  Inelastic low-energy (0--1 eV) collisions of electrons with HeH$^+$ cations
are treated theoretically, with a focus on the rovibrational excitation and
dissociative recombination (DR) channels. In an application of {\it ab initio}
multichannel quantum defect theory (MQDT), the description of both processes is
based on the Born-Oppenheimer quantum defects. The quantum defects were
determined using the R-matrix approach in two different frames of reference:
the center-of-charge and the center-of-mass frames. The results obtained in the
two reference systems, after implementing the Fano-Jungen style rovibrational
frame-transformation technique, shows differences in the rate of convergence
for these two different frames of reference. We find good agreement with the
available theoretically predicted rotationally inelastic thermal rate
coefficients. Our computed DR rate also agrees well with available experimental
results. Moreover, several computational experiments shed light on the role of
rotational and vibrational excitations in the indirect DR mechanism that
governs the low energy HeH$^+$ dissociation process. While the rotational
excitation is several orders of magnitude more probable process at the studied
collision energies, the closed-channel resonances described by the high-n,
rotationally excited neutral molecules of HeH contribute very little to the
dissociation probability. But the situation is very different for resonances
defined by the high-n, vibrationally excited HeH molecules, which are found to
dissociate with approximately 90% probability.
