Hecke algebra of a pair

In mathematical representation theory, the Hecke algebra of a pair ("g","K") is an algebra with an approximate identity, 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.