We prove that the Hilbert scheme of
Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted
 NSFPAR ID:
 10471228
 Publisher / Repository:
 Springer
 Date Published:
 Journal Name:
 Journal of Fourier Analysis and Applications
 Volume:
 29
 Issue:
 1
 ISSN:
 10695869
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract k points on ($${\mathbb {C}}^2$$ ${C}^{2}$ ) is selfdual under threedimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant Ktheory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ action. First, we find a twoparameter family$${\mathbb {C}}^\times _\hbar $$ ${C}_{\u0127}^{\times}$ of selfmirror quiver varieties of type A and study their quantum Ktheory algebras. The desired quantum Ktheory of$$X_{k,l}$$ ${X}_{k,l}$ is obtained via direct limit$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (qLanglands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$l\longrightarrow \infty $$ $l\u27f6\infty $ opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsionfree rank$$\hbar $$ $\u0127$N sheaves on with the help of a different (threeparametric) family of type A quiver varieties with known mirror dual.$${\mathbb {P}}^2$$ ${P}^{2}$ 
Abstract We construct an example of a group
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