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We relate the Bruce-Roberts numbers of a 1-form with respect to an ICIS to other invariants as the GSV-index, Tjurina and Milnor numbers.

We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $\mu_{BR}(f,X)$ is more complicated and also involves $\mu(f)$ and some lengths in terms of the ideals $I_X$ and $Jf$. We also consider the logarithmic characteristic variety, $LC(X)$, and its relative version, $LC(X)^{-}$. We show that $LC(X)^{-}$ is Cohen-Macaulay and that $LC(X)$ is Cohen-Macaulay at any point not in $X\times\{0\}$. We generalize previous results presented by the authors when $(X,0)$ has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.

We consider the relative Bruce-Roberts number $\mu_{BR}^{-}(f,X)$ of a function on an isolated hypersurface singularity $(X,0)$. We show that $\mu_{BR}^{-}(f,X)$ is equal to the sum of the Milnor number of the fibre $\mu(f^{-1}(0)\cap X,0)$ plus the difference $\mu(X,0)-\tau(X,0)$ between the Milnor and the Tjurina numbers of $(X,0)$. As an application, we show that the usual Bruce-Roberts number $\mu_{BR}(f,X)$ is equal to $\mu(f)+\mu_{BR}^{-}(f,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^-$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen-Macaulay.