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We suggest a way to study possible conformal symmetries on black holehorizons. We do this by carrying out a Kaluza-Klein like reduction of theEinstein-Hilbert action along the ignorable coordinates of stationary andaxisymmetric black holes. Rigid diffeomorphism invariance of the $m$-ignorablecoordinates then becomes a global $SL(m,R)$ gauge symmetry of the reducedaction. Related to each non-vanishing angular velocity there is a particular$SL(2,R)$ subgroup, which can be extended to the Witt algebra on the black holehorizons. The classical Einstein-Hilbert action thus has $k$-copies of infinitedimensional conformal symmetries on a given black hole horizon, with $k$ beingthe number of non-vanishing angular velocities of the black hole.

Conformal Symmetries of the Einstein-Hilbert Action on Horizons of Stationary and Axisymmetric Black Holes