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KIC~8975515: a fast-rotating ($γ$ Dor - $δ$ Sct) hybrid star with Rossby modes and a slower $δ$ Sct companion in a long-period orbit

{KIC~8975515 is a \emph{Kepler} double-lined spectroscopic binary system with hybrid pulsations. Two components have similar atmospheric properties (T$_{\rm eff}$ $\sim$ 7400~K), and one of them is a fast rotator ($v\sin i = 162$ versus 32 km/s). Our aim is to study the \emph {Kepler} light curve in great detail in order to determine the frequencies of the pulsations, to search for regular spacing patterns in the Fourier spectrum, if any, and to discuss their origin in the context of binarity and fast rotation. In this paper, we study the properties of the stellar pulsations based on a careful analysis in the low-, intermediate- and high-frequency regions of the Fourier spectrum. This is done by performing repeated frequency-search analyses with successive prewhitenings of all the significant frequencies detected in the spectrum. Moreover, we searched for regular period spacings among the $g$ modes, as well as frequency splitting among the $g$ and $p$ modes. In the low-frequency regime, five regular period spacing patterns including one series of prograde $g$ modes and four series of retrograde $r$ modes were detected. The $r$ modes are well-distributed with respect to the harmonics of the rotational frequency of the fast-rotating star $f_{\rm rot}$ = 1.647 d$^{-1}$. The dominant $g$ mode is $f_{2}$ = 2.37 d$^{-1}$. The strongest p mode, at $f_{1}$ = 13.97 d$^{-1}$, forms a singlet. In the high-frequency region, we identified two multiplets of regularly split $p$ modes with mean frequency spacings of 0.42 d$^{-1}$ and 1.65 d$^{-1}$. We detected some series of retrograde $r$ and prograde $g$ modes as well as two multiplets of $p$ modes with frequency spacings related to the stellar rotation of both components of the twin system KIC~8975515. We identified the fast-rotating component as a hybrid pulsator with $r$ modes and the slowly-rotating component as a $δ$ Sct pulsator.