  Binaries of relatively massive black holes like GW190521 have been proposed
to form in dense gas environments, such as the disks of Active Galactic Nuclei
(AGNs), and they might be associated with transient electromagnetic
counterparts. The interactions of this putative environment with the binary
could leave a significant imprint at the low gravitational wave frequencies
observable with the Laser Interferometer Space Antenna (LISA). We show that
LISA will be able to detect up to ten GW190521-like black hole binaries, with
sky position errors $\lesssim1$ deg$^2$. Moreover, it will measure directly
various effects due to the orbital motion around the supermassive black hole at
the center of the AGN, especially the Doppler modulation and the Shapiro time
delay. Thanks to a careful treatment of their frequency domain signal, we were
able to perform the full parameter estimation of Doppler and Shapiro-modulated
binaries as seen by LISA. We find that the Doppler and Shapiro effects will
allow for measuring the AGN parameters (radius and inclination of the orbit
around the AGN, central black hole mass) with up to percent-level precision.
Properly modeling these low-frequency environmental effects is crucial to
determine the binary formation history, as well as to avoid biases in the
reconstruction of the source parameters and in tests of general relativity with
gravitational waves.
