Hecke algebra

In mathematics, a Hecke algebra can be one of several algebras, similar to the algebra of Hecke operators studied by Erich Hecke. The algebra of Hecke operators can be interpreted as an algebra of double cosets, and as a result the term "Hecke algebra" is also used for several similar algebras related to double cosets. In particular it can mean:

• Iwahori–Hecke algebra of a Coxeter group.
• Hecke algebra of a pair (g,K) where g is the Lie algebra of a Lie group G and K is a compact subgroup of G.
• H(G,K), the Hecke algebra of a locally compact group G with respect to a compact subgroup K
• The Hecke algebra of a locally profinite group (such as an algebraic group over a nonarchimedean local field), given by the direct limit of the algebras H(G,K) where K ranges over compact open subgroups of G.
• The algebra generated by Hecke operators acting on modular forms
• The algebra spanned by the double cosets HgH of a finite index subgroup H of a group G.
• The centralizer algebra of an induced representation.