Conjectural Positivity for Pontryagin Product in Equivariant K-theory of Loop Groups
Shrawan Kumar
TL;DR
The paper proposes a positivity conjecture for the comultiplication structure constants in the T-equivariant K-theory of the affine Grassmannian, formulated in a basis from Schubert-ideal sheaves. Via a dual Pontryagin product and Katos modified convolution, the conjecture is shown equivalent to positivity for the multiplicative structure constants in the T-equivariant quantum K-theory of the flag variety G/B, with an explicit link between the two through a nondegenerate pairing. It provides an explicit Demazure-operator formula for the modified convolution constants and demonstrates the theory in detail for G = SL2, highlighting computable positive expressions. The results unify loop-group K-theory with quantum K-theory and connect to a broad web of positivity conjectures in quantum K-theory, situating the work within Lenart–Maeno, Buch–Mihalcea–Perrin–Xu and related literature.
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 $Ω(K)$ gives rise to a comultiplication in the topological $T$-equivariant $K$-ring $K_T^{top}(Ω(K))$. Recall that $Ω(K)$ 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 $\{ξ^w\}$ 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.