When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in the case of $A^{(2)}_{2n}$ (or from the perspective of Tits, odd unitary ramified groups).

Some symptoms of the problem:

  1. "All affine Schubert varieties are Gorenstein except for when the special parahoric is a parahoric of the of the ramified odd unitary group $SU_{2n+1}$... In this case, no Schubert variety of positive dimension in the affine flag variety is Gorenstein" -Zhu, "Coherence Conjecture of Pappas-Rapoport" page 7

  2. In Kac's construction of the twisted affine algebras from diagram automorphisms of the underlying semisimple $\mathfrak{g}$, (section 8.3 of Kac "Infinite Dimensional Lie Algebras, 3rd edition), the form of the generators is different (i.e. usually we choose $\theta_0=\frac{1}{r}(\overline{\mu}(\theta^0)+ \dots + \overline{\mu}(\theta^0))$, but to build $A_{2n}^{(2)}$ we use $\theta_0=\theta^0$). Moreover in several other places in the book, I think special care is required with calculations involving $A_{2n}^{(2)}$.

  3. Again according to Zhu in his "Coherence" paper, it seems that there have been miscalculations involving linebundles and their central charges for $Gr_{SU_{\tilde{C}/C}(2n)}$.

  4. In what is probably the closest to the "cause" rather than a symptom, we see in Tits "Reductive Groups over Local Fields", $A_{2n}^{(2)}$ (or in Tits' naming, $C-BC_n$), this is the only local Dynkin diagram where there exist special vertices which are not exchanged by a diagram automorphism.

I would be very happy to learn a unified perspective on this series of groups. I suspect that this last point 4) is the most important point, but since I'm not an expert on buildings or Bruhat-Tits theory, I would appreciate an understanding of why this feature of the Dynkin diagram (or of the building) has such serious ramifications (ha) for the associated geometry of flag varieties, Grassmannians and linebundles.

  • 5
    $\begingroup$ It is my impression that beyond the split cases or certain quasi-split, the affine buildings even for classical groups don't truly admit an approach as unified as one might like. There seems to be some inescapable combinatorial substratum that is not merely a manifestation of general principles. But perhaps I just gave up too soon myself in trying to understand. :) $\endgroup$ Commented Jul 10, 2020 at 2:59


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