The following is from this talk: <http://www.maths.usyd.edu.au/u/anthonyh/piecestalk.pdf>, Slide 14. 


The Springer correspondence gives bijections


SO2n+1 \ N(so2n+1) ↔ {(μ; ν) | μi ≥ νi − 2, νi ≥ μi+1},


Sp2n \ N(sp2n) ↔ {(μ; ν) | μi ≥ νi − 1, νi ≥ μi+1 − 1},


obtained from the previous parametrizations by taking 2-quotients.


What I don't understand, is given a partition of say, $2n$, that is symplectic (odd parts occur with even multiplicity), how to construct a bijection to the set above; and same with orthogonal partitions (even parts occur with even multiplicity). 



A bijection between "symplectic" partitions and bi-partitions via Springer correspondance
From looking at the slides, it sure looks like you wrote it in your answer: take 2-quotients.


I'm not sure if there's a standard reference for n-quotients of partitions, but they're described in [this paper](http://arxiv.org/abs/math.CO/0609175), for example.


