Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

V. V. Krylov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 2, pp. 40-65

Résumé

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$, and let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct integrable crystals $\mathbf{B}^{G}(\lambda)$, $\lambda\in\Lambda^+_G$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of $G$. We also construct tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon\mathbf{B}^{G}(\lambda_1)\otimes\mathbf{B}^{G}(\lambda_2) \to\mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ in terms of multiplication in generalized transversal slices. Let $L \subset G$> be a Levi subgroup of $G$. We describe the functor $\operatorname{Res}^G_L\colon\operatorname{Rep}(G)\to\operatorname{Rep}(L)$ of restriction to $L$ in terms of the hyperbolic localization functors for generalized transversal slices.

Détail

Keywords: affine Grassmannian, Kashiwara crystals, geometric Satake isomorphism, generalized slices.

Bibliographie

[1] A. Braverman, “Spaces of quasi-maps into the flag varieties and their applications”, ICM 2006 Proceedings, Madrid, Eur. Math. Soc., Zürich, 2006, 1145–1170 | MR | Zbl

[2] A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigen-sheaves http://math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin

[3] A. Braverman, M. Finkelberg, “Pursuing the double affine Grassmannian. I: Transversal slices via instantons on $A_k$-singularities”, Duke Math. J., 152:2 (2010), 175–206 | DOI | MR | Zbl

[4] A. Braverman, M. Finkelberg, D. Kazhdan, “Affine Gindikin–Karpelevich formula via Uhlenbeck spaces”, Springer Proc. in Math. and Statistics, 9, 2012, 17–29 | DOI | MR | Zbl

[5] A. Braverman, M. Finkelberg, H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N}=4$ gauge theories, II, arXiv: 1601.03586 | MR

[6] A. Braverman, M. Finkelberg, D. Gaitsgory, I. Mirković, “Intersection cohomology of Drinfeld's compactifications”, Selecta Math., 8:3 (2002), 381–418 | DOI | MR | Zbl

[7] A. Braverman, D. Gaitsgory, “Crystals via the affine Grassmannian”, Duke Math. J., 107:3 (2001), 561–575 | DOI | MR | Zbl

[8] T. Braden, “Hyperbolic localization of intersection cohomology”, Transformation Groups, 8:3 (2003), 209–216 | DOI | MR | Zbl

[9] M. Finkelberg, J. Kamnitzer, K. Pham, L. Rybnikov, A. Weekes, “Comultiplication for shifted Yangians and quantum open Toda lattice”, Adv. Math., 327 (2018), 349–389 ; arXiv: 1608.03331 | DOI | MR | Zbl | MR

[10] M. Finkelberg, I. Mirković, “Semi-infinite flags. I. Case of global curve $\mathbb{P}^1$”, Amer. Math. Soc. Transl. Ser. 2, 194 (1999), 81–112 | MR | Zbl

[11] V. Ginzburg, Perverse sheaves on a loop group and Langlands' duality, arXiv: 9511007 | MR

[12] A. Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 29, Springer-Verlag, Berlin, 1995 | MR | Zbl

[13] M. Kashiwara, “On crystal bases”, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995, 155–197 | MR | Zbl

[14] M. Kashiwara, T. Tanisaki, “Kazhdan–Lusztig conjecture for affine Lie algebras with negative level”, Duke Math. J., 77:1 (1995), 21–62 | DOI | MR | Zbl

[15] I. Mirković, K. Vilonen, “Geometric Langlands duality and representations of algebraic groups over commutative rings”, Ann. of Math. (2), 166:1 (2007), 95–143 | DOI | MR | Zbl