I have a bipolar (+-15V) square wave with a frequency of approx. 600 kHz. I want to amplify it to +-500V (not precisely), 50W. The load is around 1kOhm. Can anyone give me an advice on how to do it?
A transformer with 15:500 = 1:33 ratio can do this.
Be aware though that while this amplifies the voltage, it also reduces the current capability. Voltage times current is power, and you can't get power more power out of a passive device than you put in.
For example, let's say the ±15 V signal is capable of 1 A. After a 1:33 transformer, it will be ±495 V with a maximum theoretical current capability of 30 mA. In reality, there will be some loss in the transformer, so the actual output current capability will be less than 1/33 of the input current capability.
Since actively driving a signal at ±500 V is a lot more difficult than doing it at ±15 V, you should amplify the signal before the transformer. The only component that touches the high voltage output is the transformer. No other circuitry is required at the high voltage, and no high voltage power supplies are needed.
First, you need "at least" ±500V dual or 1kV single supply.
Anyway, building a long-tailed-pair with some high frequency power transistors (maximum collector-to-emitter voltage should be at least 1kV) is a way to go.
Another approach can be using the same configuration with high frequency vacuum tubes.
If output power is not an issue, you can use a 1:35 ferrite transformer as well.
Amplify and then transform is probably safest, just get power to well over 60 W at low voltage and then find suitable ignition transformers or similar that can handle the power and turns ratio.
I would go shopping for an old surplus medium wave RF transmitters, or Ham radio in the relevant band.
An alternate avenue is perhaps look for information on Tesla coil sparks to music as those folks get up to all sorts of tricks.