geometric representation theory
representation, 2-representation, β-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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:
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
where $m_{i j} = m_{j i}$ and $m_{i i} = 1$ for all $i, j \in I$. The relations may be rewritten:
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
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
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
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
For the representation theory of the degenerate affine Hecke algebra see