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In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic character theory for the unitary groups $U(n)$ and symmetric groups $S(n)$ when $n\to\infty$ (this was first done by Vershik and Kerov). It is also said the detailed exposition of the main theorem of Vershik and Kerov regarding the characters of $U(\infty)$ via machinery of shifted Schur functions will be presented in the subsequent papers (see the part 4 of the introduction). However, I cannot find the exact paper (and not sure if it actually exists).

Thus, the question is: did Okounkov and/or Olshanski publish somewhere a paper with the proof of the Vershik-Kerov theorem that uses shifted Schur functions ($s^*$-functions in the notation of Okounkov and Olshanski)?

Here is the paper of Vershik and Kerov that Okounkov and Olshanski were referring to:

Vershik, A. M.; Kerov, S. V., Characters and factor representations of the infinite unitary group, Sov. Math., Dokl. 26, 570-574 (1982); translation from Dokl. Akad. Nauk SSSR 267, 272-276 (1982). ZBL0524.22017.

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Yes, they probably mean https://arxiv.org/abs/q-alg/9709011

There, they consider the Jack generalization of the problem, but if you set $\theta=1$, then you get the theory of characters of the unitary group.

An exposition (and another proof) of the theory is also given later here

https://arxiv.org/abs/1109.1412

(I followed up on this by streamlining their computations: https://arxiv.org/abs/1208.3443)

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    $\begingroup$ Thank you very much for your answer. And a follow-up question just out of curiosity: are you aware of any similar results for Hall-Littlewood (or even Macdonald) polynomials? Since there is already Jack-generalization, one might expect others as well. (However, here I am not sure whether it would have some representation-theoretical meaning for some "big group".) $\endgroup$
    – richrow
    Commented Mar 28 at 13:29
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    $\begingroup$ Yes, use google search on terms like “Macdonald Gibbs Gelfand Tsetlin” there would be q-deformation; then also results for Macdonald. By Gorin, Olshasnki and others, in combinations. $\endgroup$
    – Leonid Petrov
    Commented Mar 29 at 17:18

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