Right, ok...yeah, it is kind of complicated but basically you aren't going to get any pure-HF (like Appaloosa) listing...it just doesn't make sense for anyone.
The fund is actually separate from the guys who advise it. So you have the fund that holds all the equities (or whatever) then you have a separate company that advises the fund. The reasons for this are fairly obvious when you think about it (the only funds where this isn't the case are unit trusts, I think), the fund has to be independent to make sure no-one is stealing money or anything. Pershing Square is a perfect example (I actually skimmed the IPO doc today)...what actually listed was a 30% share in a fund that they raised in 2007 (I think). This isn't Bill Ackman selling a bit of his company but, as you say, him getting permanent capital (the reason why he has listed a minority share is so no-one turns it and goes activist on him...this can happen and advisers can get booted off the funds they raise).
The reason for permanent capital is that it is hard to raise funds when they are most useful and most funds have to offer daily or weekly liquidity. So if I run a HF and the markets go down I might see lots of great stuff to buy but then my investors start redeeming and I end up selling into the market...not what you want. A really great case of this is Michael Burry. Another way to think about this is maturity transformation (as in banking), Ackman wants to make long-term investments but he has to offer his investors daily liquidity. To make long-term investments you need to permanent capital (this is why Rothschild and the Cazyer family have listed vehicles in London too).
Well yes, it did happen in the PE industry and it happened at IBs too when they went public. The reason why is simple...to make the most money you have to work somewhere where lazy shareholders aren't skimming the money made from your labour. Yes, liquidity is good...but, as I said already, who is going to buy Einhorn's stake when he can just turn around and do his own thing. Look at what happened to GLG, pretty much all the key people left...Coffey turned down $200m to go to Moore Capital...unsuccessfully it turned out but he isn't stupid, he did this because he knew that he could make more money there. More to the point, the guy is worth 3/4 of a billion dollars. We can assume most of this was made at GLG, do you think that today's GLG shareholders would let them take that kind of value out (the company probably isn't even worth that much today)? No way in hell. Even more to the point, why should someone like Coffey even give $1 of what he has created to someone who did absolutely nothing to deserve it.
The only way this works is if you have some kind of business, like KKR, with a brand or a repeatable formula that is ingrained in the company...then you have something to sell to the public. GLG is trying to build that up now and places like Oaktree definitely have it...there is a charismatic leader (like Schwartzman/Marks/Goettsman) but there is also a culture that ensures that you turn people with no imagination/talent/intelligence into private equity hotshots. Two other businesses to examine in this regard are Aberdeen Asset Management, very strong "team approach" that turns MBA-types into something modestly useful (and they just took over SWIP, precisely the opposite), and New Star, which had a ton of high-profile, high performing, high paid managers (it actually attracted them by paying them what they actually created, at most fund management houses the majority are subsidised by one or two people who actually know what they are doing) but flamed out in the financial crisis. The question of culture in fund management is interesting and is worth pursuing if you want to understand the biz (watch out for the partnership units when you are looking at PE though).