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In Gaitsgory - Construction of central elements in the affine Hecke algebra via nearby cycles the idea of deforming the affine flag variety into the product of the affine Grassmannian and the usual flag variety is used (for a connected reductive group over a finite field).

Has a similar idea been used in any other papers? It seems like it could be useful in other settings.

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    $\begingroup$ Deforming the affine Grassmanian into the square of the affine Grassmanian is used in geometric Satake. $\endgroup$
    – Will Sawin
    Commented Sep 11, 2021 at 13:13
  • $\begingroup$ @WillSawin, re, is there a reference that explains this nicely? $\endgroup$
    – LSpice
    Commented Sep 11, 2021 at 14:52
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    $\begingroup$ @LSpice I think any modern reference on geometric Satake will discuss this (under the name "fusion"). I don't know the best one. $\endgroup$
    – Will Sawin
    Commented Sep 11, 2021 at 15:17
  • $\begingroup$ In arxiv.org/abs/1012.5979 "On the coherence conjecture of Pappas and Rapoport" Zhu takes Gaitsgory's $Gr \times G/B \rightsquigarrow AffFl$ and picks out the subfamily $Gr \times pt \rightsquigarrow AffFl$ (smaller in the general fiber, but with the same special fiber -- weird things happen in infinite dimensions). Gaitsgory's is equivariant w.r.t. $G(\mathcal O)$ whereas Zhu's is only Iwahori-invariant. As for fusion, the other names to attach to this family are "Beilinson-Drinfeld". $\endgroup$
    – Allen Knutson
    Commented Jul 11 at 3:08

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