On the Gelfand-Kazhdan criterion and the commutativity of Hecke algebras
For a finite group G and a subgroup H, we say that (G,H) is a Gelfand pair if the decomposition of C[G/H], the G-representation of complex-valued functions on G/H, into irreducible components has multiplicity one. In this case, the Gelfand property is equivalent to the commutativity of the Hecke algebra C[H\G/H] of bi-H-invariant functions on G.
Given a reductive group G and a closed subgroup H, there are three standard ways to generalize the notion of a Gelfand pair, and a result of Gelfand and Kazhdan gives a sufficient condition under which two of these properties hold. Unfortunately, in contrast to the finite case, here the Gelfand property is not known to be equivalent to the commutativity of a Hecke algebra. In this talk we define a Hecke algebra for the pair (G,H) in the non-Archimedean case and show that if the Gelfand-Kazhdan conditions hold then it is commutative. We then explore the connection between the commutativity of this algebra and the Gelfand property of (G,H).