nLab
Hecke algebra

Hecke algebra is a term for a class of algebras.

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.

For example:

• If we consider the general linear group ${\mathrm{GL}}_n({𝔽}_q)$ acting on the set of complete flags? in ${𝔽}_q^n$, then we obtain an algebra generated by the endomorphism ${\sigma }_i$ which sends the characteristic function of one flag $F=\{F_1\subset \cdots \subset F_{n-1}\subset {𝔽}_q^n\}$ to the characteristic function of the set of flags $F\prime$ with $F_j\prime =F_j$ for all $j\ne i$ and $F_i\prime \ne F_i$. These elements satisfy the relations

  $${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}
  $${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i(\mid i-j\mid >1)$$\sigma_i\sigma_j=\sigma_j\sigma_i \qquad (|i-j|\gt1)
  $${\sigma }_i^2=(q-1)\sigma +q$$\sigma_i^2=(q-1)\sigma+q

• If we look at ${\mathrm{GL}}_n({𝔽}_q((t)))$ acting on the set of ${𝔽}_q[[t]]$ lattices in ${𝔽}_q((t){)}^n$, then we will obtain the spherical Hecke algebra.

• A variant of the Hecke algebra is the degenerate affine Hecke algebra of type $A$; this is a deformation of the semidirect product of the symmetric group $S_n$ with the polynomial ring in $n$ variables. The generators are $S_n$ and $y_1,\dots ,y_n$, with relations $\sigma y_i{\sigma }^{-1}=y_{\sigma (i)}$ and $[y_i,y_j]={\sum }_{k\ne i,j}(kij)-(kji)$; one can replace the $y_i$’s with commuting $x_i$’s with slightly messier relations. As Lusztig showed, the representation theory of the affine Hecke algebra is related to the graded or degenerate case. For the representation theory of the dAHA, see for instance Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A, by Takeshi Suzuki.