Duke Mathematical Journal

Crystals via the affine Grassmannian

Alexander Braverman and Dennis Gaitsgory

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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

Subjects
Primary: 20G05: Representation theory
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]

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


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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.
  • ––––, “Canonical bases arising from quantized enveloping algebras, II” in Common Trends in Mathematics and Quantum Field Theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl. 102, Progr. Theoret. Phys., Kyoto, 1991, 175–201.
  • ––––, An algebraic-geometric parametrization of the canonical basis, Adv. Math. 120 (1996), 173–190.
  • \lccI. Mirkovic and K. Vilonen, Perverse sheaves on loop Grassmannians and Langlands duality, preprint, http://www.arXiv.org/abs/math.AG/9703010