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Our goal of this paper is to give a complete characterization of allholomorphic invariant strongly pseudoconvex complex Finsler metrics on theclassical domains and establish a corresponding Schwarz lemma for holomorphicmaps with respect to these invariant metrics. We prove that every$\mbox{Aut}(\mathfrak{D})$-invariant strongly pseudoconvex complex Finslermetric $F$ on a classical domain $\mathfrak{D}$ of rank $\geq 2$ is aK\"ahler-Berwald metric which is not necessary Hermitian quadratic, but itenjoys very similar curvature property as that of the Bergman metric on$\mathfrak{D}$. In particular, if $F$ is Hermitian quadratic, then $F$ is mustbe a constant multiple of the Bergman metric on $\mathfrak{D}$. This actuallyanswers the $4$-th open problem posed by Bland and Kalka in [7]. We also obtaina general Schwarz lemma for holomorphic maps from a classical domain$\mathfrak{D}_1$ into another classical domain $\mathfrak{D}_2$ whenever$\mathfrak{D}_1$ and $\mathfrak{D}_2$ are endowed with arbitrary holomorphicinvariant K\"ahler-Berwald metrics $F_1$ and $F_2$, respectively. The methodused to prove the Schwarz lemma is purely geometric. Our results show that theLu constant of $(\mathfrak{D},F)$ is both an analytic invariant and a geometricinvariant which can only be more clearly seen in the complex Finsler setting.This also gives a nontrivial generalization of the classical Schwarz lemma onthe classical domains previously established case-by-case by Look [31] withrespect to the Bergman metric.

Characterization of invariant complex Finsler metrics and Schwarz lemma on the classical domains