  In this paper we study the Cauchy problem for diffusion equations associated
to a class of strongly hypoelliptic pseudo-differential operators on graded Lie
groups. To do so, we develop a global complex functional calculus on graded Lie
groups in order to analyse the corresponding energy estimates. One of the main
aspects of this complex functional calculus is that for the
$(\rho,\delta)$-Euclidean H\"ormander classes we recover the standard
functional calculus developed by Seeley [38]. In consequence the G\r{a}rding
inequality that we prove for arbitrary graded Lie groups absorbs the historical
1953's inequality due to G\r{a}rding [26].
