We will discuss how the representation theory of $PGL(d, F)$ assists in analyzing functions and operators on the associated Bruhat-Tits building (or quotients thereof). There is a certain natural algebra of functions on the group called the spherical Hecke algebra whose action on the building has a nice geometric interpretation and whose representations parametrize so-called ``spherical representations'' of $PGL(d, F)$. The spherical Hecke algebra acting on the building has certain particularly nice eigenfunctions called spherical functions which allow one to define a sort of ``Fourier transform'' on the building. In the case of $PGL(2, F)$, these topics are closely related to analyzing the eigenvalues and eigenfunctions of the adjacency operator on regular graphs/the infinite regular tree. This talk is somewhat of a continuation of last week's talk, but it will be fairly independent from that talk and it's not necessary to have attended that talk to follow this one.