Seminar

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The almost-commuting stack is the moduli of triples (g,X,q), where g is a group element, X is a Lie algebra element, and q is a scalar, such that g^{-1}Xg = qX. We compute the dg-algebra of G^\vee-invariant functions on the almost-commuting stack as a graded dg-algebra over k[q,1/q]. In other words, we calculate the pushforward of the structure sheaf to G_m/G_m. Away from roots of unity, this k[q,1/q]-dga agrees with O(T^\vee)^W[q,1/q], while at q = 1, there are derived corrections with nonzero grading degree. Unlike the related work of Li-Nadler-Yun on the multiplicative commuting stack, or of BenZvi-Chen-Helm-Nadler on coherent Springer theory, our approach does not require any form of Bezrukavnikov's equivalence. Instead, the primary technical input is a categorification of the Schur orthogonality relations / Hirzebruch-Riemann-Roch theorem, which is extracted from an adjointability criterion for doubly oplax-twisted topological field theories.