  We study the Hecke algebra modules arising from theta correspondence between
certain Harish-Chandra series for type I dual pairs over finite fields. For the
product of the pair of Hecke algebras under consideration, we show that there
is a generic Hecke algebra module whose specializations at prime powers give
the Hecke algebra modules and whose specialization at $1$ can be explicitly
described. As an application, we prove the conservation relation on the first
occurrence indices for all irreducible representations. As another application,
we generalize the results of Aubert-Michel-Rouquier and Pan on theta
correspondence between the Harish-Chandra series.
