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added an Idea-subsection on the way that the equivalence between fractional D-branes stuck at orbifold singularities, and wrapped D-branes on the blow-up resolution (hence the K-theoretic incarnation of the McKay correspondence), is supposed to appear in the worldvolume gauge theory as passage between Higgs branch and Coulomb branch. Now here.
added an Idea-section making more explicit the central special case At global linear orbifold singularities
Is Langlands duality lurking nearby?
Concerning finite groups, David Ben-Zvi once told us:
the finite groups of Lie type - eg matrix groups GL,SL,SO,Sp etc over finite fields.. this class of groups contains almost all finite simple groups (other than cyclic, where the same holds since it’s easy for abelian groups, symmetric groups where analogous stories hold – the “F_1 case” – and some of the exceptional groups). Lusztig classified the reps of this enormous class of finite groups (that certainly should have earned him a Fields medal!), and they’re classified by conjugacy classes in the dual groups. I won’t give the precise statement but I think this is not a particularly misleading simplification.. it’s a pretty amazing piece of math!
Dunno. But the self-duality of the representations in the image of $\beta \;\colon\; A(G) \to R(G)$ is what gives rise to the Hecke operations.
’Hecke operator’ appears twice on the nLab. Are these the operators mentioned towards the end of Hecke algebra?
Yes! You see the self-duality of permutation representations used in the second step of the chain of equivalences shown there.
Is it worth us importing some of John Baez’s account of Hecke algebras/operators from the groupoidification project? It seems difficult to find exposition at that straightforward level:
we prefer the case where $G$ is finite, to avoid technicalities — and also because it’s good to consider a pair of subgroups $H,K \subseteq G$ and build intertwining operators $C[G/H] \to [G/K]$ from double cosets in $H\G/K$ and to groupoidify this construction.
So does that tell us anything (physical) that for two fractional D-branes corresponding to different irreps, the ’inner product’ of the charges is zero: $\sum_g Q_V(g) \cdot Q_{V'}(g) = 0$?
There may be a better answer than the following, but from equation (4.70) to (4.71) in
the authors point out that the orthogonality relation of the characters is equivalently the consistency condition that – after passing through the K-theoretic McKay correspondence from the wrapped D-branes on the resolution of the orbifold singularity to fractional branes at the actual orbifold singularity – the orbifold RR-fields in different twisted sectors are orthogonal to each other, as it must be (since closed strings cannot change their twisted sector).
Interesting. I like these physical interpretations of long-known mathematical facts.
I have added pointer (here and in related entries) that around (137) of
it is argued that the full equivariant K-theory group $K_G(\ast)$ is in fact too large for the classification of physical D-branes, that only elements determined by low degree Chern classes should be kept.
Which brings me back to the, meanwhile, widely appreciated but still unanswered question (here) how to compute the Chern classes of the bundle corresponding to a representation in terms of its characters.
Does the discussion here require $G$ a 1-group or does it hold more generally for higher groups? Certainly we can consider propagation on a higher stack $X//\mathcal{G}$, but is the discussion the same?
You’d need to speak of cohomology theories (like K-theory) equipped with equivariance with respect to higher groups.
That’s certainly a thought worth entertaining. But essentially nothing has been worked out.
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