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".

While people commonly complain about running out of letters (despite having Roman, Greek, Cyrillic, and Hebrew to get started with), one much more quickly runs out of delimiters (), [], {}, <>. It is fantastic that a new pair has been invented.

"Reduced" as in "reduced scheme", i.e., every polynomial function that vanishes on the set of commuting pairs is a combination of the entries of AB-BA.

Flip side of my original question: is there a corresponding notion of "strongly real character"?

I'm a little late to point this out, but, in type $A$ all maximal $P$ are cominuscule so one doesn't notice the importance of this condition.

ncatlab.org/nlab/show/Beilinson-Bernstein+localization I've mostly picked it up on the street, myself, so don't have a favorite reference.

The closest I know uses the direct sum operation $Gr(a,a+b) \times Gr(c,c+d) \to Gr(a+c,a+b+c+d)$. If you use the induced map on homology to define a bigraded ring, and give that total homology an obvious basis coming from Schubert varieties, you get something that's almost the ring of symmetric functions (i.e. has structure constants = Littlewood-Richardson).