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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).

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    $\begingroup$ This question would be much more readable if you used LaTeX. See the FAQ mathoverflow.net/faq#latex Also, you could give a bit more background, and explanation of what everything is (what are $\mu$ and $\nu$?). $\endgroup$
    – Ben Webster ♦
    Dec 5, 2009 at 15:50
  • $\begingroup$ I have the same question. How to describe the springer correspondence explicitly in terms of partition and symbols? Any reference? $\endgroup$
    – Jia-jun Ma
    Jan 8, 2016 at 9:12

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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, for example.

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  • $\begingroup$ Sorry, I did try to work out what that means, but I don't know what it means. Can you please tell me what "take 2-quotients" means? $\endgroup$
    – Puraṭci Vinnani
    Dec 6, 2009 at 10:50

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