Weighted voting: Vote of representative weighted by number of election votes I was wondering if there is any literature on voting systems where voters elect representatives, and in the representatives' assembly the vote of each representative is weighted by the number of votes she was elected with.

E.g.: A special but easy example would be the US supreme court. The 'voters' here are the US senators, their 'representatives' are the supreme court judges. Instead of each judge having an equal vote their vote could have a weight equal to the number of votes they were confirmed with. This would make a difference, as consensus candidates seem to be confirmed by more votes - at least in the current system. Changing the supreme court's voting mechanism could of course affect confirmations, but this may be a way to incentivize consensus.

In case of general elections and parliamentary representatives such a weighted voting system would incentivize representatives to try to represent all the voters in their district, not just the majority. (Assuming representatives care about their voting power, but one could always tie their salary to the number of votes they were elected by.)

The idea is simple, so it has probably been raised before. Please refer me to some literature.
You ask an excellent question, as it has been partly but not fully discussed in the scientific literature.

Your exact proposal (weighting representatives by the amount of votes they got in the election) has been discussed on a blog post (in french) by Jean-François Laslier see here. 

In practice, representatives are often elected on a geographical basis, and their weight is determined by the local area they represent (typically: its population). There is a whole literature on that subject. Balinski and Young [BY] focus on how to design (integer) voting weights that are as close as possible to proportional to the populations. A consequent literature has questioned the desirability of the idea of proportionality, a representative paper in this area is Barbera and Jackson [BJ].

Besides, your proposal seems to be one feature (among other) of the liquid democracy system advocated by democracy activists. In that system, each one can delegate or vote on each decision, and delegates' weight equals the number of received delegations. The system has been experimented by the German Pirate party. It has been studied recently in computer science. One issue is that some influent members may get too many votes [GKMP].

[BY]: Balinski and Young. Fair Representation. 1982

[BJ]: Barbera and Jackson. On the Weights of Nations: Assigning Voting
Weights in a Heterogeneous Union. 2006

[GKMP]: Gölz, P., Kahng, A., Mackenzie, S., &amp; Procaccia, A. D.. The Fluid Mechanics of Liquid Democracy. 2018