iwahori–Hecke algebra''', or '''Hecke algebra''',  named for [[Erich Hecke]] and [[Nagayoshi Iwahori]], is a deformation of the [[group ring|group algebra]] of a [[Coxeter group]].

Hecke algebras are [[quotient ring|quotients]] of the group rings of [[Artin braid group]]s. This connection found a spectacular application in [[Vaughan Jones]]' construction of [[Jones polynomial|new invariants of knots]]. Representations of Hecke algebras led to discovery of [[quantum group]]s by [[Michio Jimbo]]. [[Michael Freedman]] proposed Hecke algebras as a foundation for [[topological quantum computer|topological quantum computation]].

==Hecke algebras of Coxeter groups==
Start with the following data:

* (''W'', ''S'') is a [[Coxeter system]] with the Coxeter matrix ''M'' = (''m''<sub>''st''</sub>), 
* ''R'' is a [[commutative ring]] with identity,
* {''q<sub>s</sub>'' | ''s'' &isin; ''S''} is a family of [[unit (ring theory)|unit]]s of ''R'' such that ''q<sub>s</sub>'' = ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugate in ''W'', and
* ''A'' is the ring of [[Laurent polynomial]]s over ''R'' with [[indeterminate (variable)|indeterminate]]s ''q<sub>s</sub>'' (and the above restriction that ''q<sub>s</sub>'' = ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugate), that is, ''A'' = ''R''[''q''{{su|p=±1|b=s}}].

===Multiparameter Hecke algebras===
The ''multiparameter Hecke algebra'' ''H<sub>R</sub>''(''W'', ''S'', ''q'') is a unital, [[associative algebra|associative ''R''-algebra]] with generators ''T<sub>s</sub>'' for all ''s'' &isin; ''S'' that satisfy the following relations:
* '''Braid Relations:''' for each pair ''s'', ''t'' in ''S'' for which ''m<sub>st</sub>'' < &infin; factors, a relation ''T<sub>s</sub> T<sub>t</sub> T<sub>s</sub>'' ... = ''T<sub>t</sub> T<sub>s</sub> T<sub>t</sub>'' ..., where each side has ''m<sub>st</sub>'' factors; and
* '''Quadratic Relation:''' for all ''s'' in ''S'', a relation (''T<sub>s</sub>'' − ''q<sub>s</sub>'')(''T<sub>s</sub>'' + 1) = 0.

'''Warning''': in later books and papers, Lusztig used a modified form of the quadratic relation that reads <math>(T_s-q_s^{1/2})(T_s+q_s^{-1/2})=0.</math> After extending the scalars to include the half-integer powers ''q''{{su|p=±1/2|b=s}} the resulting Hecke algebra is [[isomorphic]] to the previously defined one (but the ''T<sub>s</sub>'' here corresponds to ''q''{{su|p=−1/2|b=s}} ''T''<sub>s</sub> in our notation). While this does not change the general theory, many formulae look different.

===Generic multiparameter Hecke algebras===
The algebra ''H<sub>A</sub>''(''W'', ''S'', ''q'') is the ''generic'' multiparameter Hecke algebra. 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 ''q<sub>s</sub>'' &isin; ''A'' to the unit ''q<sub>s</sub>'' &isin; ''R''. This homomorphism turns ''R'' into a ''A''-algebra and the scalar extension ''H<sub>A</sub>''(''W'', ''S'') &otimes;<sub>''A''</sub> ''R'' is [[canonical isomorphism|canonically isomorphic]] to the Hecke algebra ''H<sub>R</sub>''(''W'', ''S'', ''q'') as constructed above. One calls this process ''specialization'' of the generic algebra.

=== One-parameter Hecke algebras ===
If one specializes every indeterminate ''q<sub>s</sub>'' to a single indeterminate ''q'' over the integers (or ''q''{{su|p=1/2|b=s}} to ''q''<sup>1/2</sup>, respectively), then one obtains the so-called generic one-parameter Hecke algebra of (''W'', ''S'').

Since in Coxeter groups with single laced Dynkin diagrams (for example Coxeter groups of type A and D) every pair of Coxeter generators is conjugated, the above-mentioned restriction of ''q<sub>s</sub>'' being equal ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugated in ''W'' forces the multiparameter and the one-parameter Hecke algebras to be equal. Therefore, it is also very common to only look at one-parameter Hecke algebras.

=== Coxeter groups with weights ===
If an integral weight function is defined on ''W'' (i.e., a map ''L'': ''W'' → '''Z''' with ''L''(''vw'') = ''L''(''v'') + ''L''(''w'') for all ''v'', ''w'' &isin; ''W'' with ''l''(''vw'') = ''l''(''v'') + ''l''(''w'')), then a common specialization to look at is the one induced by the homomorphism ''q<sub>s</sub>'' ↦ ''q''<sup>''L''(''s'')</sup>, where ''q'' is a single indeterminate over '''Z'''.

If one uses the convention with half-integer powers, then weight function ''L'': ''W'' → {{sfrac|1|2}}'''Z''' may be permitted as well. For technical reasons it is also often convenient only to consider positive weight functions.

== Properties ==
1. The Hecke algebra has a basis <math>(T_w)_{w\in W}</math> over ''A'' indexed by the elements of the Coxeter group ''W''. In particular, ''H'' is a [[free module|free ''A''-module]]. If <math>w=s_1 s_2 \ldots s_n</math> is a [[reduced decomposition]] of ''w'' &isin; ''W'', then <math>T_w = T_{s_1}T_{s_2}\ldots T_{s_n}</math>. This basis of Hecke algebra is sometimes called the '''natural basis'''. The [[neutral element]] of ''W'' corresponds to the identity of ''H'': ''T<sub>e</sub>'' = 1.

2. The elements of the natural basis are ''multiplicative'', namely, ''T<sub>yw</sub>'' = ''T<sub>y</sub> T<sub>w</sub>'' whenever ''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''{{su|p=−1|b=''s''}} = ''q''{{su|p=−1|b=''s''}} ''T<sub>s</sub>'' + (''q''{{su|p=−1|b=''s''}}−1).

4. Suppose that ''W'' is a [[finite group]] and the ground ring is the [[field (mathematics)|field]] '''C''' of [[complex number]]s. [[Jacques Tits]] has [[mathematical proof|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 algebra|semisimple]] and isomorphic to the complex group algebra '''C'''[''W''] (which also corresponds to the specialization ''q'' ↦ 1) {{Citation needed|reason=Can't find this result by Tits|date=July 2020}}.

5. More generally, if ''W'' is a finite group and the ground ring ''R'' is a field of [[characteristic (algebra)|characteristic]] zero, then the one-parameter Hecke algebra is a semisimple associative algebra over ''R''[''q''<sup>±1</sup>]. 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''<sup>±1/2</sup>]
<!--
that if ''A'' is extended to the field <math>K=R(q^{\frac12}) then the ''K''-algebra <math>H_K=H\otimes_A K</math> obtained from ''H'' by the change of scalars is isomorphic over ''K'' to the group algebra ''K[W]'' of the Coxeter group ''W''.

 Lusztig, George. On a theorem of Benson and Curtis. J. Algebra 71 (1981), no. 2, 490–498.
However, it seems excessive to give this reference in an article in an encyclopedia!
-->

== Canonical basis ==
{{main article|Kazhdan–Lusztig polynomial}}
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, ''H<sub>A</sub>''(''W'', ''S'', ''q''), has an [[involution (mathematics)|involution]] ''bar'' that maps ''q''<sup>1/2</sup> to ''q''<sup>−1/2</sup> and acts as identity on '''Z'''. Then ''H'' admits a unique ring [[automorphism]] ''i'' that is [[semilinear map|semilinear]] with respect to the bar involution of ''A'' and maps ''T<sub>s</sub>'' to ''T''{{su|p=−1|b=''s''}}. It can further be proved that this automorphism is involutive (has [[order (group theory)|order]] two) and takes any ''T<sub>w</sub>'' to <math>T^{-1}_{w^{-1}}.</math>

<blockquote> '''Kazhdan - Lusztig Theorem:''' For each ''w'' ∈ ''W'' there exists a unique element <math>C^{\prime}_w</math> which is invariant under the involution ''i'' and if one writes its expansion in terms of the natural basis:
::<math>C'_w = \left (q^{-1/2} \right )^{l(w)}\sum_{y\leq w}P_{y,w}T_y, </math>
one has the following:
* ''P''<sub>''w'',''w''</sub> = 1, 
* ''P''<sub>''y'',''w''</sub> in '''Z'''[''q''] has degree less than or equal to {{sfrac|1|2}}(''l''(''w'') − ''l''(''y'') − 1) if ''y'' < ''w'' in the [[Bruhat order]],
* ''P''<sub>''y'',''w''</sub>=0 if <math>y\nleq w.</math></blockquote>

The elements <math>C^{\prime}_w</math> 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''<sub>w</sub> | ''w'' &isin; ''W''} is obtained in a similar way. The polynomials ''P''<sub>''y'',''w''</sub>(''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 ==
{{Main|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 [[unimodular group|unimodular]] [[locally compact topological group]] ''G'' and a [[closed set|closed]] subgroup ''K'' of ''G''. Then the space of ''K''-biinvariant [[continuous function]]s of [[compact support]], ''C<sub>c</sub>''(''K''\''G''/''K''), can be endowed with a structure of an associative algebra under the operation of [[convolution]]. This algebra is denoted by ''H''(''G''//''K'') and called the '''Hecke ring''' of the pair (''G'', ''K'').

'''Example:''' If ''G'' = SL(''n'', '''Q'''<sub>''p''</sub>) and ''K'' = SL(''n'', '''Z'''<sub>''p''</sub>) then the Hecke ring is commutative and its representations were studied by [[Ian G. Macdonald]]. More generally if (''G'', ''K'') is a [[Gelfand pair]] then the resulting algebra turns out to be commutative.

'''Example:''' If ''G'' = SL(2, '''Q''') and ''K'' = SL(2, '''Z''') we get the abstract ring behind [[Hecke operators]] in the theory of [[modular form]]s, 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 ''p<sup>k</sup>'' elements, and ''B'' is its [[Borel subgroup]]. Iwahori showed that the Hecke ring ''H''(''G''//''B'') is obtained from the generic Hecke algebra ''H''<sub>q</sub> of the [[Weyl group]] ''W'' of ''G'' by specializing the indeterminate ''q'' of the latter algebra to ''p<sup>k</sup>'', 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 '''Q'''<sub>''p''</sub>, 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 group]]s 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 (mathematics)|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 [[conjecture]]d 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 representation]]s 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 representation]]s of Hecke algebras and representations at roots of unity turned out to be related with the theory of canonical bases in [[affine quantum group]]s and 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 different in many ways{{how|date=September 2020}} 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's constant term conjecture]].

== References ==
*David Goldschmidt [https://www.ams.org/online_bks/ulect4/ Group Characters, Symmetric Functions, and the Hecke Algebra] {{MR|1225799}},{{ISBN|0-8218-3220-4}} 
*Iwahori, Nagayoshi; Matsumoto, Hideya [http://www.numdam.org/item?id=PMIHES_1965__25__5_0 ''On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups.''] Publications Mathématiques de l'IHÉS, 25 (1965), pp.&nbsp;5–48. {{MR|0185016}}
* Alexander Kleshchev, ''[https://books.google.com/books?id=DRgM79-9qVcC&q=%22Iwahori%E2%80%93Hecke%22 Linear and projective representations of symmetric groups]'', Cambridge tracts in mathematics, vol. 163. Cambridge University Press, 2005. {{MR|2165457}}, {{ISBN|0-521-83703-0}}
* George Lusztig, [https://www.ams.org/bookstore-getitem/item=CRMM-18 Hecke algebras with unequal parameters], CRM monograph series, vol.18, American Mathematical Society, 2003. {{MR|1974442}}, {{ISBN|0-8218-3356-1}}
* Andrew Mathas, [https://www.ams.org/bookstore-getitem/item=ULECT-15 Iwahori-Hecke algebras and Schur algebras of the symmetric group], University Lecture Series, vol.15, American Mathematical Society, 1999. {{MR|1711316}}, {{ISBN|0-8218-1926-7}}
* Lusztig, George, ''On a theorem of Benson and Curtis'', J. Algebra 71 (1981), no. 2, 490–498. {{MR|0630610}}, {{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. {{MR|1204652}}, {{ISBN|0-691-02114-7}}

{{DEFAULTSORT:Iwahori-Hecke algebra}}
[[Category:Algebras]]
[[Category:Representation theory]]