In mathematical

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, the Hecke algebra of a pair (''g'',''K'') is an algebra with an

approximate identity In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate ...

, whose approximately unital modules are the same as ''K''-finite representations of the pairs (''g'',''K''). Here ''K'' is a compact subgroup of a Lie group with Lie algebra ''g''.

Definition

The Hecke algebra of a pair (''g'',''K'') is the algebra of ''K''-finite distributions on ''G'' with support in ''K'', with the product given by

convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

References

* Representation theory {{algebra-stub