## Duke Mathematical Journal

- Duke Math. J.
- Volume 107, Number 3 (2001), 561-575.

### Crystals via the affine Grassmannian

Alexander Braverman and Dennis Gaitsgory

#### 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 *U*_{q}($\mathfrak {g}$^{∨}) 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