Keynes' "aggregate supply function" in chapter 3 of the General Theory is just the "classical" labour demand function plus the "classical" production function. Except for the weird presentation, there is nothing new there. It is old and boring. It is Keynes' "aggregate demand function" that is new and exciting.
[Update: See Roger Farmer's response.]
Start with three equations:
1. The production function: Y=F(L). Output (Y) is a function of employment (L).
2. A "classical" labour demand curve: W/P=MPL(L). The real wage (W/P) equals the Marginal Product of Labour, which is a decreasing function of employment. This is Keynes' "first classical postulate", which he agreed with.
3. A "classical" labour supply curve: W/P=MRS(L,Y). The real wage equals the Marginal Rate of Substitution between labour (or leisure) and output (or consumption). This is Keynes' "second classical postulate", which he disagreed with (except at "full employment").
From 1 and 2, plus some tedious math, we can derive what Keynes calls "the aggregate supply function": PY/W = S(L). It shows the value of output, measured in wage units, as a function of employment. It is substantively identical to the Short Run Aggregate Supply Curve in intermediate macro textbooks that assume sticky nominal wages: Y=H(P/W), which uses the exact same equations 1 and 2, but presents the same solution differently.
From 1 and 3, plus some tedious math, we can derive a second "aggregate supply function", that is not in the General Theory: PY/W = Z(L). It is substantively identical to the short run aggregate supply curve implicit in New Keynesian models, which assume sticky P and perfectly flexible W, so the economy is always on the labour supply curve and always on the production function.
From 1 and 2 and 3, plus some tedious math, we can solve for Y, L, and W/P, and derive a third aggregate supply function: Y=Y*. This is the textbook Long Run Aggregate Supply curve. It is identical to the solution we could get if we solved for the levels of Y, L, and W/P that satisfied both the first and second "aggregate supply functions".
1. Y=log(L) production function
2. W/P=1/L "classical" labour demand function compatible with the above production function
3. W/P=Y/(1-L) "classical" labour supply function assuming utility is Cobb-Douglas in leisure and consumption.
From 1 and 2 we get YP/W=L.log(L) which is Keynes' "aggregate supply function". But this is substantively identical to the textbook SRAS function Y=log(P/W).
From 1 and 3 we get W/P=Y/(1-antilog(Y)), which tells us what is happening to the real wage in the New Keynesian model. And we could (someone could) rearrange it to solve for Y as a function of W/P, which is the second "aggregate supply function".
From 1, 2, and 3 together we get Y=(1-antilog(Y))/antilog(Y), which can be solved for Y to get the LRAS curve.
The first aggregate supply curve tells us the output that would be produced if firms could sell the output they wanted to sell, and could buy the labour they wanted to buy, as a function of W/P.
The second aggregate supply curve tells us the output that would be produced if households could sell the labour they wanted to sell, and could buy the output they wanted to buy, as a function of W/P.
The third aggregate supply curve tells us the output that would be produced if both firms and households could sell what they wanted to sell, where W/P adjusts so that both firms and households agree on how much they want to sell.
There is absolutely nothing new on the supply-side in chapter 3 of the General Theory.
It is the demand-side that is new. It is the idea that the demand for goods is a function of the quantity of labour that households are actually able to sell. If households are rationed in the labour market, that will spillover and affect their demand in the output market. Because the amount of labour they are actually able to sell, and hence the income they will earn from wages plus non-wages, depends on demand. Which means that demand depends upon demand. Demand depends on itself. That was new, and interesting.
[Update: but Keyne's weird habit of measuring output in wage units had an unfortunate result: because if YP/W=D(L) (which is Keyne's "aggregate demand function"), we can re-write that as Y=(W/P)D(L), and as Y=(W/P)D(F-1(Y)), which implies that doubling the real wage, for a given level of output and real income, would exactly double output demanded. Which contradicts Keynes' own consumption function, and only makes sense in a world where capitalists never spend any of their income, on either consumption of investment. Which makes no sense.
I can't help but think that if Keynes had assumed a simple haircut economy composed of self-employed hairdressers, where the production function is Y=L, the whole thing would have been a lot simpler and clearer. Because then W and P would be the same thing, and workers and firms would be the same thing, and the output supply function and the labour supply function would be the same thing, and my three aggregate supply functions would be the same thing. Then he could have concentrated on the output demand function, where he had something new to say, and it would be obvious that unemployment would not be caused by real wages being too high.]