Geodesic
Cohomology of a flag variety as a~Bethe algebra
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 4, pp. 16-31.

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We interpret the equivariant cohomology $H^*_{GL_n}(\mathcal{F}_{\boldsymbol\lambda},mathbb{C})$ of a partial flag variety $\mathcal{F}_{\boldsymbol\lambda}$ parametrizing chains of subspaces $0=F_0\subset F_1\subset\dots\subset F_N=\mathbb{C}^n$, $\dim F_i/F_{i-1}=\lambda_i$, as the Bethe algebra $\mathcal{B}^\infty(\mathcal{V}^\pm_{\boldsymbol\lambda})$ of the $\mathfrak{gl}_N$-weight subspace $\mathcal{V}^\pm_{\boldsymbol\lambda}$ of a $\mathfrak{gl}_N[t]$-module $\mathcal{V}^\pm$.
Keywords: Gaudin model, Bethe algebra, cohomology of flag varieties. 
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A. N. Varchenko; R. Rimányi; V. O. Tarasov; V. V. Schechtman. Cohomology of a flag variety as a~Bethe algebra. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 4, pp. 16-31. http://geodesic.mathdoc.fr/item/FAA_2011_45_4_a1/

[1] M. F. Atiyah, R. Bott, “The moment map and equivariant cohomology”, Topology, 23:1 (1984), 1–28 | DOI | MR | Zbl

[2] A. Braverman, D. Maulik, A. Okounkov, Quantum cohomology of the Springer resolution, arXiv: 1001.0056

[3] V. Chari, S. Loktev, “Weyl, Demazure and fusion modules for the current algebra of $\mathfrak{sl}_{r+1}$”, Adv. Math., 207:2 (2006), 928–960 | DOI | MR | Zbl

[4] V. Chari, A. Pressley, “Weyl modules for classical and quantum affine algebras”, Represent. Theory, 5 (2001), 191–223 (electronic) | DOI | MR | Zbl

[5] A. Chervov, D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, arXiv: hep-th/0604128

[6] M. Gaudin, “Diagonalisation d'une classe d'Hamiltoniens de spin”, J. Physique, 37:10 (1976), 1089–1098 | DOI | MR

[7] M. Goden, Volnovaya funktsiya Bete, Mir, M., 1987 | MR

[8] E. Mukhin, V. Tarasov, A. Varchenko, “Bethe eigenvectors of higher transfer matrices”, J. Stat. Mech. Theory, 2006, no. 8, P08002, 44 pp. (electronic) | MR

[9] E. Mukhin, V. Tarasov, A. Varchenko, “Schubert calculus and representations of general linear group”, J. Amer. Math. Soc., 22:4 (2009), 909–940 | DOI | MR | Zbl

[10] E. Mukhin, V. Tarasov, A. Varchenko, “Spaces of quasi-exponentials and representations of $\mathfrak{gl}_N$”, J. Phys. A, 41:19 (2008), 194017 | DOI | MR | Zbl

[11] E. Mukhin, A. Varchenko, “Critical points of master functions and flag varieties”, Commun. Contemp. Math., 6:1 (2004), 111–163 | DOI | MR | Zbl

[12] A. Okounkov, Quantum Groups and Quantum Cohomology, Lectures at the 15th Midrasha Mathematicae on: “Derived categories of algebro-geometric origin and integrable systems” (December 19–24, 2010), Jerusalem, 2010

[13] R. Rimányi, A. Varchenko, Conformal blocks in the tensor product of vector representations and localization formulas, arXiv: 0911.3253 | MR

[14] R. Rimányi, V. Schechtman, A. Varchenko, Conformal blocks and equivariant cohomology, arXiv: 1007.3155

[15] D. Talalaev, Quantization of the Gaudin system, arXiv: /hep-th/0404153

[16] A. Varchenko, “A Selberg integral type formula for an $\mathfrak{sl}_2$ one-dimensional space of conformal blocks”, Mosc. Math. J., 10:2 (2010), 469–475 | DOI | MR | Zbl

[17] E. Vasserot, “Représentations de groupes quantiques et permutations”, Ann. Sci. École Norm. Sup., 26:6 (1993), 747–773 | DOI | MR | Zbl

[18] E. Vasserot, “Affine quantum groups and equivariant $K$-theory”, Transformation groups, 3:3 (1998), 269–299 | DOI | MR | Zbl