Reduction conjecture: discrete Fourier transform from the equivariant J-function to the Givental cone of Y
Prove that for a smooth projective variety X with an algebraic T_C-action and a smooth, non-orbifold GIT quotient Y, the series I_Y defined by I_Y := Σ_{[β]∈N_1^T(X)/N_1(X)} κ(^{−β} J_X) ^β satisfies: (1) its support lies in C_{Y,N}^\vee (the monoid in N_1^T(X) corresponding to the dual cone of Y’s chamber), and (2) z I_Y belongs to the non-equivariant Givental cone L_Y of Y over C[[C_{Y,N}^\vee]]. Here J_X is the big equivariant J-function and κ is the Kirwan map.
References
We state our reduction conjecture, whose formulation has been worked out with Fumihiko Sanda . It can be thought of as a version of Teleman's conjecture stated for solutions to quantum $D$-modules.
Let $X$ be a smooth projective variety equipped with an algebraic $T_C$-action. Let $Y$ be a smooth GIT quotient of $X$ without orbifold singularities. Let $J_X = J_X(\tau,z)$ denote the big equivariant $J$-function (see \S \ref{subsec:Givental_cone_J-function}) and let $\kappa \colon H*_T(X) \to H*(Y)$ denote the Kirwan map. Define the $H*(Y)$-valued power series
I_Y:= \sum_{[\beta] \in N_1T(X)/N_1(T)} \kappa( {-\beta} J_X) \beta \in \sum_{\beta \in N_1T(X)} H*(Y)[z,z{-1}][![\tau]!] \beta
where $\beta\in N_1T(X)$ is a representative of $[\beta] \in N_1T(X)/N_1(X) \cong H_2T(,Z)$ and $\beta$ denotes the element of the group ring $C[N_1T(X)]$ associated with $\beta$; each summand does not depend on the choice of a representative. Then we have the following.
(1) Let $C_Y \subset N1_T(X)_R$ be the GIT chamber of $Y$ and let $C_{Y,N}\vee$ be the set of $\beta \in N_1T(X)$ whose image in $N_1T(X)_R$ lies in the dual cone $C_Y\vee$. The $\beta$-power series $I_Y$ is supported on $C_{Y,N}\vee$.
(2) $z I_Y$ is a point on the (non-equivariant) Givental cone $L_Y$ of $Y$ defined over the extension $C[![C_{Y,N}\vee]!]$ of the Novikov ring of $Y$.