The elliptic Weyl character formula (Q2874609)

According to the author, this paper is part of a joint program with Arun Ram, which includes a fair bit of expository material on various topics related to equivariant elliptic cohomology \(\mathcal{E}{ ll}_G\). Here \(G\) is a compact connected Lie group of rank \(n\). The theme of this paper is concerned with a statement by the author saying, \(\lq\lq\)It has long been conjectured that \(\mathcal{E}{ ll}_G\) plays the same role for representation theory of the loop group \(\mathcal{L}G\) as \(K_G\) plays for the representation theory of \(G\)''. The author tries to show that the view presented above is not necessarily completely endorsed by discussing a particular example as its ground. This is outlined as follows. Let \(H\) be a connected Lie subgroup having the same rank as \(G\) and \(T\) a maximal torus in \(H\). Then the \(K_G\)-theoretic transfer along the map \(\pi :G/H \to \text{pt}\) gives the induction map NEWLINE\[NEWLINE\text{ind} : R(H) \to R(G).NEWLINE\]NEWLINE It is proved here that, in fact, a discussion in elliptic cohomology yields the Weyl-Kac formula which allows us to calculate \(\text{ind}\). In connection with this the calculation of \(\mathcal{E}{ ll}_T(G/H)\) is dealt with previous to this discussion. Then the author shows that the argument used there is identical to that for \(K_T(G/H)\), which just means the same thing as above, and she obtains the following: NEWLINE\[NEWLINEH_T(G/H)\cong H_T(\text{pt})\otimes_{H_G(\text{pt})}H_H(\text{pt}) \quad \text{and} \quad K_T(G/H)\cong R(T)\otimes_{R(G)}R(H),NEWLINE\]NEWLINE where \(H_T(-)\) is the direct product of the even Bredon cohomology with coefficients in \(\mathbb{C}\), Incidentally, in the case here \(\mathcal{E}{ ll}_T(-)\) may be viewed as \(H_T(-)\otimes_{\mathbb{C}[z]}\mathcal{O}_0\) where \(\mathcal{O}_0\) is the local ring of germs of holomorphic functions on \(C^n\) at 0 and \(H_T(\text{pt})\cong {\mathbb C}[z]\).