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The imaginary part of the quasinormal frequencies spectrum for a static andspherically symmetric black hole is analytically known to be equally spaced,both for the highly damped and the weakly damped families of quasinormal modes.Some interesting attempts have been made in the last twenty years to understandin simple ways this level spacing for the only case of highly dampedquasinormal frequencies. Here, we show that the overtone level spacing, forboth the highly damped and weakly damped families of quasinormal modes, cansimply be understood as a fingerprint of a hidden local $SL(2,\mathbb{R})$symmetry, near different regions of the black hole spacetime, i.e. thenear-horizon and the near-photon sphere regions.

The overtone level spacing of a black hole quasinormal frequencies: a fingerprint of a local $SL(2,\mathbb{R})$ symmetry