  Highest-weight representations of infinite dimensional Lie algebras and
Hilbert schemes of points are considered, together with the applications of
these concepts to partition functions, which are most useful in physics.
Partition functions (elliptic genera) are conveniently transformed into product
expressions, which may inherit the homology properties of appropriate
(poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle
spectral functions of hyperbolic geometry in the calculation of partition
functions and associated $q$-series are discussed. Examples of these connection
in quantum field theory are considered (in particular, within the AdS/CFT
correspondence), as the AdS$_{3}$ case where one has Ruelle/Selberg spectral
functions, whereas on the CFT side, partition functions and modular forms
arise. These objects are here shown to have a common background, expressible in
terms of Euler-Poincar\'e and Macdonald identities, which, in turn, describe
homological aspects of (finite or infinite) Lie algebra representations.
Finally, some other applications of modular forms and spectral functions
(mainly related with the congruence subgroup of $SL(2, {\mathbb Z})$) to
partition functions, Hilbert schemes of points, and symmetric products are
investigated by means of homological and K-theory methods.
