Do tax cuts to the rich lower interest rates? The rationale for giving tax cuts to the income brackets is that this money is then invested in new businesses which increases jobs. 

If the saved money is being saved/invested instead of being spent, this would increase the amount of lending capital available and decrease interest rates. 

Given that changing tax rates is a distinct and frequent occurrence around the world in a variety of contexts, there must be a lot of data available to perform a regression on. 

Do tax cuts to higher income brackets cause a subsequent lowering of interest rates?
You may want to look at the literature of the effects of taxation on savings. A survey is given in: Sandmo. A. (1985) The effects of taxation on savings and risk-taking, in: Auerbach &amp; Feldstein, Handbook of Public Economics, Elsevier.

In this survey, the (simpler) problem of the overall effect of taxation on saving behavior is analyzed. A change in tax progression is much more subtle and harder to measure.

The survey states the problems associated with estimation of this simpler problem and the author comes to the conclusion that the empirical results are inconclusive. Since then there may have been some move in the literature, however.