Abstract
We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the stable envelope in equivariant K-theory (with arbitrary slope) and equivariant cohomology.
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Notes
We allow q to be non-generic with \({\mathbb {Z}}\subsetneq Hom(E,E)\).
We refer to Section 7 of [5] where definitions of the elliptic Thom sheaf and the elliptic Euler class are discussed.
We may assume \(\hbar ^{1/2}\) exists by passing to the double cover of \(\mathsf {T}\) if needed.
This expression is singular because of \(\vartheta (1)\)-factors in the denominator. We, however, are only interested in its quasi-periods which are well defined.
If a box \(\Box =(i,j)\) they are defined as
$$\begin{aligned} l_{\lambda }(\Box )=\lambda _i-j,\quad a_{\lambda }(\Box )=\lambda {'}_j-i \end{aligned}$$where \(\lambda {'}\) is the transposition of \(\lambda \). Note that these functions are well defined even if \(\Box \notin \lambda \).
For a function \(\rho _\Box \) to define the correct ordering on boxes it is enough to choose \(\epsilon <\dfrac{1}{n}\).
More precisely, in Sect. 4.3, [2] the authors show that the right side of (88) is well defined and satisfies all defining properties for stable envelopes. Thus, by uniqueness it coincides with the left side, if it exists. The map \(\text {Stab}_{\mathfrak {C}}{'}\) is the elliptic stable envelope for hypertoric varieties and thus it is well defined and exists. This way, the existence of the elliptic stable envelopes for the Nakajima varieties is proven.
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Acknowledgements
The author thanks A.Okounkov for uncountable discussions, explanations and suggestions without which the accomplishment of this project would not be possible. In particular his idea to look at the fixed points corresponding to trees in Young diagrams was the turning point in this work. We would like to thank A. Osinenko and Y. Kononov for computer checks of the results of the paper and P. Pushkar for reading its preliminary version. The author is also grateful to M. Aganagic, I. Cherednik, D. Galakhov, S. Shakirov, A. Varchenko and Z. Zhou for discussions at various stages of this project. The work is supported in part by RFBR Grant 18-01-00926.
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