Conjectural Positivity for Pontryagin Product in Equivariant K-theory of Loop Groups

Abstract

Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B\supset T$ a Borel subgroup and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group $\Omega(K)$ gives rise to a comultiplication in the topological $T$-equivariant $K$-ring $K_T^{top}(\Omega(K))$. Recall that $\Omega(K)$ is identified with the affine Grassmannian $\mathcal{X}$ (of $G$) and hence we get a comultiplication in $ K_T^{top}(\mathcal{X})$. Dualizing, one gets the Pontryagin product in the $T$-equivariant $K$-homology $K^T_0(\mathcal{X})$, which in-turn gets identified with the convolution product (due to S. Kato). Now, $ K_T^{top}(\mathcal{X})$ has a basis $\{\xi^w\}$ over the representation ring $R(T)$ given by the ideal sheaves corresponding to the finite codimension Schubert varieties $X^w$ in $\mathcal{X}$. We make a positivity conjecture on the comultiplication structure constants in the above basis. Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in $T$-equivariant quantum $K$-theory $QK_T(G/B)$ in the Schubert basis.