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The AGT relation identifies the Nekrasov functions for various N=2 SUSY gaugetheories with the 2d conformal blocks, which possess explicit Dotsenko-Fateevmatrix model (beta-ensemble) representations the latter being polylinearcombinations of Selberg integrals. The "pure gauge" limit of these matrixmodels is, however, a non-trivial multiscaling large-N limit, which requires aseparate investigation. We show that in this pure gauge limit the Selbergintegrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, theNekrasov function for pure SU(2) theory acquires a form very much reminiscentof the AMM decomposition formula for some model X into a pair of the BGWmodels. At the same time, X, which still has to be found, is the pure gaugelimit of the elliptic Selberg integral. Presumably, it is again a BGW model,only in the Dijkgraaf-Vafa double cut phase.

Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals