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 $GL_n(\mathbb{F}_q)$ acting on the set of complete flags in $\mathbb{F}_q^n$, then we obtain an algebra generated by the endomorphism $\sigma_i$ which sends the characteristic function of one flag $\mathbf{F}=\{F_1\subset \cdots \subset F_{n-1}\subset \mathbb{F}_q^n\}$ to the characteristic function of the set of flags $\mathbf{F}'$ with $F_j'=F_j$ for all $j\neq i$ and $F_i'\neq F_i$. These elements satisfy the relations
  $$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$$
  $$\sigma_i\sigma_j=\sigma_j\sigma_i \qquad (|i-j|\gt1)$$
  $$\sigma_i^2=(q-1)\sigma_i+q$$
• If we look at $GL_n(\mathbb{F}_q(\!(t)\!))$ acting on the set of $\mathbb{F}_q[\![t]\!]$ lattices in $\mathbb{F}_q(\!(t)\!)^n$, 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 $[y_i,y_j] = \sum_{k \neq i,j} (k i j)-(k j i)$; 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 $\langle s_i \rangle_{i \in I}$ and relations

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

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

$$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots$$

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

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

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

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

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 $\mathbb{F}_q$, a finite field with $q$ elements, the flag manifold $G_q/B_q \coloneqq G(\mathbb{F}_q)/B(\mathbb{F}_q)$ is also finite. One may calculate

$$\array{ \hom_{k[G_q]}(k[G_q/B_q], k[G_q/B_q]) & \cong & k[G_q/B_q]^\ast \otimes_{k[G_q]} k[G_q/B_q] \\ & \cong & k[B_q\backslash G_q] \otimes_{k[G_q]} k[G_q/B_q] \\ & \cong & k[B_q\backslash G_q \otimes_{G_q} G_q/B_q] \\ & \cong & k[B_q \backslash G_q / B_q] }$$

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

• double Hecke algebra

• double affine Hecke algebra

• Hecke correspondence

• Hecke category

• Iwahori–Hecke algebra

References

Lecture notes:

• Garth Warner: Elementary Aspects of the Theory of Hecke Operators, University of Washington (1988) [pdf, pdf]

In relation to the Knizhnik-Zamolodchikov equation:

• Ivan Cherednik, Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 [doi:10.2969/msjmemoirs/00101C010]

See also:

• Secret Blogging Seminar: Interpreting the Hecke algebra

• Secret Blogging Seminar: Interpreting the Hecke Algebra II: the sheafification

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