Hecke algebra of a locally compact group
Encyclopedia
In mathematics, a Hecke algebra of a locally compact group
is an algebra of binvariant measures under convolution.
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 bi-K-invariant continuous function
s of compact support
can be endowed with a structure of an associative algebra under the operation of convolution
. This algebra is denoted
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
'
and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald
.
On the other hand, in the case
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
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.
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is an algebra of binvariant measures under convolution.
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 bi-K-invariant continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s of compact support
- C[K\G/K]
can be endowed with a structure of an associative algebra under the operation of convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
. This algebra is denoted
- H(G//K)
and called the Hecke ring of the pair (G,K). If we start with a Gelfand pair
Gelfand pair
In mathematics, the expression Gelfand pair is a pair consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to...
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
Ian G. Macdonald
Ian Grant Macdonald is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebraic combinatorics ....
.
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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with pk elements, and B is its Borel subgroup
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
. Iwahori showed that the Hecke ring
- H(G//B)
is obtained from the generic Hecke algebra Hq of the Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
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
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
K, such as Qp, and K is what is now called an Iwahori subgroup
Iwahori subgroup
In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of...
of G. The resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra
Affine Hecke algebra
In mathematics, an affine Hecke algebra is the Hecke algebra of an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.-Definition:...
, where the indeterminate q has been specialized to the cardinality of the residue field
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
of K.