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Hecke algebra is a term for a class of algebras. They often appear as convolution algebras or as double coset spaces. For p-adic algebraic groups Hecke algebras often play a role similar a Lie algebra plays in the complex case (the Lie algebra still exists, but is too small).
Typically the term refers to an algebra which is the endomorphisms of a permutation representation of a topological group, though some liberties have been taken with this definition, and often the term means some modification of such an algebra.
- If we consider the general linear group acting on the set of complete flags? in , then we obtain an algebra generated by the endomorphism which sends the characteristic function of one flag to the characteristic function of the set of flags with for all and . These elements satisfy the relations
- If we look at acting on the set of lattices in , then we will obtain the spherical Hecke algebra.
- A variant of the Hecke algebra is the degenerate affine Hecke algebra of type ; this is a deformation of the semidirect product of the symmetric group with the polynomial ring in variables. The generators are and , with relations and ; one can replace the ‘s with commuting ’s with slightly messier relations. As George Lusztig showed, the representation theory of the affine Hecke algebra is related to the graded or degenerate case.
- There is a geometric construction of the representations of Weyl algebras when realized as certain Hecke convolution algebras by Victor Ginzburg.
Generalized Hecke algebras
To each Coxeter group one may associate a Hecke algebra, a certain deformation of the group algebra over a field , as follows. is presented by generators and relations
where and for all . The relations may be rewritten:
where each of the words in the second equation alternate in the letters , and has length , provided that . The corresponding Hecke algebra has basis , and is presented by
These relations may be interpreted structurally as follows (for simplicity, we will consider only finite, aka spherical Coxeter groups). A Coxeter group may be associated with a suitable BN-pair?; the classical example is where is an algebraic group, is a Borel subgroup (maximal solvable subgroup), and is the normalizer of a maximal torus in . Such typically arise as automorphism groups of thick -buildings, where is a stabilizer of a point of the building. The coset space may then be interpreted as a space of flags for a suitable geometry. The Coxeter group itself arises as the quotient , and under the BN-pair axioms there is a well-defined map
which is a bijection to the set of double cosets of . (In particular, the double cosets do not depend on the coefficient ring in which the points are instantiated.)
When one takes points of the algebraic group over the coefficient ring , a finite field with elements, the flag manifold is also finite. One may calculate
so that the double cosets form a linear basis of the algebra of -equivariant operators on the space of functions . This algebra is in fact the Hecke algebra.
It is a matter of interest to interpret the double cosets directly as operators on , and in particular the cosets where is a Coxeter generator.
To be continued…
References and links
- sbseminar blog: interpreting the Hecke algebra I, II
For the representation theory of the degenerate affine Hecke algebra see
- Takeshi Suzuki, Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A, math.QA/9805035