## Duke Mathematical Journal

### Crystals via the affine Grassmannian

#### Abstract

Let G be a connected reductive group over ℂ, and let $\mathfrak {g}$ be the Langlands dual Lie algebra. Crystals for $\mathfrak {g}$ are combinatorial objects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "combinatorial skeletons" of finite-dimensional representations of $\mathfrak {g}$. For every dominant weight λ of $\mathfrak {g}$ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq($\mathfrak {g}$) and then specializing it to q=0. Other (independent) constructions of B(λ) were given by G. Lusztig (cf. [9]) using the combinatorics of root systems and by P. Littelmann (cf. [7]) using the "Littelmann path model." It was also shown in [5] that the family of crystals B(λ) is unique if certain reasonable conditions are imposed (cf. Theorem 1.1).

The purpose of this paper is to give another (rather simple) construction of the crystals B(λ) using the geometry of the affine Grassmannian \mathscr {G}$G=G($\mathscr{K}$)/G($\mathscr{O}$) of the group G, where$\mathscr{K}$=ℂ((t)) is the field of Laurent power series and$\mathscr{O}$=ℂ[[t]] is the ring of Taylor series. We then check that the family B(λ) satisfies the conditions of the uniqueness theorem from [5], which shows that our crystals coincide with those constructed in the references above. It would be interesting to find these isomorphisms directly (cf., however, [10]). #### Article information Source Duke Math. J., Volume 107, Number 3 (2001), 561-575. Dates First available in Project Euclid: 5 August 2004 Permanent link to this document https://projecteuclid.org/euclid.dmj/1091737024 Digital Object Identifier doi:10.1215/S0012-7094-01-10736-9 Mathematical Reviews number (MathSciNet) MR1828302 Zentralblatt MATH identifier 1015.20030 #### Citation Braverman, Alexander; Gaitsgory, Dennis. Crystals via the affine Grassmannian. Duke Math. J. 107 (2001), no. 3, 561--575. doi:10.1215/S0012-7094-01-10736-9. https://projecteuclid.org/euclid.dmj/1091737024 #### References • \lccA. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, preprint, 2000, http://zaphod.uchicago.edu/~benzvi/ • \lccA. Braverman and D. Gaitsgory, Geometric Eisenstein series, preprint, http://www.arXiv.org/abs/math.AG/9912097 • \lccD. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, preprint, http://www.arXiv.org/abs/math.AG/9912074 • \lccV. Ginzburg, Perverse sheaves on a loop group and Langlands' duality, preprint, http://www.arXiv.org/abs/math.AG/9511007 • \lccA. Joseph, Quantum Groups and Their Primitive Ideals, Ergeb. Math. Grenzgeb. (3) 29, Springer, Berlin, 1995. • \lccM. Kashiwara, On crystal bases of the$Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. • \lccP. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), 329–346. • \lccG. Lusztig, “Singularities, character formulas, and a$q\$-analog of weight multiplicities” in Analysis and Topology on Singular Spaces (Luminy, 1981), I, II, Astérisque 101–102, Soc. Math. France, Paris, 1983, 208–229.
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