nLab
Hecke algebra

Idea

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.

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 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 $W$ one may associate a Hecke algebra, a certain deformation? of the group algebra $k[W]$ over a field $k$, as follows. $W$ is presented by generators $⟨s_i{⟩}_{i\in I}$ and relations

$$(s_i{s}_j{)}^{m_{ij}}=1$$(s_i s_j)^{m_{i j}} = 1

where $m_{ij}=m_{ji}$ and $m_{ii}=1$ for all $i,j\in I$. The relations may be rewritten:

$$s_i^2=1,s_i{s}_j\dots =s_j{s}_i\dots$$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots

where each of the words in the second equation alternate in the letters $s_i$, $s_j$ and has length $m_{ij}$, provided that $m_{ij}<\mathrm{\infty }$. The corresponding Hecke algebra has basis $W$, and is presented by

$$s_i^2=\frac{q-1}{q}s+\frac{1}{q},s_i{s}_j\dots =s_j\mathrm{sI}-\dots$$s_{i}^{2} = \frac{q-1}{q} s + \frac1{q}, \qquad s_i s_j \ldots = s_j sI- \ldots

These relations may be interpreted structurally as follows (for simplicity, we will consider only finite, aka spherical Coxeter groups). A Coxeter group $W$ may be associated with a suitable BN-pair?; the classical example is where $G$ is an algebraic group, $B$ is a Borel subgroup (maximal solvable subgroup), and $N$ is the normalizer of a maximal torus $T$ in $G$. Such $G$ typically arise as automorphism groups of thick $W$-buildings, where $B$ is a stabilizer of a point of the building. The coset space $G/B$ may then be interpreted as a space of flags for a suitable geometry. The Coxeter group itself arises as the quotient $W\cong N/T$, and under the BN-pair axioms there is a well-defined map

$$W\to B\backslash G/B:w\mapsto BwB$$W \to B\backslash G/B: w \mapsto B w B

which is a bijection to the set of double cosets of $B$. (In particular, the double cosets do not depend on the coefficient ring $R$ in which the points $G(R)$ are instantiated.)

When one takes points of the algebraic group $G$ over the coefficient ring ${𝔽}_q$, a finite field with $q$ elements, the flag manifold $G_q/B_q≔G({𝔽}_q)/B({𝔽}_q)$ is also finite. One may calculate

$$\begin{array}{ccc}{\mathrm{hom}}_{k[G_q]}(k[G_q/B_q],k[G_q/B_q])& \cong & k[G_q/B_q{]}^{*}{\otimes }_{k[G_q]}k[G_q/B_q]\\ & \cong & k[B_q\backslash G_q]{\otimes }_{k[G_q]}k[G_q/B_q]\\ & \cong & k[B_q\backslash G_q{\otimes }_{G_q}{G}_q/B_q]\\ & \cong & k[B_q\backslash G_q/B_q]\end{array}$$\array{ \hom_{k[G_q]}(k[G_q/B_q], k[G_q/B_q]) & \cong & k[G_q/B_q]^\ast \otimes_{k[G_q]} k[G_q/B_q] \\ & \cong & k[B_q\backslash G_q] \otimes_{k[G_q]} k[G_q/B_q] \\ & \cong & k[B_q\backslash G_q \otimes_{G_q} G_q/B_q] \\ & \cong & k[B_q \backslash G_q / B_q] }

so that the double cosets form a linear basis of the algebra of $G_q$-equivariant operators on the space of functions $k[G_q/B_q]$. This algebra is in fact the Hecke algebra.

It is a matter of interest to interpret the double cosets directly as operators on $k[G_q/B_q]$, and in particular the cosets $B{s}_{i}B$ where $s_i$ is a Coxeter generator.

To be continued…

Related concepts

• Iwahori–Hecke algebra

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