Hecke algebra



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(𝔽 q)GL_n(\mathbb{F}_q) acting on the set of complete flags? in 𝔽 q n\mathbb{F}_q^n, then we obtain an algebra generated by the endomorphism Οƒ i\sigma_i which sends the characteristic function of one flag F={F 1βŠ‚β‹―βŠ‚F nβˆ’1βŠ‚π”½ q n}\mathbf{F}=\{F_1\subset \cdots \subset F_{n-1}\subset \mathbb{F}_q^n\} to the characteristic function of the set of flags Fβ€²\mathbf{F}' with F jβ€²=F jF_j'=F_j for all jβ‰ ij\neq i and F iβ€²β‰ F iF_i'\neq F_i. These elements satisfy the relations
    σ iσ i+1σ i=σ i+1σ iσ i+1\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}
    Οƒ iΟƒ j=Οƒ jΟƒ i(|iβˆ’j|>1)\sigma_i\sigma_j=\sigma_j\sigma_i \qquad (|i-j|\gt1)
    Οƒ i 2=(qβˆ’1)Οƒ+q\sigma_i^2=(q-1)\sigma+q
  • If we look at GL n(𝔽 q((t)))GL_n(\mathbb{F}_q(\!(t)\!)) acting on the set of 𝔽 q[[t]]\mathbb{F}_q[\![t]\!] lattices in 𝔽 q((t)) n\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 AA; this is a deformation of the semidirect product of the symmetric group S nS_n with the polynomial ring in nn variables. The generators are S nS_n and y 1,…,y ny_1, \dots, y_n, with relations Οƒy iΟƒ βˆ’1=y Οƒ(i)\sigma y_i \sigma^{-1} = y_{\sigma(i)} and [y i,y j]=βˆ‘ kβ‰ i,j(kij)βˆ’(kji)[y_i,y_j] = \sum_{k \neq i,j} (k i j)-(k j i); one can replace the y iy_iβ€˜s with commuting x ix_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 WW one may associate a Hecke algebra, a certain deformation of the group algebra k[W]k[W] over a field kk, as follows. WW is presented by generators ⟨s i⟩ i∈I\langle s_i \rangle_{i \in I} and relations

(s is j) m ij=1(s_i s_j)^{m_{i j}} = 1

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

s i 2=1,s is j…=s js i…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 is_i, s js_j and has length m ijm_{i j}, provided that m ij<∞m_{i j} \lt \infty. The corresponding Hecke algebra has basis WW, and is presented by

s i 2=qβˆ’1qs i+1q,s is j…=s js i…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 WW may be associated with a suitable BN-pair?; the classical example is where GG is an algebraic group, BB is a Borel subgroup (maximal solvable subgroup), and NN is the normalizer of a maximal torus TT in GG. Such GG typically arise as automorphism groups of thick WW-buildings, where BB is a stabilizer of a point of the building. The coset space G/BG/B may then be interpreted as a space of flags for a suitable geometry. The Coxeter group itself arises as the quotient W≅N/TW \cong N/T, and under the BN-pair axioms there is a well-defined map

Wβ†’B\G/B:w↦BwBW \to B\backslash G/B: w \mapsto B w B

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

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

hom k[G q](k[G q/B q],k[G q/B q]) β‰… k[G q/B q] *βŠ— k[G q]k[G q/B q] β‰… k[B q\G q]βŠ— k[G q]k[G q/B q] β‰… k[B q\G qβŠ— G qG q/B q] β‰… k[B q\G q/B q]\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 qG_q-equivariant operators on the space of functions k[G q/B q]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]k[G_q/B_q], and in particular the cosets Bs iBB s_i B where s is_i is a Coxeter generator.

To be continued…

  • 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
Revised on October 15, 2015 10:37:38 by Urs Schreiber (