Crystals via the Affine Grassmannian

Let G be a connected reductive group over C and let g be the Langlands dual Lie algebra. Crystals for g are combinatoral objects, that were introduced by Kashiwara (cf. for example [5]) as certain “combinatorial skeletons” of finite-dimensional representations of g. For every dominant weight λ of g Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq(g∨) and then specializing it to q = 0. Other (independent) constructions of B(λ) were given by Lusztig (cf. [8]) using the combinatorics of root systems and by Littelmann (cf. [6]) using the “Littelmann path model”. It was also shown in [4] 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 GG = G(K)/G(O) of the group G, where K = C((t)) is the field of Laurent power series and 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 [4], which shows that our crystals coincide with those constructed in loc. cit. It would be interesting to find these isomorphisms directly (cf., however, [9]). 1. Basic results about crystals 1.1. Notation. Let G be a connected reductive group over C. Let g denote the Lie algebra of G and let g denote the Langlands dual Lie algebra (by definition, g contains a canonical Borel subalgebra b). Let l denote the rank of G (which is equal to the rank of g). Let also Rep(g) denote the category of finite-dimensional representations of the group g. Let ΛG denote the coweight lattice of G, which is the same as the weight lattice of g . We will denote by Λ+G the semi-group of dominant coweights. Let I denote the set of vertices of the Dynkin diagram corresponding to g. For i ∈ I we will denote by αi ∈ ΛG the corresponding simple coroot and by αi ∈ Λ ∨ G the corresponding simple root. For λ1, λ2 ∈ ΛG, we will write λ1 ≥ G λ2 if λ1 − λ2 is a linear combination of the αi with non-negative coefficients. Let Ei, Fi (for i ∈ I) denote the Chevalley generators of g. For every λ ∈ Λ + G we will denote by V (λ) the irreducible representation of g with highest weight λ and for μ ∈ Λ, V (λ)μ will denote the corresponding weight subspace of V (λ). 1.2. Definition. A crystal is a set B together with maps 1. wt : B → ΛG, εi, φi : B → Z, 2. ei, fi : B → B ∪ {0}, for each i ∈ I, satisfying the following axioms: A) For any b ∈ B one has φi(b) = εi(b) + 〈wt(b), αi 〉 B) Let b ∈ B. If ei · b ∈ B for some i. Then wt(ei · b) = wt(b) + αi, εi(ei · b) = εi(b)− 1, φi(ei · b) = φi(b) + 1. The work of both authors was partially supported by the National Science Foundation and by the Ellentuck Fund. 1 2 A. BRAVERMAN AND D. GAITSGORY If fi · b ∈ B for some i then wt(fi · b) = wt(b) − αi, εi(fi · b) = εi(b) + 1, φi(fi · b) = φi(b)− 1. C) For all b,b ∈ B one has b = ei · b if an only if b = fi · b. Remark. In [4] a more general definition of crystals is considered, where the maps εi and φi are allowed to assume infinite values. However, such crystals will never appear in this paper. A crystal is called normal if one has εi(b) = max{n| e n i · b 6= 0}, φi(b) = max{n| f n i · b 6= 0} (1.1) From now on we will consider only normal crystals. Thus, the maps εi and φi will be uniquely recovered from wt, ei and fi. 1.3. Tensor product of crystals. Let B1 and B2 be two crystals. Following Kashiwara ([5]) we define their tensor product B1 ⊗B2 as follows. As a set B1 ⊗B2 is just equal to B1 ×B2. The corresponding maps are defined in the following way. Let b1 ∈ B1,b2 ∈ B2. We will denote by b1 ⊗ b2 be the corresponding element in B1 ×B2. Then we set wt(b1 ⊗ b2) = wt(b1) + wt(b2), ei · (b1 ⊗ b2) = { ei · b1 ⊗ b2, if εi(b1) > φi(b2) b1 ⊗ ei · b2, otherwise fi · (b1 ⊗ b2) = { fi · b1 ⊗ b2, if εi(b1) ≥ φi(b2) b1 ⊗ fi · b2, otherwise εi(b1 ⊗ b2) = max{εi(b2), εi(b1)− φi(b2) + εi(b2)} φi(b1 ⊗ b2) = max{φi(b1), φi(b2)− εi(b1) + φi(b1)}. It is known (cf. [4] and [4]) that B1 ⊗ B2 is crystal and that ⊗ is an associative operation on crystals. Moreover, if B1 and B2 are normal then B1 ⊗B2 is normal as well. 1.4. Highest weight crystals. Let B be a crystal. We say that B is a highest weight crystal of weight λ ∈ ΛG if there exists an element bλ ∈ B, such that 1. wt(bλ) = λ. 2. ei · bλ = 0 for every i ∈ I. 3. B is generated by all the fi acting on bλ. It is clear from (1.1) that if B is a normal crystal, then one necessarily has λ ∈ Λ+G. The following lemma gives a useful reformulation of the definition of a highest weight crystal. Lemma 1.1. A crystal B is a highest weight crystal of highest weight λ if and only if there exists an element bλ ∈ B, such that 1. wt(bλ) = λ and wt(b) < λ for every b ∈ B− bλ. 2. ei · bλ = 0 for every i ∈ I. 3. For every b ∈ B− bλ there exists i ∈ I such that ei · b 6= 0. CRYSTALS VIA THE THE AFFINE GRASSMANNIAN 3 1.5. Closed families of crystals. Assume that for every λ ∈ Λ+G we are given a normal crystal B(λ) of highest weight λ. We say that the B(λ) form a closed family of crystals if for every λ, μ ∈ Λ+G there exists an embedding B(λ+ μ) →֒ B(λ) ⊗B(μ) (which necessarily sends bλ+μ to bλ ⊗ bμ). Theorem 1.1. (cf. [4], 6.4.21) Assume that G is of adjoint type. Then there exists a unique closed family of crystals B(λ). Different constructions of closed families of crystals were given by Kashiwara ([5]) using quantum groups and by Lusztig ([8]) and Littelman ([6]) using the combinatorics of the root systems. The main goal of this paper is to give another construction of the closed family B(λ), using the geometry of the affine Grassmannian. 2. Basic results about affine Grassmannian 2.1. Definition. Let K = C((t)), O = C[[t]]. By the affine Grassmannian of G we will mean the quotient GG = G(K)/G(O). It is known (cf. [1]) that GG is the set of C-points of an ind-scheme over C, which we will denote by the same symbol. The orbits of the group G(O) on GG can be described as follows. One can identify the lattice ΛG with the quotient T (K)/T (O). Fix λ ∈ Λ + G and let λ(t) denote any lift of λ to T (K). Let GG denote the G(O)-orbit of λ(t) (which clearly does not depend on the choice of h ). Then it is well-known (cf. [7]) that GG = ⊔