Bases of tensor products and geometric Satake correspondence
aa r X i v : . [ m a t h . R T ] A ug Bases of tensor products and geometric Satake correspondence
Pierre Baumann, Stéphane Gaussent and Peter Littelmann
To the memory of C. S. Seshadri
Abstract
The geometric Satake correspondence can be regarded as a geometric construction of therational representations of a complex connected reductive group G . In their study of thiscorrespondence, Mirković and Vilonen introduced algebraic cycles that provide a linear ba-sis in each irreducible representation. Generalizing this construction, Goncharov and Shendefine a linear basis in each tensor product of irreducible representations. We investigatethese bases and show that they share many properties with the dual canonical bases ofLusztig. Let G be a connected reductive group over the field of complex numbers, endowed with a Borelsubgroup and a maximal torus. Let Λ + be the set of dominant weights relative to these data.Given λ ∈ Λ + , denote by V ( λ ) the irreducible rational representation of G with highest weight λ . Given a finite sequence λ = ( λ , . . . , λ n ) in Λ + , define V ( λ ) = V ( λ ) ⊗ · · · ⊗ V ( λ n ) . A construction due to Mirković and Vilonen [39] in the context of the geometric Satake corre-spondence endows the spaces V ( λ ) with linear bases. Specifically, V ( λ ) is identified with theintersection homology of a parabolic affine Schubert variety, and Mirković and Vilonen’s basisis the datum of a family Z ( λ ) of algebraic cycles in this variety. In [22], Goncharov and Shenextend this construction to the tensor products V ( λ ) . In the present paper, we investigate theproperties of these bases, which we call MV bases.We show that MV bases are compatible with the isotypical filtration of the representations V ( λ ) . They are also compatible with the action of the Chevalley generators of the Lie algebra g of G , in the sense that the leading terms of this action define on Z ( λ ) the structure of a1rystal in the sense of Kashiwara; this result is due to Braverman and Gaitsgory [8] in thecase n = 1 . With the help of the path model [33], we prove that there is a natural crystalisomorphism Z ( λ ) ∼ = Z ( λ ) ⊗ · · · ⊗ Z ( λ n ) (Braverman and Gaitsgory implicitly treated the case n = 2 ).We study the transition matrix between the MV basis of a tensor product V ( λ ) and the tensorproduct of the MV bases of the different factors V ( λ ) , . . . , V ( λ n ) . Using the fusion productin the sense of Beilinson and Drinfeld [5], we explain that the entries of this transition matrixcan be computed as intersection multiplicities. As a consequence, the transition matrix isunitriangular with nonnegative integral entries.The properties stated above are analogues of results obtained by Lusztig about the dual canon-ical bases. To be specific, Lusztig [37] defines a notion of based module (module endowed witha basis enjoying certain specific properties) over the quantized enveloping algebra U v ( g ) andshows the following facts: • A simple module over U v ( g ) , endowed with its canonical basis, is a based module. • The tensor product of finitely many based modules can be endowed with a basis thatmakes it a based module. This basis is constructed from the tensor product of the basesof the factors by adding corrective terms in an unitriangular fashion. • The basis of a based module is compatible with the decreasing isotypical filtration ofthe module. Each subquotient in this filtration, endowed with the induced basis, isisomorphic as a based module to the direct sum of copies of a simple module endowedwith its canonical basis.The dual canonical bases for the representations V ( λ ) (see [14]) can then be defined by dualizingLusztig’s construction and specializing the quantum parameter at v = 1 .Because of its compatibility with the isotypical filtration, the dual canonical basis of a tensorproduct V ( λ ) yields a linear basis of the invariant subspace V ( λ ) G . Just as well, the MV basisprovides a linear basis (sometimes called the Satake basis) of V ( λ ) G . The Satake basis andthe dual canonical basis of V ( λ ) G generally differ (the paper [12] provides a counterexample);nonetheless we show that the Satake basis enjoys the first two items in Khovanov and Kuper-berg’s list of properties for the dual canonical basis [30]. In particular, after restriction to theinvariant subspaces, the signed permutation V ( λ ) ⊗ V ( λ ) ⊗ · · · ⊗ V ( λ n ) ≃ −→ V ( λ ) ⊗ · · · ⊗ V ( λ n ) ⊗ V ( λ ) maps the Satake basis of the domain to that of the codomain.2s explained in [2], the MV bases of the irreducible representations V ( λ ) can be glued togetherto produce a basis of the algebra C [ N ] of regular functions on the unipotent radical N of B .Of particular interest would be any relation with the cluster algebra structure on C [ N ] [6, 21].The methods developed in the present paper allow for explicit computations; for instance weshow that the cluster monomials attached to certain seeds belong to the MV basis. However C [ N ] is not of finite cluster type in general; in other words cluster monomials do not span thewhole space. In type D we compare the MV basis element with the dual canonical and thedual semicanonical elements at the first position not covered by cluster monomials. Acknowledgements.
P.B. and S.G.’s research is in part supported by the ANR project GeoLie,ANR-15-CE40-0012. P.L.’s research is in part supported by the SFB/TRR 191 ‘SymplecticStructures in Geometry, Algebra and Dynamics’. P.B. thanks Joel Kamnitzer for several dis-cussions about the fusion product, which led to the main result of sect. 5. The authors warmlythank Simon Riche for help with several technicalities. P.B. also gratefully acknowledges thehospitality of the Centre de Recherches Mathématiques in Montreal, where research reportedin sect. 7.1 was carried out.
In the whole paper G is a connected reductive group over C , endowed with a Borel subgroup B and a maximal torus T ⊆ B . We denote by Λ the character lattice of T , by Φ ⊆ Λ theroot system of ( G, T ) , by Φ ∨ the coroot system, and by W the Weyl group. The datum of B determines a set of positive root in Φ ; we endow Λ with the dominance order ≤ relative tothis and denote the cone of dominant weights by Λ + . We denote by ρ : Λ → Q the half-sumof the positive coroots; so h ρ, α i = 1 for each simple root α . The Langlands dual of G is theconnected reductive group G ∨ over C built from the dual torus T ∨ = Λ ⊗ Z G m and the rootsystem Φ ∨ ; the positive coroots define a Borel subgroup B ∨ ⊆ G ∨ . The geometric Satake correspondence was devised by Lusztig [36] and given its definitiveshape by Beilinson and Drinfeld [5] and Mirković and Vilonen [39]. Additional references forthe material exposed in this section are [46] and [3].As recalled in the introduction of [41], loop groups appear under several guises across mathe-matics: there is the differential-geometric variant, the algebraic-geometric one, etc. We adoptthe framework of Lie theory and Kac-Moody groups [31]. For instance, though the affine Grass-mannian is a (generally non-reduced) ind-scheme, we will only look at its complex ind-variety3tructure.Let O = C [[ z ]] be the ring of formal power series in z with complex coefficients and let K = C (( z )) be the fraction field of O . The affine Grassmannian of the Langlands dual G ∨ is thehomogeneous space Gr = G ∨ ( K ) /G ∨ ( O ) . This space, a partial flag variety for an affine Kac-Moody group, is endowed with the structure of an ind-variety.Each weight λ ∈ Λ gives a point z λ in T ∨ ( K ) , whose image in Gr is denoted by L λ . The G ∨ ( O ) -orbit through L λ in Gr is denoted by Gr λ ; this is a smooth connected simply-connectedvariety of dimension ρ ( λ ) . The Cartan decomposition implies that Gr = G λ ∈ Λ + Gr λ ; moreover Gr λ = G µ ∈ Λ + µ ≤ λ Gr µ . Let
Perv(Gr) be the category of G ∨ ( O ) -equivariant perverse sheaves on Gr (for the middleperversity) supported on finitely many G ∨ ( O ) -orbits, with coefficients in C . This is an abeliansemisimple category; the simple objects in Perv(Gr) are the intersection cohomology sheaves I λ = IC (cid:16) Gr λ , C (cid:17) . (By convention, IC sheaves are shifted so as to be perverse.)Let θ ∈ Λ be a dominant and regular weight. The embedding C × θ −→ T ∨ ( C ) → G ∨ ( K ) gives rise to an action of C × on Gr . For each µ ∈ Λ , the point L µ is fixed by this action; wedenote its stable and unstable sets by S µ = n x ∈ Gr (cid:12)(cid:12)(cid:12) lim c → θ ( c ) · x = L µ o and T µ = n x ∈ Gr (cid:12)(cid:12)(cid:12) lim c →∞ θ ( c ) · x = L µ o and denote the inclusion maps by s µ : S µ → Gr and t µ : T µ → Gr .Given µ ∈ Λ and A ∈ Perv(Gr) , Mirković and Vilonen identify the homology groups H c (cid:0) S µ , ( s µ ) ∗ A (cid:1) and H (cid:0) T µ , ( t µ ) ! A (cid:1) via Braden’s hyperbolic localization, show that these groups are concentrated in degree ρ ( µ ) ,and define F µ ( A ) = H ρ ( µ ) (cid:0) T µ , ( t µ ) ! A (cid:1) and F ( A ) = M µ ∈ Λ F µ ( A ) . F is an exact and faithful functor from Perv(Gr) to the category of finite dimensional Λ -graded C -vector spaces. Mirković and Vilonen prove that F induces an equivalence F from Perv(Gr) to the category
Rep( G ) of finite dimensional rational representations of G , the Λ -graduation on F ( A ) giving rise to the decomposition of F ( A ) into weight subspaces. In thecourse of the proof, it is shown that F maps I λ to the irreducible highest weight representa-tion V ( λ ) .The map G ∨ ( K ) → Gr is a principal G ∨ ( O ) -bundle. From the G ∨ ( O ) -space Gr , we form theassociated bundle Gr = G ∨ ( K ) × G ∨ ( O ) Gr . This space is called the -fold convolution variety and has the structure of an ind-variety. Theaction of G ∨ ( K ) on Gr defines a map m : Gr → Gr of ind-varieties. Let p : G ∨ ( K ) → Gr and q : G ∨ ( K ) × Gr → Gr be the quotient maps. Given two equivariant perverse sheaves A and A on Gr , there is a unique equivariant perverse sheaf A e ⊠ A on Gr such that p ∗ A ⊠ A = q ∗ (cid:0) A e ⊠ A (cid:1) in the equivariant derived category of constructible sheaves on G ∨ ( K ) × Gr . We then definethe convolution product of A and A to be A ∗ A = m ∗ (cid:0) A e ⊠ A (cid:1) . Using Beilinson and Drinfeld’s fusion product, one defines associativity and commutativityconstraints and obtains a monoidal structure on
Perv(Gr) . Then F is a tensor functor; inparticular, the fusion product imparts an explicit identification of Λ -graded vector spaces F ( A ∗ A ) ∼ = F ( A ) ⊗ F ( A ) for any ( A , A ) ∈ Perv(Gr) . In this paper we study tensor products V ( λ ) = V ( λ ) ⊗ · · · ⊗ V ( λ n ) , where λ = ( λ , . . . , λ n ) is a sequence of dominant weights. Accordingly, we want to consider convolution products I λ = I λ ∗ · · · ∗ I λ n and therefore need the n -fold convolution variety Gr n = G ∨ ( K ) × G ∨ ( O ) · · · × G ∨ ( O ) G ∨ ( K ) | {z } n factors G ∨ ( K ) / G ∨ ( O ) .
5s customary, we denote elements in Gr n as classes [ g , . . . , g n ] of tuples of elements in G ∨ ( K ) .Plainly Gr = Gr ; in consequence, we write the quotient map G ∨ ( K ) → Gr as g [ g ] . Wedefine a map m n : Gr n → Gr by setting m n ([ g , . . . , g n ]) = [ g . . . g n ] .Given G ∨ ( O ) -stable subsets K , . . . , K n of Gr , we define K e × · · · e × K n = (cid:8) [ g , . . . , g n ] ∈ Gr n (cid:12)(cid:12) [ g ] ∈ K , . . . , [ g n ] ∈ K n (cid:9) . Alternatively, K e × · · · e × K n can be defined as b K × G ∨ ( O ) · · · × G ∨ ( O ) b K n / G ∨ ( O ) where each b K j is the preimage of K j under the quotient map G ∨ ( K ) → Gr .For λ = ( λ , . . . , λ n ) in (Λ + ) n , we set Gr λ n = Gr λ e × · · · e × Gr λ n and | λ | = λ + · · · + λ n . Viewing Gr λ n as an iterated fibration with base Gr λ and successive fibers Gr λ , . . . , Gr λ n , weinfer that it is a smooth connected simply-connected variety of dimension ρ ( | λ | ) ; also Gr n = G λ ∈ (Λ + ) n Gr λ n and Gr λ n = Gr λ e × · · · e × Gr λ n = G µ ∈ (Λ + ) n µ ≤ λ , ..., µ n ≤ λ n Gr µ n . Proposition 2.1
Let λ ∈ (Λ + ) n . Then I λ is the direct image by m n of the intersectioncohomology sheaf of Gr λ n with trivial local system, to wit I λ = ( m n ) ∗ IC (cid:16) Gr λ n , C (cid:17) , and the cohomology sheaves H k IC (cid:16) Gr λ n , C (cid:17) vanish unless k and ρ ( | λ | ) have the same parity.Proof. We content ourselves with the case n = 2 ; the proof is the same in the general case butrequires more notation. Working out the technicalities explained in [3], §1.16.4, we get IC (cid:16) Gr ( λ ,λ )2 , C (cid:17) = IC (cid:16) Gr λ , C (cid:17) e ⊠ IC (cid:16) Gr λ , C (cid:17) . Applying ( m ) ∗ then gives the announced equality, while the parity property follows from [36],sect. 11. (cid:3) λ ∈ Λ + and µ ∈ Λ , all the irreducible components of Gr λ ∩ S µ (respectively, Gr λ ∩ T µ ) have dimension ρ ( λ + µ ) (respectively, ρ ( λ − µ ) ) ([39], Theorem 3.2). We need asimilar result for the intersections Gr λ n ∩ ( m n ) − ( S µ ) and Gr λ n ∩ ( m n ) − ( T µ ) inside the n -foldconvolution variety.Let N ∨ be the unipotent radical of B ∨ . Then S µ is the N ∨ ( K ) -orbit through L µ ; this well-known fact follows from the easily proved inclusion N ∨ ( K ) L µ ⊆ S µ and the Iwasawa decom-position G ∨ ( K ) = G µ ∈ Λ N ∨ ( K ) z µ G ∨ ( O ) . (1)We record that S µ = ( N ∨ ( K ) z µ ) / N ∨ ( O ) and that for each λ ∈ Λ + , the action of the connectedsubgroup N ∨ ( O ) leaves stable the intersection Gr λ ∩ S µ , hence leaves stable each irreduciblecomponent of this intersection.The construction of the n -fold convolution variety is functorial in the group G ∨ ; applied to theinclusion B ∨ → G ∨ , this remark provides a natural map Ψ : G ( µ ,...,µ n ) ∈ Λ n (cid:0) N ∨ ( K ) z µ (cid:1) × N ∨ ( O ) · · · × N ∨ ( O ) (cid:0) N ∨ ( K ) z µ n (cid:1) / N ∨ ( O ) → Gr n . Using (1), we easily see that Ψ is bijective.Given weights µ , . . . , µ n and N ∨ ( O ) -stable subsets Z ⊆ S µ , . . . , Z n ⊆ S µ n , we define Z ⋉ · · · ⋉ Z n = e Z × N ∨ ( O ) · · · × N ∨ ( O ) e Z n / N ∨ ( O ) where each e Z j is the preimage of Z j under the quotient map N ∨ ( K ) z µ j → S µ j . With thisnotation, the bijectivity of Ψ implies that Gr n = G ( µ ,...,µ n ) ∈ Λ n Ψ (cid:0) S µ ⋉ · · · ⋉ S µ n (cid:1) . If Z , . . . , Z n are varieties, then Z ⋉ · · · ⋉ Z n is an iterated fibration with basis Z andsuccessive fibers Z , . . . , Z n and Ψ induces an homeomorphism from Z ⋉ · · · ⋉ Z n onto itsimage.We use the symbol Irr( − ) to designate the set of irreducible components of its argument. For λ ∈ Λ + , λ ∈ (Λ + ) n , and µ ∈ Λ , we define ∗ Z ( λ ) µ = Irr (cid:16) Gr λ ∩ S µ (cid:17) and ∗ Z ( λ ) µ = Irr (cid:16) Gr λ n ∩ ( m n ) − ( S µ ) (cid:17) . roposition 2.2 Let λ = ( λ , . . . , λ n ) in (Λ + ) n and let µ ∈ Λ .(i) All the irreducible components of Gr λ n ∩ ( m n ) − ( S µ ) have dimension ρ ( | λ | + µ ) .(ii) The map ( Z , . . . , Z n ) Ψ( Z ⋉ · · · ⋉ Z n ) induces a bijection G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ ∗ Z ( λ ) µ × · · · × ∗ Z ( λ n ) µ n ≃ −→ ∗ Z ( λ ) µ . (The bar above Ψ( Z ⋉ · · · ⋉ Z n ) means closure in ( m n ) − ( S µ ) .)Proof. One easily checks that ( m n ◦ Ψ) (cid:0) S µ ⋉ · · · ⋉ S µ n (cid:1) ⊆ S µ + ··· + µ n , whence ( m n ) − ( S µ ) = G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Ψ (cid:0) S µ ⋉ · · · ⋉ S µ n (cid:1) for any µ ∈ Λ . Adding λ = ( λ , . . . , λ n ) to the mix, we see that Gr λ n ∩ ( m n ) − ( S µ ) = G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Ψ (cid:16)(cid:16) Gr λ ∩ S µ (cid:17) ⋉ · · · ⋉ (cid:16) Gr λ n ∩ S µ n (cid:17)(cid:17) is the disjoint union over ( µ , . . . , µ n ) of an iterated fibration with base Gr λ ∩ S µ and successivefibers Gr λ ∩ S µ , . . . , Gr λ n ∩ S µ n . Statement (i) follows directly from this observation andMirković and Vilonen’s dimension estimates. Statement (ii) requires the additional argumentthat the spaces Gr λ j ∩ S µ j are contractible (because of the C × -action), so the monodromy isnecessarily trivial. (cid:3) For λ ∈ Λ + , λ ∈ (Λ + ) n , and µ ∈ Λ , we similarly define Z ( λ ) µ = Irr (cid:16) Gr λ ∩ T µ (cid:17) and Z ( λ ) µ = Irr (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) (cid:17) . Then all cycles in Z ( λ ) µ have dimension ρ ( | λ | − µ ) and there is a natural bijection G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Z ( λ ) µ × · · · × Z ( λ n ) µ n ≃ −→ Z ( λ ) µ . (2)Elements in ∗ Z ( λ ) µ , Z ( λ ) µ , ∗ Z ( λ ) µ or Z ( λ ) µ are called Mirković-Vilonen (MV) cycles. Forfuture use, we note that the map Z Z ∩ Gr λ n provides a bijection from Z ( λ ) µ onto the set ofirreducible components of Gr λ n ∩ ( m n ) − ( T µ ) . (Each Z ∈ Z ( λ ) µ meets the open subset Gr λ n of Gr λ n , because the dimension of (cid:0) Gr λ n \ Gr λ n (cid:1) ∩ ( m n ) − ( T µ ) is smaller than the dimension of Z .)8 .3 Mirković-Vilonen bases Following Goncharov and Shen ([22], sect. 2.4), we now define the MV basis of the tensorproduct representations V ( λ ) = F ( I λ ) = M µ ∈ Λ F µ ( I λ ) . Let λ ∈ (Λ + ) n and let µ ∈ Λ . By base change in the Cartesian square Gr λ n ∩ ( m n ) − ( T µ ) f / / (cid:15) (cid:15) Gr λ nm n (cid:15) (cid:15) T µ t µ / / Gr we compute F µ ( I λ ) = H ρ ( µ ) (cid:16) T µ , ( t µ ) ! ( m n ) ∗ IC (cid:16) Gr λ n , C (cid:17)(cid:17) = H ρ ( µ ) (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) , f ! IC (cid:16) Gr λ n , C (cid:17)(cid:17) . Let j : Gr λ n → Gr λ n and g : Gr λ n ∩ ( m n ) − ( T µ ) → Gr λ n be the inclusion maps. We can then lookat the sequence of maps F µ ( I λ ) → H ρ ( µ ) (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) , f ! j ∗ j ∗ IC (cid:16) Gr λ n , C (cid:17)(cid:17) = H ρ ( µ ) (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) , g ! C Gr λ n (cid:2) dim Gr λ n (cid:3)(cid:17) ∩ [Gr λ n ] −−−−→ H BM2 ρ ( | λ |− µ ) (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) (cid:17) . Here the first two maps carry out the restriction to Gr λ n (technically, an adjunction followedby a base change) and the last map is the Alexander duality ∗ .We claim that these maps are isomorphisms. For the Alexander duality, this comes fromthe smoothness of Gr λ n . For the restriction, consider a stratum Gr η n ⊆ Gr λ n with η = λ ;denoting the inclusion map by i and using the perversity condition, the parity property inProposition 2.1, and the dimension estimate for Gr η n ∩ ( m n ) − ( T µ ) , one checks that H k (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) , f ! i ∗ i ! IC (cid:16) Gr λ n , C (cid:17)(cid:17) vanishes if k < ρ ( µ ) + 2 ; therefore the stratum Gr η n does not contribute to F µ ( I λ ) . ∗ Specifically, the generalization presented in [15], sect. 19.1, equation (3) or in [24], Theorem IX.4.7.
9o sum up there is a natural isomorphism F µ ( I λ ) ≃ −→ H BM2 ρ ( | λ |− µ ) (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) (cid:17) . (3)The irreducible components Gr λ n ∩ ( m n ) − ( T µ ) have all dimension ρ ( | λ | − µ ) and their fun-damental classes provide a basis of the Borel-Moore homology group above. Gathering thesebases for all weights µ ∈ Λ produces what we call the MV basis of V ( λ ) . L -perfect bases In this section we consider a general setup, which captures properties shared by both theMV bases and the dual canonical bases. As before, G is a connected reductive group over C endowed with a Borel subgroup B and a maximal torus T ⊆ B , Λ is the character lattice of T , and Φ and Φ ∨ are the root and coroot systems of ( G, T ) . We denote by { α i | i ∈ I } theset of simple roots defined by B and by { α ∨ i | i ∈ I } the set of simple coroots. We endow Λ with the dominance order ≤ and denote the cone of dominant weights by Λ + . We regard theWeyl group W as a subgroup of Aut(Λ) ; for i ∈ I , we denote by s i the simple reflection alongthe root α i . When needed, we choose simple root vectors e i and f i of weights ± α i in the Liealgebra of G in such a way that [ e i , f i ] = − α ∨ i . We start by recalling the following definitions due to Kashiwara [28]. A semi-normal crystal isa set B endowed with a map wt : B → Λ and, for each “color” i ∈ I , with a partition into acollection of finite oriented strings. This latter structure is recorded by the datum of operators ˜ e i : B → B ⊔ { } and ˜ f i : B → B ⊔ { } which move an element of b ∈ B upwards and downwards, respectively, along the string ofcolor i to which b belongs. The special value is assigned to ˜ e i ( b ) or ˜ f i ( b ) when b is at the upperor lower end of a string of color i . For convenience one usually further sets ˜ e i (0) = ˜ f i (0) = 0 .The position of b in its string of color i is recorded by functions ε i and ϕ i defined as follows: ε i ( b ) = max (cid:8) n ≥ (cid:12)(cid:12) ˜ e i ( b ) = 0 (cid:9) , ϕ i ( b ) = max (cid:8) n ≥ (cid:12)(cid:12) ˜ f i ( b ) = 0 (cid:9) . Two compatibility conditions between the weight map wt and the datum of the partitions intooriented strings are required: first, wt( b ) increases by α i when b moves upwards the string ofcolor i to which it belongs; second, ϕ i ( b ) − ε i ( b ) = h α ∨ i , wt( b ) i for any b ∈ B and any i ∈ I .10hese conditions imply that the image of a string of color i by the map wt is stable under theaction of the simple reflection s i . As a consequence, the set { wt( b ) | b ∈ B } is stable under theaction of the Weyl group W .The direct sum of two semi-normal crystals B and B is defined to be just the disjoint unionof the underlying sets. The tensor product B ⊗ B is defined to be the Cartesian productof the sets endowed with the maps given in [28], §7.3. Notably, for each color, the strings in B ⊗ B are created from the strings contained in B and in B by the process illustrated bythe picture below. B ⊇ ⊆ B L -perfect bases To a subset J ⊆ I we attach the standard Levi subgroup M J of G , the cone Λ + J = (cid:8) λ ∈ Λ (cid:12)(cid:12) ∀ j ∈ J, h α ∨ j , λ i ≥ (cid:9) of dominant weights for M J , and the J -dominance order ≤ J on Λ defined by µ ≤ J λ ⇐⇒ λ − µ ∈ span N { α j | j ∈ J } . Given λ ∈ Λ + J we denote by V J ( λ ) the irreducible rational representation of M J with highestweight λ . Given a finite sequence λ = ( λ , . . . , λ n ) in Λ + J we define V J ( λ ) = V J ( λ ) ⊗ · · · ⊗ V J ( λ n ) . For J = I we recover the conventions previously used by dropping the decoration J in thenotation Λ + J , ≤ J or V J ( λ ) .Let V be a rational representation of G . With respect to the action of M J the space V can beuniquely written as a direct sum of isotypical components V = M µ ∈ Λ + J V J,µ V J,µ is the sum of all subrepresentations of res GM J ( V ) isomorphic to V J ( µ ) . We define V J, ≤ J µ = M ν ∈ Λ + J ν ≤ J µ V J,ν . We say that a linear basis B of V is L -perfect † if for each J ⊆ I and each µ ∈ Λ + J :(P1) The subspace V J, ≤ J µ is spanned by a subset of B .(P2) The induced basis on the quotient V J, ≤ J µ /V J,< J µ ∼ = V J,µ is compatible with a decompo-sition of the isotypical component as a direct sum of irreducible representations.Taking J = ∅ , we see that an L -perfect basis of V consists of weight vectors (note that ≤ ∅ isthe trivial order on Λ ). Now let i ∈ I , and for each nonnegative integer ℓ , define V { i } , ≤ ℓ = M µ ∈ Λ0 ≤h α ∨ i ,µ i≤ ℓ V { i } ,µ , the sum of all irreducible subrepresentations of res GM { i } ( V ) of dimension at most ℓ + 1 . If B satisfies the conditions (P1) and (P2) for J = { i } , then V { i } , ≤ ℓ is spanned by B ∩ V { i } , ≤ ℓ andthe induced basis on the quotient V { i } , ≤ ℓ /V { i } , ≤ ℓ − is compatible with a decomposition as adirect sum of irreducible representations. Therefore ( B ∩ V { i } , ≤ ℓ ) \ ( B ∩ V { i } , ≤ ℓ − ) decomposesas the disjoint union of oriented strings of length ℓ , in such a way that the simple root vector e i or f i acts on a basis vector of V { i } , ≤ ℓ /V { i } , ≤ ℓ − by moving it upwards or downwards alongthe string that contains it, up to a scalar.We can sum up the discussion in the previous paragraph as follows: if B satisfies (P1) and(P2) for all J of cardinality ≤ , then B is endowed with the structure of crystal and is perfectin the sense of Berenstein and Kazhdan ([7], Definition 5.30). Lemma 3.1
Let B be an L -perfect basis of a rational representation V of G and let B ′ ⊆ B .Assume that the linear space V ′ spanned by B ′ is a subrepresentation of V . Then B ′ is an L -perfect basis of V ′ and (the image of ) B \ B ′ is an L -perfect basis of the quotient V /V ′ . † L stands for Levi. This notion of L -perfect basis appears unnamed (and in a dual form) in Braverman andGaitsgory’s paper ([8], sect. 4.3). roof. Let J ⊆ I and µ ∈ Λ + J . Then V ′ J,µ = V ′ ∩ V J,µ and V ′ J, ≤ J µ = V ′ ∩ V J, ≤ J µ . Now bothspaces V ′ and V J, ≤ J µ are spanned by a subset of the basis B , so their intersection V ′ J, ≤ J µ isspanned by a subset of B , namely B ′ ∩ V J, ≤ J µ .Let C be the image of ( B ∩ V J, ≤ J µ ) \ ( B ∩ V J,< J µ ) in the quotient V J, ≤ J µ /V J,< J µ ∼ = V J,µ . Then C can be viewed as a basis of V J,µ and it can be partitioned into disjoint subsets C , . . . , C n so that each C k spans an irreducible subrepresentation. By construction, the subspace V ′ J,µ is spanned by a subset C ′ ⊆ C . Each subset C k can be either contained in C ′ or disjointfrom C ′ , depending whether the subrepresentation that it spans is contained in V ′ J,µ or meetstrivially V ′ J,µ . Therefore C ′ is the disjoint union of some C k ; in other words C ′ is compatiblewith a decomposition of V ′ J,µ as a direct sum of irreducible subrepresentations.Thus, B ′ satisfies both conditions (P1) and (P2), and is therefore an L -perfect basis of V ′ . Theproof that B \ B ′ yields an L -perfect basis of the quotient V /V ′ rests on similar arguments andis left to the reader. (cid:3) Under the assumptions of Lemma 3.1, the subset B ′ is a subcrystal of B ; in other words, thecrystal structure on B is the direct sum of the crystal structures on B ′ and B \ B ′ . Proposition 3.2
Let V be a rational representation of G . Up to isomorphism, the crystal ofan L -perfect basis of V depends only on V , and not on the basis.Proof. Let B be an L -perfect basis of V . The conditions imposed on B with the choice J = I imply the existence in V of a composition series compatible with B . By Lemma 3.1, the crystal B is the direct sum of the crystals of the L -perfect bases induced by B on the subquotients.It thus suffices to prove the desired uniqueness property in the particular case where V is anirreducible representation, which in fact is just Theorem 5.37 in [7]. (cid:3) In particular, the crystal of an L -perfect basis of an irreducible representation V ( λ ) is unique.We use henceforth the notation B ( λ ) for the associated crystal. Remark 3.3.
The crystal B ( λ ) of an irreducible representation V ( λ ) was introduced by Kashi-wara in the context of representations of quantum groups. The definition via crystallization at q = 0 and the definition via the combinatorics of L -perfect bases yield the same crystal; thisfollows from [26], sect. 5.Fortunately this nice little theory is not empty. As mentioned in the introduction, any tensorproduct of irreducible representations has an L -perfect basis, namely its dual canonical basis.Another example for a L -perfect basis: it can be shown, in the case where G is simply laced,that the dual semicanonical basis of an irreducible representation is L -perfect.13 heorem 3.4 The MV basis of a tensor product of irreducible representations is L -perfect. The end of sect. 3 is devoted to the proof of this result. The case of an irreducible representationis Proposition 4.1 in [8]. The proof for an arbitrary tensor product follows the same lines; itis only sketched in loc. cit. , and we add quite a few details to Braverman and Gaitsgory’sexposition.
Consider a subset J ⊆ I . In this section we recall Beilinson and Drinfeld’s geometric construc-tion of the restriction functor res GM J ([5], sect. 5.3). Additional details can be found in [38],sect. 8.6 and [3], sect. 1.15.Define the root and coroot systems Φ J = Φ ∩ span Z { α j | j ∈ J } and Φ ∨ J = Φ ∨ ∩ span Z { α ∨ j | j ∈ J } and denote by ρ J : Λ → Q the half-sum of the positive coroots in Φ ∨ J . Then ρ − ρ J vanisheson Z Φ J so induces a linear form ρ I,J : Λ / Z Φ J → Q .To J we also attach the standard Levi subgroup M ∨ J of G ∨ . Choose a dominant θ J ∈ Λ suchthat h α ∨ j , θ J i = 0 for each j ∈ J and h α ∨ i , θ J i > for each i ∈ I \ J . The embedding C × θ J −→ T ∨ ( C ) → G ∨ ( K ) gives rise to an action of C × on Gr . Then the set Gr J of fixed points under this action canbe identified with the affine Grassmannian for M ∨ J . We denote by Perv(Gr J ) the category of M ∨ J ( O ) -equivariant sheaves on Gr J supported on finitely many M ∨ J ( O ) -orbits.Let ζ ∈ Λ / Z Φ J be a coset; then all the points L µ for µ ∈ ζ belong to the same connectedcomponent of Gr J , which we denote by Gr J,ζ . The map ζ Gr J,ζ is a bijection from Λ / Z Φ J onto π (Gr J ) . We denote the stable and unstable sets of Gr J,ζ with respect to the C × -actionby Gr + J,ζ and Gr − J,ζ and form the diagram
GrGr + J,ζs Jζ > > ⑤⑤⑤⑤⑤⑤⑤ p J,ζ ❅❅❅❅❅❅ Gr − J,ζt Jζ ` ` ❇❇❇❇❇❇❇ q J,ζ ~ ~ ⑦⑦⑦⑦⑦⑦ Gr J,ζ (4)14here s Jζ and t Jζ are the inclusion maps and the maps p J,ζ and q J,ζ are defined by p J,ζ ( x ) = lim c → θ J ( c ) · x and q J,ζ ( x ) = lim c →∞ θ J ( c ) · x. Given ζ ∈ Λ / Z Φ J and A ∈ Perv(Gr) , Beilinson and Drinfeld identify the two sheaves ( q J,ζ ) ∗ ( t Jζ ) ! A and ( p J,ζ ) ! ( s Jζ ) ∗ A on Gr J,ζ via Braden’s hyperbolic localization and show that they live in perverse degree ρ I,J ( ζ ) . Then they define a functor r IJ : Perv(Gr) → Perv(Gr J ) by r IJ ( A ) = M ζ ∈ Λ / Z Φ J ( q J,ζ ) ∗ ( t Jζ ) ! A [2 ρ I,J ( ζ )] . For µ ∈ Λ , let T J,µ be the analog of the unstable subset T µ for the affine Grassmannian Gr J .Let ζ be the coset of µ modulo Z Φ J and let t J,µ : T J,µ → Gr J,ζ be the inclusion map. Usingthe Iwasawa decomposition, one checks that T µ ⊆ Gr − J,ζ and T µ = ( q J,ζ ) − ( T J,µ ) . (5)Performing base change in the Cartesian square T µ / / / / t µ ( ( (cid:15) (cid:15) Gr − J,ζ t Jζ / / q J,ζ (cid:15) (cid:15) Gr T J,µ t
J,µ / / Gr J,ζ we obtain, for any sheaf A ∈ Perv(Gr) , a canonical isomorphism H ρ ( µ ) (cid:0) T µ , ( t µ ) ! A (cid:1) ∼ = H ρ J ( µ ) (cid:0) T J,µ , ( t J,µ ) ! r IJ ( A ) (cid:1) . (6)For B ∈ Perv(Gr J ) , define F J,µ ( B ) = H ρ J ( µ ) (cid:0) T J,µ , ( t J,µ ) ! B (cid:1) and F J ( B ) = M µ ∈ Λ F J,µ ( B ) . Then (6) can be rewritten as F µ = F J,µ ◦ r IJ . This equality can be refined in the followingstatement: the functor F J induces an equivalence F J from Perv(Gr J ) to the category Rep( M J ) of finite dimensional rational representations of M J and the following diagram commutes. Perv(Gr) F / / r IJ (cid:15) (cid:15) Rep( G ) res GMJ (cid:15) (cid:15)
Perv(Gr J ) F J / / Rep( M J ) .4 The J -decomposition of an MV cycle We fix a subset J ⊆ I . We denote by P − , ∨ J the parabolic subgroup of G ∨ containing M ∨ J andthe negative root subgroups.The group P − , ∨ J ( K ) certainly acts on Gr ; it also acts on Gr J via the quotient morphism P − , ∨ J ( K ) → M ∨ J ( K ) . Given µ ∈ Λ + J , we denote by Gr µJ the orbit of L µ under the action of M ∨ J ( O ) (or P − , ∨ J ( O ) ) on Gr J . Noting that lim a →∞ θ J ( a ) g θ J ( a ) − = 1 for all g in the unipotent radical of P − , ∨ J , we see that for any ζ ∈ Λ / Z Φ J , the connectedcomponent Gr J,ζ of Gr J and the unstable subset Gr − J,ζ in Gr are stable under the action of P − , ∨ J ( O ) and that the map q J,ζ is equivariant.Let λ ∈ (Λ + ) n , let µ ∈ Λ + J and let ζ be the coset of µ modulo Z Φ J . We consider the followingdiagram. ( m n ) − (cid:0) Gr − J,ζ (cid:1) m n / / (cid:127) _ (cid:15) (cid:15) Gr − J,ζ q
J,ζ / / t Jζ (cid:15) (cid:15) Gr J,ζ Gr n m n / / Gr The group G ∨ ( K ) acts on Gr n by left multiplication on the first factor and the action of thesubgroup G ∨ ( O ) leaves Gr λ n stable. Let H be the stabilizer of L µ with respect to the action of P − , ∨ J ( O ) on Gr J ; it acts on E = Gr λ n ∩ ( q J,ζ ◦ m n ) − ( L µ ) . Since q J,ζ ◦ m n is equivariant underthe action of P − , ∨ J ( O ) , we can make the identification P − , ∨ J ( O ) × H E ∼ = / / (cid:15) (cid:15) Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) q J,ζ ◦ m n (cid:15) (cid:15) P − , ∨ J ( O ) /H ∼ = / / Gr µJ where the left vertical arrow is the projection along the first factor. We thereby see that theright vertical arrow is a locally trivial fibration.In particular, all the fibers Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) with x ∈ Gr µJ are isomorphic varieties.Remembering that ( q J,ζ ) − ( L µ ) ⊆ T µ , we find the following bound for their dimension: dim (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) (cid:17) = dim E ≤ dim (cid:16) Gr λ n ∩ ( m n ) − ( T µ ) (cid:17) = ρ ( | λ | − µ ) . dim (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) (cid:17) ≤ dim Gr µJ + ρ ( | λ | − µ ) = 2 ρ J ( µ ) + ρ ( | λ | − µ ) . (7)Since Gr µJ is connected and simply-connected, the fibration induces a bijection between the setof irreducible components of Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) and the set of irreducible componentsof any fiber Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) .We define Z J ( λ ) µ = n Z ∈ Irr (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) (cid:17) (cid:12)(cid:12)(cid:12) dim Z = 2 ρ J ( µ ) + ρ ( | λ | − µ ) o . For ν ∈ Λ , we define Z J ( µ ) ν = Irr (cid:16) Gr µJ ∩ T J,ν (cid:17) ; as we saw in sect. 2.2, the map Z Z ∩ Gr µJ is a bijection from Z J ( µ ) ν onto the set ofirreducible components of Gr µJ ∩ T J,ν .Fix now λ ∈ (Λ + ) n and ν ∈ Λ . Following Braverman and Gaitsgory’s method, we define abijection Z ( λ ) ν ∼ = G µ ∈ Λ + J Z J ( λ ) µ × Z J ( µ ) ν . The union above is in fact restricted to those weights µ such that µ − ν ∈ Z Φ J , for otherwise Z J ( µ ) ν is empty. Let ζ denote the coset of ν modulo Z Φ J .First choose µ ∈ Λ + J ∩ ζ and a pair ( Z J , Z J ) ∈ Z J ( λ ) µ × Z J ( µ ) ν . Using (5) and the fibrationabove, we see that Z J ∩ ( q J,ζ ◦ m n ) − ( Z J ∩ Gr µJ ) is an irreducible subset of Gr λ n ∩ ( q J,ζ ◦ m n ) − ( T J,ν ) = Gr λ n ∩ ( m n ) − ( T ν ) of dimension dim Z J − dim Gr µJ + dim Z J = ρ ( | λ | − µ ) + ρ J ( µ − ν ) = ρ ( | λ | − ν ) . Therefore there is a unique Z ∈ Z ( λ ) ν that contains Z J ∩ ( q J,ζ ◦ m n ) − ( Z J ∩ Gr µJ ) as a densesubset.Conversely, start from Z ∈ Z ( λ ) ν . Then Z ⊆ T ν ⊆ Gr − J,ζ . We can thus partition Z intolocally closed subsets as follows. Z = G µ ∈ Λ + J ∩ ζ (cid:16) Z ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) (cid:17) Z is irreducible, there is a unique µ ∈ Λ + J ∩ ζ such that Z ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) is opendense in Z . That subset is certainly irreducible, hence contained in an irreducible component Z J of Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) . Also, q J,ζ ◦ m n maps Z ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) to an irreduciblesubset of Gr µJ ∩ T J,ν , which in turn is contained in an irreducible component Z J ∈ Z J ( µ ) ν .Then Z ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) ⊆ Z J ∩ ( q J,ζ ◦ m n ) − ( Z J ) . The left-hand side has dimension ρ ( | λ | − ν ) and the right-hand side has dimension dim Z J − ρ J ( µ ) + dim Z J = dim Z J − ρ J ( µ ) + ρ ( µ − ν ); combining with the bound (7) we get dim Z J = ρ ( | λ | − µ ) + 2 ρ J ( µ ) and therefore Z J ∈ Z J ( λ ) µ .These two constructions define mutually inverse bijections; in particular, Z J ∩ ( q J,ζ ◦ m n ) − ( Z J ∩ Gr µJ ) = Z ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) . We record that to each MV cycle Z ∈ Z ( λ ) ν is assigned a weight µ ∈ Λ + J characterized bythe conditions ( q J,ζ ◦ m n )( Z ) ⊆ Gr µJ and ( q J,ζ ◦ m n )( Z ) ∩ Gr µJ = ∅ ; in the sequel this weight will be denoted by µ J ( Z ) . L -perfect We now give the proof of Theorem 3.4, properly speaking. We fix a positive integer n and atuple λ ∈ (Λ + ) n . We need two ingredients besides the constructions explained in sects. 2.3and 3.3.(A) Take A ∈ Perv(Gr) and write the sheaf B = r IJ ( A ) = M ζ ∈ Λ / Z Φ J ( q J,ζ ) ∗ ( t Jζ ) ! A [2 ρ I,J ( ζ )] in Perv(Gr J ) as a direct sum of isotypical components B = M µ ∈ Λ + J IC (cid:16) Gr µJ , L µ (cid:17) . (8)18he local systems L µ on Gr µJ that appear in (8) can be expressed as L µ = H k h ! B where h : Gr µJ → Gr J is the inclusion map and k = − dim Gr µJ = − ρ J ( µ ) . With e : { x } → Gr µJ theinclusion of a point and ζ the coset of µ modulo Z Φ J , the fiber of L µ is ( L µ ) x ∼ = e ! L µ (cid:2) µJ (cid:3) ∼ = H ρ J ( µ ) (cid:0) { x } , e ! h ! B (cid:1) . For the specific case A = I λ = ( m n ) ∗ IC (cid:16) Gr λ n , C (cid:17) , noting the equality ρ J ( µ ) + ρ I,J ( ζ ) = ρ ( µ ) , we get ( L µ ) x ∼ = H ρ ( µ ) (cid:16) ( q J,ζ ◦ m n ) − ( x ) , f ! IC (cid:16) Gr λ n , C (cid:17)(cid:17) where f is the injection map depicted in the Cartesian diagram below. ( q J,ζ ◦ m n ) − ( x ) f / / (cid:15) (cid:15) Gr nm n (cid:15) (cid:15) ( q J,ζ ) − ( x ) / / (cid:15) (cid:15) Gr − J,ζ t Jζ / / q J,ζ (cid:15) (cid:15) Gr { x } e / / Gr µJ h / / Gr J,ζ
The same reasoning as in sect. 2.3 proves that only the stratum Gr λ n contributes to this co-homology group. Denoting by g : Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) → Gr λ n the inclusion map, thisobservation leads to an isomorphism ( L µ ) x = H ρ J ( µ )+2 ρ I,J ( ζ ) (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) , g ! C Gr λ n (cid:2) dim Gr λ n (cid:3)(cid:17) ∩ [Gr λ n ] −−−−→ H BM2 ρ ( | λ |− µ ) (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − ( x ) (cid:17) . Thus the local systems L µ appearing in (8) have a natural basis, namely the set Z J ( λ ) µ .We record the following consequence of this discussion: given ( λ , µ, ν ) ∈ (Λ + ) n × Λ + J × Λ suchthat µ − ν ∈ Z Φ J , we have dim H ρ J ( ν ) (cid:16) T J,ν , ( t J,ν ) ! IC (cid:16) Gr µJ , L µ (cid:17)(cid:17) = rank L µ × dim H ρ J ( ν ) (cid:16) T J,ν , ( t J,ν ) ! IC (cid:16) Gr µJ , C (cid:17)(cid:17) = Card Z J ( λ ) µ × Card Z J ( µ ) ν = Card (cid:8) Z ∈ Z ( λ ) ν (cid:12)(cid:12) µ J ( Z ) = µ (cid:9) . (9)19B) Now let us start with a sheaf B in Perv(Gr J ) and a weight µ ∈ Λ + J . Let us denote by i : Gr µJ → Gr J the inclusion map. By [4], Amplification 1.4.17.1, the largest subobject of B in Perv(Gr J ) supported on Gr µJ is B ≤ J µ = p τ ≤ i ∗ i ! B , where p τ ≤ is the truncation functor forthe perverse t -structure. From the distinguished triangle p τ ≤ i ∗ i ! B → i ∗ i ! B → p τ > i ∗ i ! B + −→ in the bounded derived category of constructible sheaves on Gr J , we deduce the long exactsequence H ρ J ( ν ) − (cid:0) T J,ν , ( t J,ν ) ! p τ > i ∗ i ! B (cid:1) → H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! p τ ≤ i ∗ i ! B (cid:1) → H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! i ∗ i ! B (cid:1) → H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! p τ > i ∗ i ! B (cid:1) . Theorem 3.5 in [39] implies that the two extrem terms vanish, and therefore F J,ν (cid:0) B ≤ J µ (cid:1) = H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! i ∗ i ! B (cid:1) . Let us patch all these pieces together. We take ( λ , µ, ν ) ∈ (Λ + ) n × Λ + J × Λ such that µ and ν belong to the same coset ζ ∈ Z/ Z Φ J and we consider I λ = ( m n ) ∗ IC (cid:16) Gr λ n , C (cid:17) and B = r IJ ( I λ ) . Composing the isomorphisms given in (6) and (3), we get H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! B (cid:1) ∼ = H ρ ( ν ) (cid:0) T ν , ( t ν ) ! I λ (cid:1) ∼ = H BM2 ρ ( | λ |− ν ) (cid:16) Gr λ n ∩ ( m n ) − ( T ν ) (cid:17) . (10)To save place we set S = ( q J,ζ ) − (cid:0) Gr µJ (cid:1) ∩ T ν and denote by s : S → T ν the inclusion map.Chasing in the three-dimensional figure Gr Gr nm n o o Gr λ n o o Gr J,ζ Gr − J,ζq
J,ζ o o t Jζ O O Gr λ n ∩ ( m n ) − (cid:0) Gr − J,ζ (cid:1) o o O O Gr µJ i ♦♦♦♦♦♦♦♦♦♦ ( q J,ζ ) − (cid:0) Gr µJ (cid:1) o o ♦♦♦♦♦♦♦ Gr λ n ∩ ( q J,ζ ◦ m n ) − (cid:0) Gr µJ (cid:1) o o ♦♦♦♦♦♦ T J,ν O O T ν o o O O Gr λ n ∩ ( m n ) − ( T ν ) o o O O Gr µJ ∩ T J,ν O O ♦♦♦♦♦♦♦♦ S o o O O s ♦♦♦♦♦♦♦♦♦♦♦♦ Gr λ n ∩ ( m n ) − ( S ) o o O O ♦♦♦♦♦♦♦ (11)20e complete (10) in the following commutative diagram. H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! i ∗ i ! B (cid:1) ≃ / / (cid:15) (cid:15) H ρ ( ν ) (cid:0) S, ( t ν s ) ! I λ (cid:1) ≃ / / (cid:15) (cid:15) H BM2 ρ ( | λ |− ν ) (cid:16) Gr λ n ∩ ( m n ) − ( S ) (cid:17) (cid:15) (cid:15) H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! B (cid:1) ≃ / / H ρ ( ν ) (cid:0) T ν , ( t ν ) ! I λ (cid:1) ≃ / / H BM2 ρ ( | λ |− ν ) (cid:16) Gr λ n ∩ ( m n ) − ( T ν ) (cid:17) As explained in (B), the left vertical arrow of this diagram is the inclusion map F J,ν ( B ≤ J µ ) → F J,ν ( B ) . If an MV cycle Z ∈ Z ( λ ) ν satisfies µ J ( Z ) ≤ J µ , then it is contained in ( m n ) − ( S ) , so thefundamental class of Z ∩ Gr λ n belongs to F J,ν ( B ≤ J µ ) . Looking at equation (9), we see thatthere are just enough such MV cycles to span this subspace. Going through the geometricSatake correspondence, we conclude that the MV basis of V ( λ ) satisfies the condition (P1) forbeing L -perfect.Eying now to the condition (P2), we consider the diagram below, consisting of inclusion maps. Gr µJ \ Gr µJ f / / Gr µJi (cid:15) (cid:15) Gr µJg o o h | | ②②②②②②②②② Gr J,ζ
For any sheaf F ∈ Perv(Gr J ) supported on Gr µJ \ Gr µJ , we have Hom (cid:0) F , p τ ≤ (cid:0) h ∗ h ! B (cid:1)(cid:1) = Hom (cid:0) F , h ∗ h ! B (cid:1) = Hom (cid:0) h ∗ F , h ! B (cid:1) = 0 in the bounded derived category of constructible sheaves over Gr J (the first two equalitiesby adjunction, the last one because h ∗ F = 0 ). Since h ∗ h ! B is concentrated in nonnegativeperverse degrees ([4], Proposition 1.4.16), the sheaf p τ ≤ (cid:0) h ∗ h ! B (cid:1) is perverse, and from thesemisimplicity of Perv(Gr J ) we conclude that Hom (cid:0) p τ ≤ (cid:0) h ∗ h ! B (cid:1) , F (cid:1) = 0 . Again, in the distinguished triangle i ∗ f ∗ f ! i ! B → i ∗ i ! B → i ∗ g ∗ g ! i ! B + −→ p H the first ho-mology group for the perverse t -structure, we obtain the exact sequence → p τ ≤ (cid:0) i ∗ f ∗ f ! i ! B (cid:1) → p τ ≤ (cid:0) i ∗ i ! B (cid:1) → p τ ≤ (cid:0) h ∗ h ! B (cid:1) → p H (cid:0) i ∗ f ∗ f ! i ! B (cid:1) . The perverse sheaf on the right is supported on Gr µJ \ Gr µJ , so the right arrow is zero by theprevious step. The resulting short exact sequence can be identified with → B < J µ → B ≤ J µ → B ≤ J µ / B < J µ → . With the same arguments as in the point (B) above, we deduce that F J,ν (cid:0) B ≤ J µ / B < J µ (cid:1) = F J,ν (cid:0) p τ ≤ h ∗ h ! B (cid:1) = H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! h ∗ h ! B (cid:1) . In (11), we replace Gr µJ by Gr µJ ; the same chasing as before now leads to the isomorphism H ρ J ( ν ) (cid:0) T J,ν , ( t J,ν ) ! h ∗ h ! B (cid:1) ≃ −→ H BM2 ρ ( | λ |− ν ) (cid:16) Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) ∩ ( m n ) − ( T ν ) (cid:17) . (12)Now the point (A) at the beginning of this section explains that H k h ! B , where k = − ρ J ( µ ) ,is the local system L µ and that it comes with a natural basis, namely Z J ( λ ) µ . This basisinduces a decomposition of B ≤ J µ / B < J µ into a sum of simple objects in Perv(Gr J ) . On theone hand, this decomposition can be followed through the geometric Satake correspondence,where it gives a decomposition of the subquotient of the isotypical filtration of res GM J V ( λ ) intoa direct sum of irreducible representations. On the other hand, it can also be tracked throughthe isomorphism (12): F J,ν (cid:0) B ≤ J µ / B < J µ (cid:1) ∼ = M Y ∈ Z J ( λ ) µ H BM2 ρ ( | λ |− ν ) (cid:16) Gr λ n ∩ Y ∩ ( m n ) − ( T ν ) (cid:17) . (13)From sect. 3.4, we see that the irreducible components of Gr λ n ∩ ( q J,ζ ◦ m n ) − (Gr µJ ) ∩ ( m n ) − ( T ν ) of dimension ρ ( | λ | − ν ) are the cycles Z J ∩ ( q J,ζ ◦ m n ) − ( Z J ∩ Gr µJ ) , with ( Z J , Z J ) ∈ Z J ( λ ) µ × Z J ( µ ) ν . The basis of the right-hand side of (12) afforded by the fundamental classes of theseirreducible components is thus compatible with the decomposition (13). Therefore, the MVbasis of V ( λ ) satisfies the condition (P2) for being L -perfect.The proof of Theorem 3.4 is now complete. Remark 3.5.
The proof establishes that the MV basis of V ( λ ) satisfies a stronger property than(P2): there exists an isomorphism of the isotypical component V ( λ ) J,µ with a direct sum ofcopies of the irreducible representation V J ( µ ) such that the induced basis on V ( λ ) J,µ matchesthe direct sum of the MV bases of the summands.22 .6 Crystal structure on MV cycles
Let λ ∈ (Λ + ) n . The MV basis of V ( λ ) defined in sect. 2.3 is indexed by Z ( λ ) = G ν ∈ Λ Z ( λ ) ν and is L -perfect. Thus, the set Z ( λ ) is endowed with the structure of a crystal, as explainedin sect. 3.2. Obviously the weight of an MV cycle Z ∈ Z ( λ ) ν is simply wt( Z ) = ν . The aimof this section is to characterize the action on Z ( λ ) of the operators ˜ e i and ˜ f i .In semisimple rank , one can give an explicit analytical description of the MV cycles, asfollows. Proposition 3.6
Assume that G has semisimple rank and denote by α and α ∨ the positiveroot and coroot. Let y : G a → G ∨ be the additive one-parameter subgroup for the root − α ∨ .Let ( µ, ν ) ∈ Λ + × Λ and set r = h α ∨ , µ i . Then Gr µ ∩ T ν is nonempty if and only if thereexists p ∈ { , , . . . , r } such that ν = µ − pα ; in this case, the map a y ( az p − r ) L ν inducesan isomorphism of algebraic varieties O /z p O ≃ −→ Gr µ ∩ T ν so Gr µ ∩ T ν is an affine space of dimension p and Z ( µ ) ν is a singleton. We skip the proof since this proposition is well-known; compare for instance with [1], Propo-sition 3.10. We can now describe the crystal structure on Z ( λ ) , which extends [1], Proposi-tion 4.2. Proposition 3.7
Let ( λ , ν ) ∈ (Λ + ) n × Λ , let i ∈ I and let Z ∈ Z ( λ ) ν .(i) We have wt( Z ) = ν , ε i ( Z ) = (cid:10) α ∨ i , µ { i } ( Z ) − ν (cid:11) and ϕ i ( Z ) = (cid:10) α ∨ i , µ { i } ( Z ) + ν (cid:11) .(ii) Let Y ∈ Z ( λ ) ν + α i . Then Y = ˜ e i Z if and only if Y ⊆ Z and µ { i } ( Y ) = µ { i } ( Z ) .Proof. Let λ , ν , i , Z as in the statement and set µ = µ { i } ( Z ) . By definition, the MV cycles ˜ e i Z and ˜ f i Z (if nonzero) are obtained by letting the Chevalley generators e i and f i act on(the basis element indexed by) Z in the appropriate subquotient of the isotypical filtration of res GM { i } V ( λ ) . According to (13), this entails that µ { i } ( Z ) = µ { i } (˜ e i Z ) = µ { i } ( ˜ f i Z ) and Z { i } = (˜ e i Z ) { i } = ( ˜ f i Z ) { i } .
23n addition, Z { i } ( µ ) ν + α i and Z { i } ( µ ) ν − α i are empty or singletons, and the MV cycles (˜ e i Z ) { i } and ( ˜ f i Z ) { i } in the affine Grassmannian Gr { i } are uniquely determined by weight considera-tions.The statements can be deduced from these facts by using the explicit description provided byProposition 3.6 and the construction of the map ( Z J , Z J ) Z in sect. 3.4. (cid:3) In the previous section, we defined the structure of a crystal on the set Z ( λ ) . In this section,we turn to Littelmann’s path model [33] to study this structure. This combinatorial devicecan be used to effectively assemble MV cycles. Our construction is inspired by the resultspresented in [17] but is more flexible, for it relaxes the restriction to minimal galleries.In this paper, piecewise linear means continuous piecewise linear. We keep the notation set upin the header of sect. 3. Let Λ R = Λ ⊗ Z R be the real space spanned by the weight lattice and let Λ + R be the dominantcone inside Λ R .A path is a piecewise linear map π : [0 , → Λ R such that π (0) = 0 and π (1) ∈ Λ . We denoteby e Π the set of all paths. The concatenation π ∗ η of two paths is defined in the usual way: π ∗ η ( t ) = π (2 t ) for ≤ t ≤ , and π ∗ η ( t ) = η (2 t −
1) + π (1) for ≤ t ≤ .In [33], the third author associates to each simple root α of Φ a pair ( e α , f α ) of “root operators”from e Π to e Π ⊔ { } and shows that the construction yields a semi-normal crystal structure on e Π . Here the weight map is given by wt( π ) = π (1) . To agree with the notation in sect. 3.1, wewill write ˜ e i and ˜ f i instead of e α i and f α i for each i ∈ I .Let ℓ : [0 , → R be a piecewise linear function. We say that p ∈ R is a local absolute minimumof ℓ if there exists a compact interval [ a, b ] ⊆ [0 , over which ℓ takes the value p , and thereexists an ǫ > such that ℓ ( x ) > p for all x ∈ ( a − ǫ, a ) ∩ [0 , and all x ∈ ( b, b + ǫ ) ∩ [0 , .Given π ∈ e Π , we denote by A π the set of all paths η ∈ e Π that can be obtained from π byapplying a finite sequence of root operators ˜ e i or ˜ f i . We say that a path π ∈ e Π is integral iffor each η ∈ A π and each i ∈ I , all local absolute minima of the function t
7→ h α ∨ i , η ( t ) i areintegers. 24e denote the set of all integral paths by Π . Obviously, Π is a subcrystal of e Π . Moreover,the root operators have a simpler form on Π , since integral paths need only to be cut intothree parts: the initial part is left invariant, the second part is reflected, and the third part istranslated. Specifically, given ( π, η ) ∈ Π and i ∈ I , we have η = ˜ e i π if and only if there exista negative integer p ∈ Z and two reals a and b with ≤ a < b ≤ , such that the function t
7→ h α ∨ i , π ( t ) i is weekly decreasing on [ a, b ] , and for each t ∈ [0 , : • if t ≤ a , then h α ∨ i , π ( t ) i ≥ p + 1 and η ( t ) = π ( t ) ; • if t = a , then h α ∨ i , π ( t ) i = p + 1 ; • if a < t < b , then p ≤ h α ∨ i , π ( t ) i < p + 1 and η ( t ) = π ( t ) − ( h α ∨ i , π ( t ) i − p − α i ; • if t = b , then h α ∨ i , π ( t ) i = p ; • if t ≥ b , then h α ∨ i , π ( t ) i ≥ p and η ( t ) = π ( t ) + α i .We say that an integral path π ∈ Π is dominant if its image is contained in Λ + R . Remark 4.1.
Let Γ be the group of all strictly increasing piecewise linear maps from [0 , ontoitself, the product being the composition of functions. The group acts on the set of all pathsby right composition: π π ◦ γ for a path π and γ ∈ Γ . We say that π ◦ γ is obtainedfrom π by a piecewise linear reparameterization. Visibly, the set Π of integral paths is stableunder this action, the weight map wt is invariant, and the root operators are equivariant. Wecan thus safely consider all our previous constructions modulo this action. In the sequel wesometimes implicitly assume that this quotient has been performed, i.e. among the possibleparameterizations we choose one which is appropriate for the application in view.The first two items in the following proposition ensure that there is an abundance of integralpaths. Proposition 4.2 (i) A dominant path π is integral as soon as for each i ∈ I , the function t
7→ h α ∨ i , π ( t ) i is weakly increasing.(ii) The set Π is stable under concatenation of paths and the map π ⊗ η π ∗ η is a strictmorphism of crystals from Π ⊗ Π to Π .(iii) Let π ∈ Π . Then A π contains a unique dominant path η and is isomorphic as a crystalto B (wt( η )) . roof. For ν ∈ Λ R let π ν be the map [0 , → Λ R , t tν . A path in Proposition 4.2 (i) is ofthe form π = π ν ∗ . . . ∗ π ν s , where ν , . . . , ν s ∈ Λ + R are dominant and ν + · · · + ν s ∈ Λ + . Sucha path can be approximated by a rational dominant path, i.e. a path η = π µ ∗ . . . ∗ π µ s suchthat µ , . . . , µ s ∈ Λ + Q and µ + · · · + µ s = ν . By Lemma 4.4 below, a rational dominant pathcan be approximated by a locally integral concatenation ([35], Definition 5.3). These paths areintegral by loc. cit. , Lemma 5.6 and Proposition 5.9. The integrality property in (i) followsnow by the continuity of the root operators loc. cit. , property (v) continuity .It remains to prove the other two statements. The endpoint of a path is by definition anelement of the lattice, so the concatenation of integral paths is an integral path. Moreover, byLemma 6.12 in loc. cit. , concatenation defines a strict morphism of crystals Π ⊗ Π → Π . Thisshows (ii). Statement (iii) follows by Lemma 6.11 in loc. cit. (cid:3) For a rational dominant path π = π µ ∗ · · · ∗ π µ s , the property of being a locally integralconcatenation is equivalent to:( ∗ ) For all j = 1 , . . . , s , the affine line passing through µ + · · · + µ j − and µ + · · · + µ j meets at least two lattice points.An equivalent formulation: the affine line meets at least one rational point and one latticepoint.To prove the approximation property used in the proof above, we need the following elementarygeometric construction. We fix a scalar product ( · , · ) on Λ R and let d ( · , · ) be the correspondingdistance function. We fix a basis B of Λ and let L ⊂ Λ R be the associated unit cube, i.e. theset of points in Λ R which can be written as a linear combination of B with coefficients in theinterval [0 , . Let M be the maximal distance between two points in L . Let P ∈ Λ + Q be adominant rational point and let S ( P, be the sphere with center P and radius . Let g be aray starting in P and let g be the intersection point of this ray with the sphere S ( P, . Lemma 4.3
One can find for any ǫ > a ray f starting in P such that ( f \ { P } ) ∩ Λ = ∅ ,and for { f } = f ∩ S ( P, we have d ( g , f ) < ǫ .Proof. Parametrize g by g ( t ) = P + t ( g − P ) for t ≥ . Choose t ≫ and pick λ ∈ Λ suchthat g ( t ) ∈ λ + L . Let f be the ray starting in P passing through λ . Let f be the intersectionpoint of this ray with S ( P, ; then f ( t ) = P + t ( f − P ) for t ≥ is a parameterization of f .Set g = g ( t ) and f = f ( t ) . Noting that d ( P, f ) = t = d ( P, g ) and using the triangularinequality, we get d ( f , λ ) = | d ( P, f ) − d ( P, λ ) | = | d ( P, g ) − d ( P, λ ) | ≤ d ( g , λ ) ≤ M d ( g , f ) ≤ M . By the intercept theorem d ( f , g ) /d ( g , f ) = 1 /t , and hence d ( f , g ) ≤ (2 M ) /t . For t large enough we obtain d ( g , f ) < ǫ . (cid:3) Lemma 4.4
A dominant rational path can be approximated by a locally integral concatenation.Proof.
Let π = π λ ∗ · · · ∗ π λ s be a dominant rational path ending in λ ∈ Λ + . We define thesupport of an element µ ∈ Λ R as the subset of simple roots such that h α ∨ , µ i 6 = 0 . We canassume that the support of each λ j is the same as the support of λ ; otherwise we approximate π by a path we get by slightly perturbing λ , . . . , λ s , for instance by replacing λ i by λ i + ǫ ( λ/s − λ i ) for some rational < ǫ ≪ . We can also assume the support of λ is I , otherwise we workwithin the subspace T i ∈ I \ supp( λ ) (ker α ∨ i ) .Under these assumptions, small perturbations λ ′ , . . . , λ ′ s − of the directions λ , . . . , λ s − remaindominant, and so does λ ′ s = λ − ( λ ′ + · · · + λ ′ s − ) . By Lemma 4.3, one can perturb in such away that the new path η := π λ ′ ∗ · · · ∗ π λ ′ s is a rational dominant path and the first s − linesegments of η satisfy the affine line condition ( ∗ ) . The last line segment of η meets the latticepoint λ and a rational point, and thus satisfies the affine line condition ( ∗ ) too. Hence η is alocally integral concatenation, approximating the dominant rational path π . (cid:3) We need additional terminology before we proceed to the main construction of this section.To each coroot α ∨ ∈ Φ ∨ corresponds an additive one-parameter subgroup x α ∨ : G a → G ∨ .Given additionally an integer p ∈ Z , we define a map x ( α ∨ , p ) : C → G ∨ ( K ) , a x α ∨ ( az p ) . An affine coroot is a pair ( α ∨ , p ) consisting of a coroot α ∨ ∈ Φ ∨ and an integer p ∈ Z . Thedirection of an affine coroot ( α ∨ , p ) is α ∨ . An affine coroot is said to be positive if its directionis so. We denote the set of affine coroots by Φ ∨ a and the set of positive affine coroots by Φ ∨ , + a .To an affine coroot β , besides the map x β defined above, we attach an hyperplan H β and anegative closed half-space H − β in Λ R as follows: H ( α ∨ , p ) = (cid:8) x ∈ Λ R (cid:12)(cid:12) h α ∨ , x i = p (cid:9) , H − ( α ∨ , p ) = (cid:8) x ∈ Λ R (cid:12)(cid:12) h α ∨ , x i ≤ p (cid:9) . Let s β be the reflection across the hyperplane H β ; concretely s ( α ∨ , p ) ( x ) = x − ( h α ∨ , x i − p ) α x ∈ Λ R . In addition, we denote by τ λ the translation x x + λ by the element λ ∈ Λ .The subgroup of Aut(Λ R ) generated by all the reflections s β is the affine Weyl group W a ; whenwe add the translations τ λ , we obtain the extended affine Weyl group f W a . Then τ λ ∈ W a ifand only if λ ∈ Z Φ .The group f W a acts on the set Φ ∨ a of affine roots: one demands that w ( H − β ) = H − wβ for eachelement w ∈ f W a and each affine coroot β ∈ Φ ∨ a . Then for each β ∈ Φ ∨ a and each λ ∈ Λ , wehave x τ λ β ( a ) = z λ x β ( a ) z − λ for all a ∈ C .We denote by H the arrangement formed by the hyperplanes H β , where β ∈ Φ ∨ a . It dividesthe vector space Λ R into faces. The closure of a face is the disjoint union of faces of smallerdimension. Endowed with the set of all faces, Λ R becomes a polysimplicial complex, called theaffine Coxeter complex.For each face f of the affine Coxeter complex, we denote by N ∨ ( f ) the subgroup of N ∨ ( K ) generated by the elements x α ∨ ( az p ) , where a ∈ O and ( α ∨ , p ) is a positive affine coroot suchthat f ⊆ H − ( α ∨ , p ) . We note that N ∨ ( τ λ f ) = z λ N ∨ ( f ) z − λ for each face f and each λ ∈ Λ andthat N ∨ ( f ) ⊆ N ∨ ( O ) if f ⊆ Λ + R .For x ∈ Λ R , we denote by f x the face in the affine Coxeter complex that contains the point x . We use the symbol Q ′ to denote the restricted product of groups, consisting of familiesinvolving only finitely many nontrivial terms. Also, remember the notation introduced insect. 2.2 about the n -fold convolution variety Gr n .With these conventions, given ( π , . . . , π n ) ∈ Π n , we define ˚ Z ( π ⊗ · · · ⊗ π n ) as the subset of Gr n of all elements Y t ∈ [0 , v ,t z wt( π ) , . . . , Y t n ∈ [0 , v n,t n z wt( π n ) with (( v ,t ) , . . . , ( v n,t n )) ∈ Y ′ t ∈ [0 , N ∨ (cid:0) f π ( t ) (cid:1) × · · · × Y ′ t n ∈ [0 , N ∨ (cid:0) f π n ( t n ) (cid:1) . The resort to the restricted infinite products is here merely cosmetic; we could as well usefinite products, since the paths π j meet only finitely many faces. The point that matters is tocompute products of elements in the various groups N ∨ ( f π j ( t ) ) in the order indicated by thepaths. Proposition 4.5
Let ( π , . . . , π n ) ∈ Π n .(i) The set ˚ Z ( π ⊗ · · · ⊗ π n ) is stable under left multiplication by N ∨ ( O ) . ii) Let µ = wt( π ) + · · · + wt( π n ) ; then the set ˚ Z ( π ⊗ · · · ⊗ π n ) is an irreducible constructiblesubset of ( m n ) − ( S µ ) .(iii) We have ˚ Z ( π ⊗ · · · ⊗ π n ) = Ψ (cid:16) ˚ Z ( π ) ⋉ · · · ⋉ ˚ Z ( π n ) (cid:17) and ˚ Z ( π ∗ · · · ∗ π n ) = m n (cid:16) ˚ Z ( π ⊗ · · · ⊗ π n ) (cid:17) . (iv) Let i ∈ I and compute η ⊗ · · · ⊗ η n = ˜ e i ( π ⊗ · · · ⊗ π n ) in the crystal Π ⊗ n , assuming thatthis operation is doable. Then ˚ Z ( π ⊗· · ·⊗ π n ) is contained in the closure of ˚ Z ( η ⊗· · ·⊗ η n ) in Gr n .Proof. Assertion (i) is a direct consequence of the equality N ∨ (cid:0) f π (0) (cid:1) = N ∨ ( f ) = N ∨ ( O ) .Assertion (ii) comes from general principles once we have replaced the restricted infinite productby a finite one.The first equation in (iii) is tautological. In the second one, we view the concatenation π = π ∗ · · · ∗ π n as a map from [0 , n ] to Λ R , each path π , . . . , π n being travelled at nominal speed.For j ∈ { , . . . , n } , we set ν j − = wt( π ) + · · · + wt( π j − ) ; then π ( t + ( j − ν j − + π j ( t ) forall t ∈ [0 , , and accordingly N ∨ (cid:0) f π ( t +( j − (cid:1) = z ν j − N ∨ (cid:0) f π j ( t ) (cid:1) z − ν j − . A banal calculationthen yields the desired result.The proof of assertion (iv) is much more involved. We defer its presentation to sect. 4.5. (cid:3) For ( π , . . . , π n ) ∈ Π n , we denote by Z ( π ⊗ · · · ⊗ π n ) the closure of ˚ Z ( π ⊗ · · · ⊗ π n ) in ( m n ) − ( S µ ) , where µ = wt( π ) + · · · + wt( π n ) . Theorem 4.6
Let ( π , . . . , π n ) ∈ Π n , set µ = wt( π ) + · · · + wt( π n ) and for j ∈ { , . . . , n } , let λ j be the weight of the unique dominant path in A π j . Then Z ( π ⊗ · · · ⊗ π n ) is an MV cycle;specifically Z ( π ⊗ · · · ⊗ π n ) ∈ ∗ Z ( λ , . . . , λ n ) µ .Proof. We start with the particular case n = 1 . Let π ∈ Π , let η be the unique dominant path in A π , set λ = wt( η ) and µ = wt( π ) , and set p = 2 ρ ( λ ) and k = ρ ( λ + µ ) . By Proposition 4.2 (iii),the crystal A π is isomorphic to B ( λ ) , so it contains a unique lowest weight element ξ . Then π can be reached by applying a sequence of root operators ˜ f i to η or by applying a sequence ofroot operators ˜ e i to ξ . Thus, there exists a finite sequence ( π , . . . , π p ) of elements in A π suchthat π = ξ , π k = π , π p = η and such that each π j +1 is obtained from π j by applying a rootoperator ˜ e i . 29ince η is dominant, each face f η ( t ) is contained in Λ + R , so each group N ∨ (cid:0) f η ( t ) (cid:1) is contained in N ∨ ( O ) . Then by construction ˚ Z ( η ) ⊆ N ∨ ( O ) L λ , and therefore Z ( η ) (the closure of Z ( η ) in Gr ) is contained in Gr λ . Also, Proposition 4.5 (iv) implies that Z ( π ) ⊆ Z ( π ) ⊆ · · · ⊆ Z ( π p ) . (14)These inclusions are strict because Z ( π j ) is contained in the closure of Gr λ ∩ S wt( π j ) , which isdisjoint from S wt( π j +1 ) by [39], Proposition 3.1 (a), while Z ( π j +1 ) is contained in S wt( π j +1 ) . Thus(14) is a strictly increasing chain of closed irreducible subsets of Gr λ . As Gr λ has dimension p , we see that each Z ( π j ) has dimension j .In particular, Z ( π ) has dimension k . But Z ( π ) is locally closed, because it is a closed subsetof S µ which is locally closed. So Z ( π ) has dimension k . At this point, we know that Z ( π ) isa closed irreducible subset of Gr λ ∩ S µ of dimension k = ρ ( λ + µ ) . Therefore Z ( π ) belongsto Z ( λ ) µ .The reasoning above establishes the case n = 1 of the Theorem. The general case then followsfrom Propositions 2.2 (ii) and 4.5 (iii). (cid:3) For the proofs in the following sections, it will be convenient to have a more economicalpresentation of the sets ˚ Z ( π ⊗ · · · ⊗ π n ) . We need a few additional pieces of notation.When f and f ′ are two faces of the affine Coxeter complex such that f is contained in the closure f ′ of f ′ , we denote by Φ ∨ , + a ( f , f ′ ) the set of all positive affine coroots β such that f ⊆ H β and f ′ H − β . We denote by N ∨ ( f , f ′ ) the subgroup of N ∨ ( K ) generated by the elements x β ( a ) with β ∈ Φ ∨ , + a ( f , f ′ ) and a ∈ C .The following result is Proposition 19 (ii) in [1]. Lemma 4.7
Let f and f ′ be two faces of the affine Coxeter complex such that f ⊆ f ′ . Then N ∨ ( f ) is the bicrossed product N ∨ ( f , f ′ ) ⊲⊳ N ∨ ( f ′ ) ; in other words, the product induces a bijection N ∨ ( f , f ′ ) × N ∨ ( f ′ ) ≃ −→ N ∨ ( f ) . In addition, the map C Φ ∨ , + a ( f , f ′ ) → N ∨ ( f , f ′ ) , ( a β ) Y β ∈ Φ ∨ , + a ( f , f ′ ) x β ( a β ) is bijective, whichever order on Φ ∨ , + a ( f , f ′ ) is used to compute the product. π ∈ Π and t ∈ [0 , , we denote by f π ( t +0) the face in the affine Coxeter complex that con-tains the points π ( t + h ) for all small enough h > . Obviously, its closure meets, hence contains,the face f π ( t ) . We set Φ ∨ , + a ( π, t ) = Φ ∨ , + a (cid:0) f π ( t ) , f π ( t +0) (cid:1) and N ∨ ( π, t ) = N ∨ (cid:0) f π ( t ) , f π ( t +0) (cid:1) .Concretely, Φ ∨ , + a ( π, t ) is the set of all β ∈ Φ ∨ , + a such that π quits the half-space H − β at time t and N ∨ ( π, t ) is the subgroup of N ∨ ( K ) generated by the elements x β ( a ) with β ∈ Φ ∨ , + a ( π, t ) and a ∈ C . Note that Φ ∨ , + a ( π, t ) is empty save for finitely many t . Proposition 4.8
Let ( π , . . . , π n ) ∈ Π n . Then ˚ Z ( π ⊗ · · · ⊗ π n ) is the set of all elements Y t ∈ [0 , v ,t z wt( π ) , . . . , Y t n ∈ [0 , v n,t n z wt( π n ) with (( v ,t ) , . . . , ( v n,t n )) ∈ Y t ∈ [0 , N ∨ ( π , t ) × · · · × Y t n ∈ [0 , N ∨ ( π n , t n ) . Proof.
Let π ∈ Π . Let ( t , . . . , t m ) be the ordered list of all elements t ∈ [0 , such that Φ ∨ , + a ( π, t ) = ∅ and set t m +1 = 1 .Pick ℓ ∈ { , . . . , m } ; between the times t ℓ and t ℓ +1 , the path π never quits the half-space H − β of a positive affine coroot β ; as a consequence, the map t N ∨ (cid:0) f π ( t ) (cid:1) is non-decreasing on theinterval ] t ℓ , t ℓ +1 ] . This map is also non-decreasing on the interval [0 , t ] if t > . However when t goes past a point t ℓ , the group N ∨ (cid:0) f π ( t ) (cid:1) slims down by dropping the subgroup N ∨ ( π, t ℓ ) ,in the sense brought by the bicrossed product N ∨ (cid:0) f π ( t ℓ ) (cid:1) = N ∨ ( π, t ℓ ) ⊲⊳ N ∨ (cid:0) f π ( t ℓ +0) (cid:1) .It follows that for any family ( u t ) in Q ′ t ∈ [ t ℓ ,t ℓ +1 ] N ∨ (cid:0) f π ( t ) (cid:1) , there exists ( v, u ) ∈ N ∨ ( π, t ℓ ) × N ∨ (cid:0) f π ( t ℓ +1 ) (cid:1) such that Q t ∈ [ t ℓ ,t ℓ +1 ] u t = vu . To see this, one decomposes u t ℓ as vu ′ accordingto the bicrossed product and one defines u as the product of u ′ and of the u t for t ∈ ] t ℓ , t ℓ +1 ] .Assembling these pieces (and an analogous statement over the interval [0 , t ] if t > ) fromleft to right, and noting that N ∨ (cid:0) f π (1) (cid:1) stabilizes L wt( π ) , we deduce that ˚ Z ( π ) is the image ofthe map Y t ∈ [0 , N ∨ ( π, t ) → Gr , ( v t ) Y t ∈ [0 , v t L wt( π ) . This proves our statement in the case of just one path. The general case then follows fromProposition 4.5 (iii). (cid:3) .4 Isomorphisms of crystals In the previous section we explained how to build elements in ∗ Z ( λ ) µ , while in sect. 3 we weredealing with MV cycles in Z ( λ ) . This clumsiness is due to a mismatch between the definitionof the path model and the conventions in [39] and [2]. To mitigate the disagreement, we definea crystal structure on ∗ Z ( λ ) = G µ ∈ Λ ∗ Z ( λ ) µ . Remember the setup of sect. 3.3: we consider a subset J ⊆ I , define an action of C × on Gr given by a special dominant weight θ J , and get the diagram (4). Now let ( λ , ν ) ∈ (Λ + ) n × Λ ,let ζ be the coset of ν modulo Z Φ J , and let Z ∈ ∗ Z ( λ ) ν . Then m n ( Z ) ⊆ S ν ⊆ Gr + J,ζ andthere is a unique weight µ ∈ Λ + J characterized by the conditions ( p J,ζ ◦ m n )( Z ) ⊆ Gr µJ and ( p J,ζ ◦ m n )( Z ) ∩ Gr µJ = ∅ . We denote this weight by µ J ( Z ) .By analogy with Proposition 3.7, we can then claim the existence of a crystal structure on ∗ Z ( λ ) such that for all ν ∈ Λ , i ∈ I and Z ∈ Z ( λ ) ν : • We have wt( Z ) = ν , ε i ( Z ) = (cid:10) α ∨ i , µ { i } ( Z ) − ν (cid:11) and ϕ i ( Z ) = (cid:10) α ∨ i , µ { i } ( Z ) + ν (cid:11) . • Let Y ∈ ∗ Z ( λ ) ν + α i . Then Y = ˜ e i Z if and only if Y ⊇ Z and µ { i } ( Y ) = µ { i } ( Z ) . Theorem 4.9
Let ( λ , . . . , λ n ) ∈ (Λ + ) n , and for each j ∈ { , . . . , n } choose a subcrystal Π j of Π isomorphic to B ( λ j ) . Then the map ( π , . . . , π n ) Z ( π ⊗ · · · ⊗ π n ) is an isomorphismof crystals Π ⊗ · · · ⊗ Π n ≃ −→ ∗ Z ( λ , . . . , λ n ) . Proof.
Let i ∈ I and let ( π , . . . , π n ) ∈ Π × · · · × Π n . Set ν = wt( π ) + · · · + wt( π n ) , let ζ bethe coset of ν modulo Z α i , and set π = π ∗ · · · ∗ π n , p = min (cid:8) h α ∨ i , π ( t ) i (cid:12)(cid:12) t ∈ [0 , (cid:9) and q = h α ∨ i , ν i = h α ∨ i , π (1) i . For any a ∈ C [ z, z − ] and any positive coroot α ∨ , we have lim c → θ { i } ( c ) x α ∨ ( a ) θ { i } ( c ) − = ( x α ∨ ( a ) if α ∨ = α ∨ i , otherwise.32sing Proposition 4.8, we see that p { i } ,ζ (cid:16) ˚ Z ( π ) (cid:17) is the set of all elements of the form lim c → Y t ∈ [0 , Y β ∈ Φ ∨ , + a ( π,t ) θ { i } ( c ) x β ( a t,β ) θ { i } ( c ) − L ν where a t,β are complex numbers. All factors in the product disappear in the limit c → ,except those for the affine roots β of direction α ∨ i . Let ( α ∨ i , p ) , . . . , ( α ∨ i , p s ) be these affineroots. Since the function t
7→ h α ∨ i , π ( t ) i assumes the value p and thereafter reaches the value q , the path π must, at some point, quit each half-space H − ( α ∨ i , p ) , H − ( α ∨ i , p +1) , . . . , H − ( α ∨ i , q − , so { p, p + 1 , . . . , q − } ⊆ { p , . . . , p s } ⊆ { p, p + 1 , . . . } . We conclude that p { i } ,ζ (cid:16) ˚ Z ( π ) (cid:17) = (cid:8) x α ∨ i ( az p ) L ν (cid:12)(cid:12) a ∈ O /z q − p O (cid:9) . Proposition 4.5 (iii) and a variant of Proposition 3.6 then yield ( p { i } ,ζ ◦ m n ) (cid:16) ˚ Z ( π ⊗ · · · ⊗ π n ) (cid:17) = p { i } ,ζ (cid:16) ˚ Z ( π ) (cid:17) = Gr µ { i } ∩ S { i } ,ν where µ = ν − pα i . Thus, µ { i } ( Z ( π ⊗ · · · ⊗ π n )) = ν − pα i (15)and ε i ( Z ( π ⊗ · · · ⊗ π n )) = − p and ϕ i ( Z ( π ⊗ · · · ⊗ π n )) = q − p. These latter equations show that the map ( π , . . . , π n ) Z ( π ⊗ · · · ⊗ π n ) is compatible withthe functions ε i and ϕ i .Now compute η ⊗ · · · ⊗ η n = ˜ e i ( π ⊗ · · · ⊗ π n ) in the crystal Π ⊗ · · · ⊗ Π n , assuming this operation to be doable. By Proposition 4.5 (iv), Z ( η ⊗ · · · ⊗ η n ) ⊇ Z ( π ⊗ · · · ⊗ π n ) . (16)Let η = η ∗ · · · ∗ η n . Then η = ˜ e i π by Proposition 4.2 (ii), and therefore wt( η ) = ν + α i and min (cid:8) h α ∨ i , η ( t ) i (cid:12)(cid:12) t ∈ [0 , (cid:9) = p + 1 . Repeating the arguments above, we get µ { i } ( Z ( η ⊗ · · · ⊗ η n )) = ( ν + α i ) − ( p + 1) α i = ν − pα i . Z ( η ⊗ · · · ⊗ η n ) = ˜ e i Z ( π ⊗ · · · ⊗ π n ) . We conclude that the map ( π , . . . , π n ) Z ( π ⊗ · · · ⊗ π n ) has the required compatibility withthe operations ˜ e i . (cid:3) Corollary 4.10
Let ( λ , . . . , λ n ) ∈ (Λ + ) n . Then the map ( Z , . . . , Z n ) Ψ( Z ⋉ · · · ⋉ Z n ) from Proposition 2.2 (ii) is an isomorphism of crystals ∗ Z ( λ ) ⊗ · · · ⊗ ∗ Z ( λ n ) ≃ −→ ∗ Z ( λ , . . . , λ n ) . The crystals Z ( λ ) enjoy a factorization property analogous to Corollary 4.10; one musthowever use the opposite tensor product on crystals. Remark 4.11.
The plactic algebra [32] is an algebraic combinatorial tool invented by Lascouxand Schützenberger long before the notion of a crystal basis of a representation was introduced.Loosely speaking, for a complex reductive algebraic group, the plactic algebra is the algebrahaving as basis the union S λ ∈ Λ + B ( λ ) of the crystal bases B ( λ ) for all irreducible representa-tions, the product being given by the tensor product of crystals. For G = SL n ( C ) , Lascouxand Schützenberger give a description of such an algebra in terms of the word algebra modulothe Knuth relations, and it was shown later that this algebra is isomorphic to the one givenby the crystal basis. A combinatorial Lascoux-Schützenberger type description for the othertypes was given in [34]; this description uses the path model.It is natural to ask whether it is possible to do the same with MV cycles: endow the set ofall MV cycles for all dominant weights λ ∈ Λ + with the structure of a crystal and define(in a geometric way) a multiplication on the cycles which mimics the plactic algebra. For G = SL n ( C ) , a positive answer was given in [18]. This approach was adapted to the symplecticcase in [44].The results in this section can be naturally viewed as a generalization of [18] to arbitraryconnected reductive groups. Using [34] and Proposition 4.2, one can use the set Π to constructthe plactic algebra so that it has as basis equivalence classes (generalized Knuth relations) ofelements in Π . The sets ˚ Z ( π ⊗ · · · ⊗ π n ) (Proposition 4.5) replace in the general setting theBiałynicki-Birula cells in [18]. By combining Proposition 4.5 (iii) and Theorem 4.9, we see thatthe closure of m n (cid:16) ˚ Z ( π ⊗ · · · ⊗ π n ) (cid:17) is an MV cycle which depends only on the class of thepath π ∗ · · · ∗ π n modulo the generalized Knuth relations. In particular, the main result of [18]follows as a special case. 34 different approach to this problem was taken by Xiao and Zhu [45]. They define a set of‘elementary Littelmann paths’, modeled over minuscule or quasi-minuscule representations, usethe methods from [40] to assign an MV cycle to each concatenation of elementary Littelmannpaths, and show that the resulting map factorizes through the generalized Knuth relations. This section can be skipped without substantial loss for the appreciation of our main storyline.We follow the same method as in [1], proof of Proposition 5.11.The group G ∨ ( C ) is generated by elements x α ∨ ( a ) and c λ , where ( a, α ∨ ) ∈ C × Φ ∨ and ( c, λ ) ∈ C × × Λ , which obey the following relations: • For any ( a, α ∨ ) ∈ C × Φ ∨ and any ( c, λ ) ∈ C × × Λ , c λ x α ∨ ( a ) c − λ = x α ∨ (cid:0) c h α ∨ ,λ i a (cid:1) . • Given two linearly independent elements α ∨ and β ∨ in Φ ∨ , there exist constants C i,j such that x α ∨ ( a ) x β ∨ ( b ) x α ∨ ( a ) − x β ∨ ( b ) − = Y ( i,j ) x iα ∨ + jβ ∨ (cid:0) C i,j a i b j (cid:1) (17)for any ( a, b ) ∈ C . The product in the right-hand side is taken over all pairs of positiveintegers ( i, j ) for which iα ∨ + jβ ∨ ∈ Φ ∨ , in order of increasing i + j .Further, the one-parameter subgroups x α ∨ can be normalized so that for any root α ∈ Φ : • For any ( a, b ) ∈ C such that − ab = 0 , x α ∨ ( a ) x − α ∨ ( b ) = x − α ∨ ( b/ (1 − ab )) (1 − ab ) α x α ∨ ( a/ (1 − ab )) . (18) • There exists an element s α ∈ G ∨ ( C ) such that for any a ∈ C × , x α ∨ ( a ) x − α ∨ ( a − ) x α ∨ ( a ) = x − α ∨ ( a − ) x α ∨ ( a ) x − α ∨ ( a − ) = a α s α = s α a − α . (19)This element s α lifts in the normalizer of T ∨ ( C ) the reflection s α ∈ W along the root α . Allthe above relations also hold for scalars b , c in K , provided of course that we regard themin G ∨ ( K ) .The Chevalley commutation relation (17) implies the following easy lemma.35 emma 4.12 Let f be a face of the affine Coxeter complex, let ( α ∨ , p ) and ( β ∨ , q ) be twopositive affine coroots, and let ( a, b ) ∈ O . Assume that α ∨ is simple, that α ∨ = β ∨ , and that f ⊆ H − ( − α ∨ , − p ) ∩ H − ( β ∨ , q ) . Then x − α ∨ ( az − p ) x β ∨ ( bz q ) x − α ∨ ( − az − p ) ∈ N ∨ ( f ) . Proof.
We consider the situation set forth in the statement of the lemma. Using (17), we write x − α ∨ ( az − p ) x β ∨ ( bz q ) x − α ∨ ( − az − p ) x β ∨ ( − bz q ) = Y ( i,j ) x − iα ∨ + jβ ∨ (cid:0) C i,j a i b j z − ip + jq (cid:1) (20)where the product in the right-hand side is taken over all pairs of positive integers ( i, j ) forwhich − iα ∨ + jβ ∨ is a coroot.Consider such a pair ( i, j ) . In view of our assumptions, the coroot − iα ∨ + jβ ∨ is necessarilypositive. Moreover for any x ∈ f we have h− iα ∨ + jβ ∨ , x i = i h− α ∨ , x i + j h β ∨ , x i ≤ i ( − p ) + jq, so f ⊆ H − ( − iα ∨ + jβ ∨ , − ip + jq ) . It follows that the right-hand side of (20) lies in N ∨ ( f ) , whichreadily implies the statement. (cid:3) Given g ∈ N ∨ ( K ) , there is a unique tuple ( a i ) ∈ K I such that g ≡ Y i ∈ I x α ∨ i ( a i ) mod ( N ∨ ( K ) , N ∨ ( K )); looking at a specific i ∈ I , we denote by a i,p ( g ) the coefficient of z p in the Laurent series a i .This procedure defines a morphism of groups a i,p : N ∨ ( K ) → C for each pair ( i, p ) ∈ I × Z . Lemma 4.13
Let π ∈ Π and let ( t , . . . , t m ) be the ordered list of all elements t ∈ [0 , suchthat Φ ∨ , + a ( π, t ) = ∅ . Set t m +1 = 1 . Let i ∈ I and set p = min (cid:8) h α ∨ i , π ( t ) i (cid:12)(cid:12) t ∈ [0 , (cid:9) . Let r ∈ { , . . . , m + 1 } and let ( v ℓ ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) . i) Let r + be the smallest element in (cid:8) ℓ ∈ { r, . . . , m } (cid:12)(cid:12) ( α ∨ i , p ) ∈ Φ ∨ , + a ( π, t ℓ ) (cid:9) , contingent on this set to be nonempty. Then for any u ∈ N ∨ (cid:0) f π ( t r ) (cid:1) there exists ( v ′ ℓ ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) such that v ′ r · · · v ′ m L wt( π ) = u v r · · · v m L wt( π ) and a i,p ( v ′ ℓ ) = ( a i,p ( u ) + a i,p ( v ℓ ) if ℓ = r + , a i,p ( v ℓ ) for all other ℓ ∈ { r, . . . , m } .(ii) For any c ∈ z O and any λ ∈ Λ , there exists ( v ′ ℓ ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) such that v ′ r · · · v ′ m L wt( π ) = c λ v r · · · v m L wt( π ) and a i,p ( v ′ ℓ ) = a i,p ( v ℓ ) for all ℓ ∈ { r, . . . , m } .(iii) For any b ∈ C not in (cid:8) (cid:9) ∪ (cid:8) a i,p ( v r ) + · · · + a i,p ( v ℓ ) (cid:12)(cid:12) ℓ ∈ { r, . . . , m } (cid:9) , there exists ( v ′ ℓ ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) such that v ′ r · · · v ′ m L wt( π ) = x ( − α ∨ i , − p ) (1 /b ) v r · · · v m L wt( π ) . Proof.
The lemma is trivial for r = m + 1 . Proceeding by decreasing induction, we choose r ∈ { , . . . , m } , assume that statements (i), (ii) and (iii) hold for r + 1 , and show that theyalso hold for r . We recall (see the proof of Proposition 4.8) that N ∨ (cid:0) f π ( t r ) (cid:1) = N ∨ ( π, t r ) ⊲⊳ N ∨ (cid:0) f π ( t r +0) (cid:1) and N ∨ (cid:0) f π ( t r +0) (cid:1) ⊆ N ∨ (cid:0) f π ( t r +1 ) (cid:1) . Let ( v ℓ ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) .We start with (i). Let u ∈ N ∨ (cid:0) f π ( t r ) (cid:1) . We can write uv r ∈ N ∨ (cid:0) f π ( t r ) (cid:1) as a product v ′ r u ′ with ( v ′ r , u ′ ) ∈ N ∨ ( π, t r ) × N ∨ (cid:0) f π ( t r +0) (cid:1) . Then a i,p ( u ) + a i,p ( v r ) = a i,p ( v ′ r ) + a i,p ( u ′ ) . Noting that u ′ ∈ N ∨ (cid:0) f π ( t r +1 ) (cid:1) , we make use of the inductive assumption: there exists ( v ′ ℓ ) ∈ Q mℓ = r +1 N ∨ ( π, t ℓ ) such that v ′ r +1 · · · v ′ m L wt( π ) = u ′ v r +1 · · · v m L wt( π ) a i,p ( v ′ ℓ ) = ( a i,p ( u ′ ) + a i,p ( v ℓ ) if ℓ = ( r + 1) + , a i,p ( v ℓ ) for all other ℓ ∈ { r + 1 , . . . , m } .We distinguish two cases. If ( α ∨ i , p ) ∈ Φ ∨ , + a ( π, t r ) , then f π ( t r +0) H − ( α ∨ i , p ) , whence a i,p ( u ′ ) =0 ; also r + = r in this case. If ( α ∨ i , p ) / ∈ Φ ∨ , + a ( π, t r ) , then a i,p ( v r ) = a i,p ( v ′ r ) = 0 ; here r + = ( r + 1) + . In both cases, routine checks conclude the proof of (i).We now turn to statement (ii). Let c ∈ z O and let λ ∈ Λ . One easily checks thatany subgroup of the form N ∨ ( f ) , in particular N ∨ (cid:0) f π ( t r ) (cid:1) , is stable under conjugation by c λ .Additionally, for any v ∈ N ∨ (cid:0) f π ( t r ) (cid:1) , when we write v ≡ Y i ∈ I x α ∨ i ( a i ) mod ( N ∨ ( K ) , N ∨ ( K )) , the Laurent series a i has valuation at least p ; this series is multiplied by c h α ∨ i ,λ i when oneconjugates v by c λ ; looking at the coefficient of z p then gives a i,p ( v ) = a i,p ( c λ vc − λ ) .Write c λ v r c − λ ∈ N ∨ (cid:0) f π ( t r ) (cid:1) as a product v ′ r u with ( v ′ r , u ) ∈ N ∨ ( π, t r ) × N ∨ (cid:0) f π ( t r +0) (cid:1) . Then a i,p ( v r ) = a i,p ( c λ v r c − λ ) = a i,p ( v ′ r ) + a i,p ( u ) . By induction, there exists ( v ′ ℓ ) ∈ Q mℓ = r +1 N ∨ ( π, t ℓ ) such that v ′ r +1 · · · v ′ m L wt( π ) = uc λ v r +1 · · · v m L wt( π ) and a i,p ( v ′ ℓ ) = ( a i,p ( u ) + a i,p ( v ℓ ) if ℓ = ( r + 1) + , a i,p ( v ℓ ) for all other ℓ ∈ { r + 1 , . . . , m } .Again we distinguish two cases. If ( α ∨ i , p ) ∈ Φ ∨ , + a ( π, t r ) , then f π ( t r +0) H − ( α ∨ i , p ) and therefore a i,p ( u ) = 0 . If ( α ∨ i , p ) / ∈ Φ ∨ , + a ( π, t r ) , then a i,p ( v r ) = a i,p ( v ′ r ) = 0 and anew a i,p ( u ) = 0 . Thus, a i,p ( u ) = 0 holds unconditionally, which concludes the proof of (ii).Lastly, let us deal with statement (iii). We distinguish three cases.Suppose first that ( α ∨ i , p ) ∈ Φ ∨ , + a ( π, t r ) . We write v r = x ( α ∨ i , p ) ( a ) e v r where a = a i,p ( v r ) and e v r is a product of elements x β ( a β ) with β ∈ Φ ∨ , + a ( π, t r ) \ { ( α ∨ i , p ) } and a β ∈ C . From (18) we get x ( − α ∨ i , − p ) (1 /b ) x ( α ∨ i , p ) ( a ) = (1 − a/b ) − α i x ( α ∨ i , p ) ( a (1 − a/b )) x ( − α ∨ i , − p ) (1 / ( b − a )) .
38y Lemma 4.12, x ( − α ∨ i , − p ) (1 / ( b − a )) e v r x ( − α ∨ i , − p ) ( − / ( b − a )) belongs to N ∨ (cid:0) f π ( t r ) (cid:1) ; we write it as a product e v ′ r u with ( e v ′ r , u ) ∈ N ∨ ( π, t r ) × N ∨ (cid:0) f π ( t r +0) (cid:1) .By induction, there exists ( v ′ ℓ ) ∈ Q mℓ = r +1 N ∨ ( π, t ℓ ) such that v ′ r +1 · · · v ′ m L wt( π ) = u x ( − α ∨ i , − p ) (1 / ( b − a )) v r +1 · · · v m L wt( π ) . Then x ( − α ∨ i , − p ) (1 /b ) v r · · · v m L wt( π ) = (1 − a/b ) − α i (cid:2) x ( α ∨ i , p ) ( a (1 − a/b )) e v ′ r (cid:3) v ′ r +1 · · · v ′ m L wt( π ) . Denoting the element between square brackets above by v ′ r , we get the desired expression, upto the inconsequential left multiplication by (1 − a/b ) − α i .Suppose now that there exists q > p such that ( α ∨ i , q ) ∈ Φ ∨ , + a ( π, t r ) ; then a i,p ( v r ) = 0 .We write v r = x ( α ∨ i , q ) ( a ) e v r where a ∈ C and e v r is a product of elements x β ( a β ) with β ∈ Φ ∨ , + a ( π, t r ) \ { ( α ∨ i , q ) } and a β ∈ C . Let c be a square root in t O of − ( a/b ) t q − p . From (18)we get x ( − α ∨ i , − p ) (1 /b ) x ( α ∨ i , q ) ( a ) = c − α i x ( α ∨ i , q ) ( a ) x ( − α ∨ i , − p ) (1 /b ) c − α i . By Lemma 4.12, x ( − α ∨ i , − p ) (1 /b ) (cid:0) c − α i e v r c α i (cid:1) x ( − α ∨ i , − p ) ( − /b ) belongs to N ∨ (cid:0) f π ( t r ) (cid:1) ; we write it as a product e v ′ r u with ( e v ′ r , u ) ∈ N ∨ ( π, t r ) × N ∨ (cid:0) f π ( t r +0) (cid:1) .By induction, there exists ( v ′ ℓ ) ∈ Q mℓ = r +1 N ∨ ( π, t ℓ ) such that v ′ r +1 · · · v ′ m L wt( π ) = u x ( − α ∨ i , − p ) (1 /b ) c − α i v r +1 · · · v m L wt( π ) . Then x ( − α ∨ i , − p ) (1 /b ) v r · · · v m L wt( π ) = c − α i (cid:2) x ( α ∨ i , q ) ( a ) e v ′ r (cid:3) v ′ r +1 · · · v ′ m L wt( π ) . Denoting the element between square brackets above by v ′ r , we get the desired expression, upto the inopportune left multiplication by c − α i ; the latter can however be wiped off by a furtheruse of the inductive assumption.Last, suppose that no affine coroot of direction α ∨ i occurs in Φ ∨ , + a ( π, t r ) ; then a i,p ( v r ) = 0 . ByLemma 4.12, x ( − α ∨ i , − p ) (1 /b ) v r x ( − α ∨ i , − p ) ( − /b ) belongs to N ∨ (cid:0) f π ( t r ) (cid:1) ; we write it as a product v ′ r u with ( v ′ r , u ) ∈ N ∨ ( π, t r ) × N ∨ (cid:0) f π ( t r +0) (cid:1) .By induction, there exists ( v ′ ℓ ) ∈ Q mℓ = r +1 N ∨ ( π, t ℓ ) such that v ′ r +1 · · · v ′ m L wt( π ) = u x ( − α ∨ i , − p ) (1 /b ) v r +1 · · · v m L wt( π ) . x ( − α ∨ i , − p ) (1 /b ) v r · · · v m L wt( π ) = v ′ r v ′ r +1 · · · v ′ m L wt( π ) , as desired, which concludes the proof of (iii). (cid:3) Let us now consider i ∈ I and two integral paths π and η related by the equation η = ˜ e i π . Wedenote by p the minimum of the function t
7→ h α ∨ i , π ( t ) i over the interval [0 , and by a and b the two points in time where π is bent to produce η . Noting that the conditions spelled outin sect. 4.1 do not uniquely determine b , we choose it to be the largest possible: either b = 1 or h α ∨ i , π ( b + h ) i > p for all small enough h > .Let ( t , . . . , t m ) be the ordered list of all elements in [0 , such that Φ ∨ , + a ( π, t ) = ∅ . We set t m +1 = 1 . The set Φ ∨ , + a ( π, a ) may be empty; if this happens, we insert a in the list ( t , . . . , t m ) ,for it will simplify the notation hereafter. On the contrary, the above condition imposed on b ensures that either b = 1 or ( α ∨ i , p ) ∈ Φ ∨ , + a ( π, b ) , so b automatically appears in the list ( t , . . . , t m +1 ) . We denote by r and s the indices in { , . . . , m + 1 } such that a = t r and b = t s ;by design t r = a < t r +1 ≤ t s = b . Lemma 4.14
In the setting described in the two paragraphs above, let ( v ℓ ) ∈ Q mℓ =1 N ∨ ( π, t ℓ ) .Assume that a i,p ( v s ) + · · · + a i,p ( v ℓ ) = 0 for each ℓ ∈ { s, . . . , m } . Then for any h ∈ C × , onecan construct ( w ℓ ) ∈ Q mℓ =1 N ∨ ( η, t ℓ ) such that v · · · v r − x ( − α ∨ i , − p − ( h ) v r · · · v m L wt( π ) = w . . . w m L wt( η ) . Proof.
Let ( v ℓ ) be as in the statement and let h ∈ C × . We set A = v · · · v r − and B = v r · · · v m . We note that f π ( t r ) ⊆ H ( α ∨ i , p +1) , so x ( α ∨ i , p +1) ( − /h ) ∈ N ∨ (cid:0) f π ( t r ) (cid:1) .Using Lemma 4.13 (i), we find ( v ′ r +1 , . . . , v ′ m ) ∈ Q mℓ = r N ∨ ( π, t ℓ ) such that x ( α ∨ i , p +1) ( − /h ) B L wt( π ) = v ′ r · · · v ′ m L wt( π ) and a i,p ( v ′ ℓ ) = a i,p ( v ℓ ) for all ℓ ∈ { r, . . . , m } . We set c = a i,p ( v s ) and write v ′ s = x ( α ∨ i , p ) ( c ) e v ′ s ;then e v ′ s ∈ N ∨ ( π, t s ) and a i,p ( e v ′ s ) = 0 . We also set C = v ′ r · · · v ′ s − and D = e v ′ s v ′ s +1 · · · v ′ m . Using Lemma 4.13 (iii), we find ( e v ′′ s , v ′′ s +1 , . . . , v ′′ m ) ∈ Q mℓ = s N ∨ ( π, t ℓ ) such that x ( − α ∨ i , − p ) ( − /c ) D L wt( π ) = e v ′′ s v ′′ s +1 · · · v ′′ m L wt( π ) . E = x ( α ∨ i , p ) ( c ) x ( − α ∨ i , − p ) (1 /c ) x ( α ∨ i , p ) ( c ) ,F = x ( α ∨ i , p ) ( − c ) e v ′′ s v ′′ s +1 · · · v ′′ m ,K = x ( − α ∨ i , − p − ( h ) x ( α ∨ i , p +1) (1 /h ) . Then
A x ( − α ∨ i , − p − ( h ) B L wt( π ) = AKCEF L wt( π ) . (21)Observing that Φ ∨ , + a ( η, t ℓ ) = Φ ∨ , + a ( π, t ℓ ) if ≤ ℓ < r , { ( α ∨ i , p + 1) } ⊔ s ( α ∨ i , p +1) (cid:0) Φ ∨ , + a ( π, t r ) (cid:1) if ℓ = r , s ( α ∨ i , p +1) (cid:0) Φ ∨ , + a ( π, t ℓ ) (cid:1) if r < ℓ < s , τ α i (cid:0) Φ ∨ , + a ( π, t ℓ ) (cid:1) if s ≤ ℓ ≤ m ,we check that the sequence (cid:16) v , . . . , v r − , x ( α ∨ i , p +1) ( − h ) (cid:0) z ( p +1) α i s i (cid:1) v ′ r (cid:0) z ( p +1) α i s i (cid:1) − , (cid:0) z ( p +1) α i s i (cid:1) v ′ r +1 (cid:0) z ( p +1) α i s i (cid:1) − , . . . , (cid:0) z ( p +1) α i s i (cid:1) v ′ s − (cid:0) z ( p +1) α i s i (cid:1) − ,z α i x ( α ∨ i , p ) ( − c ) e v ′′ s z − α i , z α i v ′′ s +1 z − α i , . . . , z α i v ′′ m z − α i (cid:17) (22)belongs to Q mℓ =1 N ∨ ( η, t ℓ ) . In addition, the product of the elements in this sequence is A x ( α ∨ i , p +1) ( − h ) (cid:0) z ( p +1) α i s i (cid:1) C (cid:0) z ( p +1) α i s i (cid:1) − z α i F z − α i . We now apply two transformations to the sequence (22): we conjugate the last m − s + 1 termsby ( − c ) − α i , and we conjugate the last m − r + 1 by h − α i . The resulting sequence, denoted by ( w ℓ ) , still belongs to Q mℓ =1 N ∨ ( η, t ℓ ) , because all our constructions are T ∨ ( C ) -equivariant.Observing that K = h − α i x ( α ∨ i , p +1) ( − h ) (cid:0) z ( p +1) α i s i (cid:1) and E = (cid:0) z ( p +1) α i s i (cid:1) − ( − c ) − α i z α i (see equation (19)), we obtain w · · · w m = AKCEF z − α i ( − ch ) α i , and a comparison with (21) yields A x ( − α ∨ i , − p − ( h ) B L wt( π ) = AKCEF z − α i L wt( η ) = w · · · w m L wt( η ) , as desired. (cid:3)
41e can now prove Proposition 4.5 (iv). We consider the situation η ⊗ · · · ⊗ η n = ˜ e i ( π ⊗ · · · ⊗ π n ) in the crystal Π ⊗ n , and our aim is to show that ˚ Z ( π ⊗ · · · ⊗ π n ) is contained in the closure of ˚ Z ( η ⊗ · · · ⊗ η n ) in Gr n .As in the proof of Proposition 4.5 (iii), we regard the concatenation π = π ∗ · · · ∗ π n as amap from [0 , n ] to Λ R , each path π , . . . , π n being travelled at nominal speed, and ditto for η = η ∗ · · · ∗ η n . Thus, for each j ∈ { , . . . , n } the restriction of π to the interval [ j − , j ] is π j , up to the obvious shifts in time and space.By Proposition 4.2 (ii), we have η = ˜ e i π . We denote by a and b the two points in time where π is bent to produce η . Let ( t , . . . , t m ) be the ordered list of all elements in [0 , n [ such that Φ ∨ , + a ( π, t ) = ∅ . We insert a in this list if it does not already appear there. We set t = 0 and t m +1 = n . We denote by r and s the indices in { , . . . , m + 1 } such that a = t r and b = t s .There is a unique integer k ∈ { , . . . , n } such that a and b both belong to [ k − , k ] ; plainly, η k = ˜ e i π k and η j = π j for all j ∈ { , . . . , n } \ { k } . We record that η ∗ · · · ∗ η j = ˜ e i ( π ∗ · · · ∗ π j ) if j ∈ { k, . . . , n } .For j ∈ { , . . . , n } , we set ν j = wt( π ) + · · · + wt( π j ) and denote by m j the largest element ℓ ∈ { , . . . , m } such that t ℓ ∈ [0 , j [ . Then ˚ Z ( π ⊗ · · · ⊗ π n ) is the set of all elements m Y ℓ =1 v ℓ z ν , z − ν m Y ℓ = m +1 v ℓ z ν , . . . , z − ν n − m n Y ℓ = m n − +1 v ℓ z ν n (23)with ( v ℓ ) ∈ Q mℓ =1 N ∨ ( π, t ℓ ) .Now assume that ( v ℓ ) is chosen so that a i,p ( v s ) + · · · + a i,p ( v ℓ ) = 0 for each ℓ ∈ { s, . . . , m } andpick h ∈ C × . Lemma 4.14 provides us with a sequence ( w ℓ ) ∈ Q mℓ =1 N ∨ ( η, t ℓ ) such that v · · · v r − x ( − α ∨ i , − p − ( h ) v r · · · v m L wt( π ) = w · · · w m L wt( η ) . However ( w ℓ ) satisfies more equations: for j ∈ { , . . . , n } , we have ( v · · · v m j L ν j = w · · · w m j L ν j if j < k , v · · · v r − x ( − α ∨ i , − p − ( h ) v r · · · v m j L ν j = w · · · w m j L ν j + α i if j ≥ k , (24)in the first case because w ℓ = v ℓ for all ℓ ∈ { , . . . , m k − } , in the second case becauseLemma 4.14 would have returned the subsequence ( w ℓ ) ≤ ℓ ≤ m j if we had fed it with the paths π ∗ · · · ∗ π j and η ∗ · · · ∗ η j and the datum ( v ℓ ) ≤ ℓ ≤ m j and h .42he system (24) translates to a single equation in Gr n , which manifests that the elementobtained by inserting x ( − α ∨ i , − p − ( h ) just before v r in (23) belongs to ˚ Z ( η ⊗ · · · ⊗ η n ) . Letting h tend to , we conclude that (23) lies in the closure of this set. To be sure, this conclusion hasbeen reached under the assumption that a i,p ( v s ) + · · · + a i,p ( v ℓ ) = 0 for each ℓ ∈ { s, . . . , m } ,but this restriction can be removed by a small perturbation of a i,p ( v s ) .Thus, Proposition 4.5 (iv) is, at last, fully proven. We keep the notation from sect. 2. Let λ = ( λ , . . . , λ n ) in (Λ + ) n . The tensor product V ( λ ) can be endowed on the one hand with its MV basis (sect. 2.3), on the other hand with thetensor product of the MV bases of the factors V ( λ ) , . . . , V ( λ n ) . In this section, we comparethese two bases through the explicit identification F ( I λ ∗ · · · ∗ I λ n ) ∼ = F ( I λ ) ⊗ · · · ⊗ F ( I λ n ) afforded by Beilinson and Drinfeld’s fusion product. We show that the transition matrix isupper unitriangular and that its entries are intersection multiplicities. The order relationneeded to convey the triangularity involves the inclusion of cycles. The Beilinson-Drinfeld Grassmannian G r BD is a relative version of the affine Grassmannianwhere the base is the space of effective divisors on a smooth curve. The choice of the affineline amply satisfies our needs and offers three advantages: there is a natural global coordinateon A , every G -torsor on A is trivializable, and the monodromy of any local system is trivial.Rather than looking for more generality, we will pragmatically stick with this choice. Consistentwith sect. 2, the coordinate on A is denoted by z .Formally, the Beilinson-Drinfeld Grassmannian G r BD n is defined as the functor on the categoryof commutative C -algebras that assigns to an algebra R the set of isomorphism classes oftriples ( x , . . . , x n ; F , β ) , where ( x , . . . , x n ) ∈ A n ( R ) , F is a G ∨ -torsor over A R and β is atrivialization of F away from the points x , . . . , x n ([5], sect. 5.3.10; [43], Definition 3.3; [46],Definition 3.1.1). We denote by π : G r BD n → A n the morphism to the base, which forgets F and β . It is known that G r BD n is representable by an ind-scheme and that π is ind-proper.We are only interested in the set of C -points, endowed with its ind-variety structure. Usinga trivialization of F , we can thus adopt the following simplified definition: G r BD n is the set of43airs ( x , . . . , x n ; [ β ]) , where ( x , . . . , x n ) ∈ C n and [ β ] belongs to the homogenenous space G ∨ (cid:0) C (cid:2) z, ( z − x ) − , . . . , ( z − x n ) − (cid:3)(cid:1) / G ∨ (cid:0) C [ z ] (cid:1) . This set is endowed with the structure of an ind-variety.
Example 5.1. ([5], Remark in sect. 5.3.10.) We consider the case G ∨ = GL N . Here the datumof [ β ] is equivalent to the datum of the C [ z ] -lattice β ( L ) in C ( z ) N , where L = C [ z ] N is thestandard lattice. Let us write x for the point ( x , . . . , x n ) and set f x = ( z − x ) · · · ( z − x n ) ;then a lattice L is of this form β ( L ) if and only if there exists a positive integer k such that f k x L ⊆ L ⊆ f − k x L . For each positive integer k , define (cid:0) G r BD n (cid:1) k to be the subset of G r BD n consisting of all pairs ( x ; L ) with f k x L ⊆ L ⊆ f − k x L . We identify C [ z ] / ( f k x ) with the vectorspace V of polynomials of degree strictly less than kn , and subsequently identify L /f k x L with V N . The space (cid:0) G r BD n (cid:1) k can then be realized as a Zariski-closed subset of C n × knN [ d =0 G d (cid:0) V N (cid:1) where G d (cid:0) V N (cid:1) denotes the Grassmannian of d -planes in V N . In this way, G r BD n is the inductivelimit of a system of algebraic varieties and closed embeddings, in other words, an ind-variety.We also want to deform the n -fold convolution variety Gr n . Accordingly, we define G r n as theset of pairs ( x , . . . , x n ; [ β , . . . , β n ]) , where ( x , . . . , x n ) ∈ C n and [ β , . . . , β n ] belongs to G ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) × G ∨ ( C [ z ]) · · · × G ∨ ( C [ z ]) G ∨ (cid:0) C (cid:2) z, ( z − x n ) − (cid:3)(cid:1) / G ∨ (cid:0) C [ z ] (cid:1) (see [43], Definition 3.8, or [46], (3.1.21)). This set G r n is endowed with the structure of anind-variety; it comes with a map m n : G r n → G r BD n defined by m n ( x , . . . , x n ; [ β , . . . , β n ]) = ( x , . . . , x n ; [ β · · · β n ]) . Example 5.2.
We again consider the case G ∨ = GL N . Then an element in G r n is the datumof a point ( x , . . . , x n ) ∈ C n and a sequence ( L , . . . , L n ) of C [ z ] -lattices in C ( z ) N for whichthere exists a positive integer k such that ( z − x j ) k L j − ⊆ L j ⊆ ( z − x j ) − k L j − for all j ∈ { , . . . , n } ; here again L = C [ z ] N is the standard lattice and L j = ( β · · · β j )( L ) .44n the above example, we can partition G r n into cells by specifying the relative positions of thepairs of lattices ( L j − , L j ) in terms of invariant factors. This construction can be generalizedto an arbitrary group G as follows: given λ = ( λ , . . . , λ n ) in (Λ + ) n , we define G r λ n as thesubset of G r n consisting of all pairs ( x , . . . , x n ; [ β , . . . , β n ]) with β j ∈ G ∨ ( C [ z ]) ( z − x j ) λ j G ∨ ( C [ z ]) for j ∈ { , . . . , n } . The Cartan decomposition G ∨ (cid:0) C (cid:2) z, ( z − x j ) − (cid:3)(cid:1) = G λ j ∈ Λ + G ∨ ( C [ z ]) ( z − x j ) λ j G ∨ ( C [ z ]) yields G r n = G λ ∈ (Λ + ) n G r λ n and it can be checked that G r λ n = G µ ∈ (Λ + ) n µ ≤ λ , ..., µ n ≤ λ n G r µ n . (25)In addition, the maps ( x , . . . , x j ; [ β , . . . , β j ]) ( x , . . . , x j − ; [ β , . . . , β j − ]) exhibit G r λ n asthe total space of an iterated fibration with base G r λ and successive fibers G r λ , . . . , G r λ n ; itfollows that G r λ n is a smooth connected variety of dimension ρ ( | λ | ) + n .Let us now investigate the fibers of the map π ◦ m n : G r n → C n . Given x ∈ C , we set O x = C [[ z − x ]] and K x = C (( z − x )) ; thus K x is the completion of C ( z ) at the place defined by x . We identify O and K with O x and K x by means of the map z z − x .We fix x = ( x , . . . , x n ) in C n . Let supp( x ) be the set of values y ∈ C that appear in thetuple x . For y ∈ supp( x ) , denote by m y the number of indices j ∈ { , . . . , n } such that x j = y and choose an increasing sequence ( p = 0 , p , p , . . . , p m y = n ) in a way that each interval [ p k − + 1 , p k ] contains exactly one index j such that x j = y . For β = [ β , . . . , β n ] in the fiber ( G r n ) x = G ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) × G ∨ ( C [ z ]) · · · × G ∨ ( C [ z ]) G ∨ (cid:0) C (cid:2) z, ( z − x n ) − (cid:3)(cid:1) / G ∨ (cid:0) C [ z ] (cid:1) , we define Θ( β ) y as the point [( β · · · β p ) , ( β p +1 · · · β p ) , . . . , ( β p my − +1 · · · β n )] in G ∨ ( K y ) × G ∨ ( O y ) · · · × G ∨ ( O y ) G ∨ ( K y ) | {z } m y factors G ∨ ( K y ) / G ∨ ( O y ) ∼ = Gr m y (note that Θ( β ) y does not depend on this choice, because β j ∈ G ∨ ( O y ) if x j = y ).45 roposition 5.3 The map β (Θ( β ) y ) is a bijection ( G r n ) x ≃ −→ Y y ∈ supp( x ) Gr m y . Proof.
Combining the Iwasawa decomposition (1) with the easily proven equality N ∨ ( K x ) = N ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) N ∨ ( O x ) , (26)we obtain the well-known equality G ∨ ( K x ) = G ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) G ∨ ( O x ) , for each x ∈ C .The case n = 1 of the proposition is banal. Assume that n ≥ , and for y ∈ supp( x ) , pick γ y ∈ Gr m y . Set x ′ = ( x , . . . , x n − ) and m = m x n , write γ x n = [ γ , . . . , γ m ] . Reasoning byinduction on n , we know that there is a unique β ′ = [ β , . . . , β n − ] in ( G r n − ) x ′ such that Θ( β ′ ) y = ( γ y if x n = y, [ γ , . . . , γ m − ] if x n = y. The elements γ , . . . , γ m belong to G ∨ ( K ) , which we identify to G ∨ ( K x n ) . We choose β n ∈ G ∨ (cid:0) C (cid:2) z, ( z − x n ) − (cid:3)(cid:1) such that ( β . . . β n − ) − ( γ . . . γ m ) ∈ β n G ∨ ( O x n ) . Then [ β , . . . , β n − , β n ] is the unique element β in ( G r n ) x such that Θ( β ) y = γ y for all y . (cid:3) Keep the notation above for x and the integers m y and let λ = ( λ , . . . , λ n ) in (Λ + ) n . Foreach y ∈ supp( x ) , define λ y ∈ (Λ + ) m y as the ordered tuple formed by the weights λ j , for j ∈ { , . . . , n } such that x j = y . Then, under the bijection given in Proposition 5.3, the fiber (cid:0) G r λ n (cid:1) x identifies with Y y ∈ supp( x ) Gr λ y m y . Recall our notation N ∨ for the unipotent radical of B ∨ . For µ ∈ Λ and x ∈ C , we define e S µ | x = ( z − x ) µ N ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) = N ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) ( z − x ) µ . N ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) / N ∨ ( C [ z ]) → N ∨ ( K x ) /N ∨ ( O x ) is bijective; composing with the natural map N ∨ ( K ) /N ∨ ( O ) → Gr , we obtain, after leftmultiplication by ( z − x ) µ , a bijection e S µ | x / N ∨ ( C [ z ]) ≃ −→ S µ . For ( µ , . . . , µ n ) ∈ Λ n , let S µ ∝ · · · ∝ S µ n be the set of all pairs ( x , . . . , x n ; [ β , . . . , β n ]) with ( x , . . . , x n ) in C n and [ β , . . . , β n ] in e S µ | x × N ∨ ( C [ z ]) · · · × N ∨ ( C [ z ]) e S µ n | x n / N ∨ ( C [ z ]) . Rewriting the Iwasawa decomposition as G ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) = G µ ∈ Λ N ∨ (cid:0) C (cid:2) z, ( z − x ) − (cid:3)(cid:1) ( z − x ) µ G ∨ ( C [ z ]) , we then see that the natural map Ψ : G ( µ ,...,µ n ) ∈ Λ n S µ ∝ · · · ∝ S µ n → G r n is bijective. Here Ψ is regarded as the calligraphic variant of the letter Ψ used in sect. 2.2;these two glyphs may be hard to distinguish, but hopefully this choice will not lead to anyconfusion.More generally, given ( µ , . . . , µ n ) ∈ Λ n and N ∨ ( O ) -stable subsets Z ⊆ S µ , . . . , Z n ⊆ S µ n , wedefine Z ∝ · · · ∝ Z n to be the subset of all pairs ( x , . . . , x n ; [ β , . . . , β n ]) with ( x , . . . , x n ) ∈ C n and [ β , . . . , β n ] ∈ e Z | x × N ∨ ( C [ z ]) · · · × N ∨ ( C [ z ]) e Z n | x n / N ∨ ( C [ z ]) where each e Z j | x j is the preimage of Z j under the map e S µ j | x j → S µ j .For µ ∈ Λ , we define ˙ S µ = [ ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Ψ (cid:0) S µ ∝ · · · ∝ S µ n (cid:1) . Proposition 5.4
Let λ = ( λ , . . . , λ n ) in (Λ + ) n and let µ ∈ Λ .(i) All the irreducible components of G r λ n ∩ ˙ S µ have dimension ρ ( | λ | + µ ) + n . ii) The map ( Z , . . . , Z n ) Ψ ( Z ∝ · · · ∝ Z n ) induces a bijection G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ ∗ Z ( λ ) µ × · · · × ∗ Z ( λ n ) µ n ≃ −→ Irr (cid:16) G r λ n ∩ ˙ S µ (cid:17) . (The bar above Ψ ( Z ∝ · · · ∝ Z n ) means closure in ˙ S µ .)Proof. Let ( µ , . . . , µ n ) ∈ Λ n be such that µ + · · · + µ n = µ and let ( Z , . . . , Z n ) ∈ ∗ Z ( λ ) µ ×· · · × ∗ Z ( λ n ) µ n . Then the set Ψ ( Z ∝ · · · ∝ Z n ) is irreducible. By Proposition 5.3 and its proof,the fiber of this set over a point x ∈ C n is isomorphic to the product, over all y ∈ supp( x ) , ofcycles Ψ( Z j ⋉ · · · ⋉ Z j m ) ⊆ Gr m where j , . . . , j m are the indices j ∈ { , . . . , n } such that x j = y . We remark that if weset λ y = ( λ j , . . . , λ j m ) and µ y = µ j + · · · + µ j m , then this cycle belongs to ∗ Z ( λ y ) µ y . ByProposition 2.2 (i), the dimension of the fiber of Ψ ( Z ∝ · · · ∝ Z n ) over x is therefore X y ∈ supp( x ) ρ ( | λ y | + µ y ) = ρ ( | λ | + µ ) and we conclude that Ψ ( Z ∝ · · · ∝ Z n ) has dimension ρ ( | λ | + µ ) + n .To finish the proof, we observe that these sets Ψ ( Z ∝ · · · ∝ Z n ) cover G r λ n ∩ ˙ S µ and are notredundant. (cid:3) Our MV bases are defined with the help of the unstable subsets T µ instead of the stable subsets S µ . We can easily adapt the constructions of this subsection to this case by replacing the Borelsubgroup B ∨ with its opposite with respect to T ∨ , and replacing similarly its unipotent radical N ∨ . We shall do this while keeping the notation ∝ and Ψ . Note that when we replace ˙ S µ by ˙ T µ = [ ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Ψ (cid:0) T µ ∝ · · · ∝ T µ n (cid:1) in Proposition 5.4, ρ ( | λ | + µ ) + n must be replaced by ρ ( | λ | − µ ) + n and the sets ∗ Z ( λ j ) µ j must be replaced by their unstarred counterparts.48 .3 The fusion product For any x ∈ C , the fibers of G r BD n and G r n over ( x, . . . , x ) are isomorphic to Gr and Gr n ,respectively. Thus, G r BD n (cid:12)(cid:12) ∆ ≃ −→ ∆ × Gr and G r n (cid:12)(cid:12) ∆ ≃ −→ ∆ × Gr n , where ∆ is the small diagonal, defined as the image of the map x ( x, . . . , x ) from C to C n .In the other extreme, the morphism m n : G r n → G r BD n is an isomorphism after restriction tothe open locus U ⊆ C n of points with pairwise different coordinates ([46], Lemma 3.1.23), andby Proposition 5.3, G r n (cid:12)(cid:12) U is isomorphic to U × (Gr) n . We define maps τ , i , j and ζ accordingto the diagram below. Gr n ∆ × Gr n (cid:15) (cid:15) τ o o i / / G r nm n (cid:15) (cid:15) U × (Gr) n ≃ (cid:15) (cid:15) j o o ζ / / (Gr) n ∆ × Gr (cid:15) (cid:15) G r BD nπ (cid:15) (cid:15) G r BD n (cid:12)(cid:12) U (cid:15) (cid:15) ∆ / / C n U o o Let λ ∈ (Λ + ) n and µ ∈ Λ , set B ( λ ) = IC (cid:16) G r λ n , C (cid:17) , d = dim G r λ n = 2 ρ ( | λ | ) + n, k = 2 ρ ( µ ) − n and denote the inclusion ˙ T µ → G r n by ˙ t µ . The next statement is due to Mirković and Vilonen. Proposition 5.5 (i) There are natural isomorphisms i ! B ( λ )[ n ] ∼ = τ ! IC (cid:16) Gr λ n , C (cid:17) and j ! B ( λ )[ n ] ∼ = ζ ! (cid:16) IC (cid:16) Gr λ , C (cid:17) ⊠ · · · ⊠ IC (cid:16) Gr λ n , C (cid:17)(cid:17) . (ii) Each cohomology sheaf of ( π ◦ m n ) ∗ B ( λ ) is a local system on C n .(iii) The complex of sheaves ( π ◦ m n ◦ ˙ t µ ) ∗ ( ˙ t µ ) ! B ( λ ) is concentrated in degree k and its k -thcohomology sheaf is a local system on C n . roof. To prove statement (i), one follows the reasoning in [3], sect. 1.7.5, noting that B ( λ ) and IC (cid:16) Gr λ n , C (cid:17) are the sheaves denoted by (cid:0) τ ◦ I λ (cid:1) e ⊠ · · · e ⊠ (cid:0) τ ◦ I λ n (cid:1) and I λ e ⊠ · · · e ⊠ I λ n in loc. cit. Statement (ii) is [39], (6.4). Statement (iii) is contained in the proof of [39],Proposition 6.4, up to a base change in the Cartesian square ˙ T µ ˙ t µ / / (cid:15) (cid:15) G r nm n (cid:15) (cid:15) T µ ( A n ) k µ / / G r BD n . (cid:3) Combining Propositions 2.1 and 5.5 (i), we see that the total cohomology of the stalk of thecomplex ( π ◦ m n ) ∗ B ( λ ) identifies with F ( I λ ) at any point in ∆ , and with F ( I λ ) ⊗· · ·⊗ F ( I λ n ) at any point in U . Statement (ii) in Proposition 5.5 thus provides the identification F ( I λ ) ∼ = F ( I λ ) ⊗ · · · ⊗ F ( I λ n ) required to compare the two bases of V ( λ ) . Statement (iii) further identifies the weight spaces F µ ( I λ ) ∼ = M ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ F µ ( I λ ) ⊗ · · · ⊗ F µ n ( I λ n ) . We keep the setup introduced in the previous section, in particular λ ∈ (Λ + ) n , µ ∈ Λ , ˙ t µ : ˙ T µ → G r n , B ( λ ) = IC (cid:16) G r λ n , C (cid:17) , d = dim G r λ n = 2 ρ ( | λ | ) + n, k = 2 ρ ( µ ) − n. In addition, we denote by L µ ( λ ) = H k ( π ◦ m n ◦ ˙ t µ ) ∗ ( ˙ t µ ) ! B ( λ ) the local system appearing in Proposition 5.5 (iii).50or each point x ∈ C n , we define maps as indicated below (cid:0) G r λ n ∩ ˙ T µ (cid:1) x / / g ′ (cid:15) (cid:15) h ′ $ $ ❍❍❍❍❍ (cid:0) ˙ T µ (cid:1) x ˙ t ′ µ (cid:15) (cid:15) i ′ $ $ ❍❍❍❍❍❍ G r λ n ∩ ˙ T µ / / g (cid:15) (cid:15) ˙ T µ ˙ t µ (cid:15) (cid:15) (cid:0) G r λ n (cid:1) x j ′ / / h $ $ ❍❍❍❍❍ ( G r n ) x i $ $ ❍❍❍❍❍❍ / / { x } i $ $ ❍❍❍❍❍❍ G r λ n j / / G r n π ◦ m n / / C n where for instance (cid:0) G r λ n (cid:1) x is the fiber of G r λ n over x . (The notation i and j does not designatethe same maps as in the previous subsection.) We then construct the following diagram,referred to as ( ♥ ) in the sequel. H k (cid:16) ˙ T µ , ( ˙ t µ ) ! B ( λ ) (cid:17) ≃ / / ≃ (cid:15) (cid:15) H k + d (cid:16) G r λ n ∩ ˙ T µ , g ! C G r λ n (cid:17) ≃∩ (cid:2) G r λ n (cid:3) / / (cid:15) (cid:15) H BM d − k (cid:16) G r λ n ∩ ˙ T µ (cid:17) ( g ∗ u x ) ∩ (cid:15) (cid:15) H k (cid:16) ( ˙ T µ ) x , ( ˙ t ′ µ ) ! i ∗ B ( λ ) (cid:17) ≃ / / H k + d (cid:16)(cid:0) G r λ n ∩ ˙ T µ (cid:1) x , g ′ ! C ( G r λ n ) x (cid:17) ∩ (cid:2) ( G r λ n ) x (cid:3) / / H BM d − k − n (cid:16)(cid:0) G r λ n ∩ ˙ T µ (cid:1) x (cid:17) The left vertical arrow in ( ♥ ) is the restriction of the cohomology with support in ˙ T µ from G r n to ( G r n ) x ; in other words, it is the image by the functor H k (cid:0) ˙ T µ , ( ˙ t µ ) ! − (cid:1) of the adjunctionmorphism B ( λ ) → i ∗ i ∗ B ( λ ) . Lemma 5.7 below implies that it is an isomorphism. Likewise,the middle vertical arrow is the restriction from G r λ n to (cid:0) G r λ n (cid:1) x , afforded by the adjunctionmorphism j ∗ B ( λ ) → h ∗ h ∗ j ∗ B ( λ ) .On the top line, the left arrow is the restriction from G r n to G r λ n , fulfilled by the adjunctionmorphism B ( λ ) → j ∗ j ∗ B ( λ ) = j ∗ C G r n [ d ] . On the bottom line, it is the restriction from ( G r n ) x to (cid:0) G r λ n (cid:1) x , achieved by i ∗ B ( λ ) → ( j ′ ) ∗ ( j ′ ) ∗ i ∗ B ( λ ) . Mirković and Vilonen’s argument(reproduced in sect. 2.3) shows that these two arrows are isomorphisms.The two paths around the left square in ( ♥ ) are two different expressions for the restrictionfrom G r n to (cid:0) G r λ n (cid:1) x ; therefore this square commutes.In both lines of ( ♥ ) the right arrow is Alexander duality; we note that H BM d − k (cid:0) G r λ n ∩ ˙ T µ (cid:1) and H BM d − k − n (cid:0)(cid:0) G r λ n ∩ ˙ T µ (cid:1) x (cid:1) are the top-dimensional Borel-Moore homology groups.The map h is a regular embedding of codimension n . Its orientation class (generalized Thom51lass) is an element u x ∈ H n (cid:16)(cid:0) G r λ n (cid:1) x , h ! C G r λ n (cid:17) . The right vertical arrow in ( ♥ ) is the cap product with g ∗ u x ∈ H n (cid:16)(cid:0) G r λ n ∩ ˙ T µ (cid:1) x , ( h ′ ) ! C G r λ n ∩ ˙ T µ (cid:17) , the restriction of u x to G r λ n ∩ ˙ T µ . Lemma 5.6
In the diagram ( ♥ ), the square on the right commutes.Proof. Applying formula IX.4.9 in [24], we get u x ∩ (cid:2) G r λ n (cid:3) = (cid:2) ( G r λ n ) x (cid:3) .Formula (8) in [15], sect. 19.1 (or formula IX.3.4 in [24]) asserts that given a topologicalmanifold X and inclusions of closed subsets a : A → X and b : B → X , for any α ∈ H • ( A, a ! C X ) , β ∈ H • ( B, b ! C X ) and C ∈ H BM • ( X ) one has ( b ∗ α ) ∩ ( β ∩ C ) = ( α ∪ β ) ∩ C. (27)Using the six operations formalism, one checks without much trouble that this result is alsovalid if A and B are only locally closed.Now pick ξ ∈ H k + d (cid:16) G r λ n ∩ ˙ T µ , g ! C G r λ n (cid:17) . Applying (27) twice and using that u x has even degree, we compute ( h ∗ ξ ) ∩ (cid:16) u x ∩ (cid:2) G r λ n (cid:3)(cid:17) = ( ξ ∪ u x ) ∩ (cid:2) G r λ n (cid:3) = ( u x ∪ ξ ) ∩ (cid:2) G r λ n (cid:3) = ( g ∗ u x ) ∩ (cid:16) ξ ∩ (cid:2) G r λ n (cid:3)(cid:17) . This equality means precisely that ξ has the same image under the two paths in ( ♥ ) thatcircumscribe the square on the right. (cid:3) Lemma 5.7
There are natural isomorphisms H k (cid:16) ˙ T µ , ( ˙ t µ ) ! B ( λ ) (cid:17) ∼ = H (cid:0) C n , L µ ( λ ) (cid:1) and H k (cid:16) ( ˙ T µ ) x , ( ˙ t ′ µ ) ! i ∗ B ( λ ) (cid:17) ∼ = (cid:0) L µ ( λ ) (cid:1) x and the left vertical arrow in ( ♥ ) is the stalk map H (cid:0) C n , L µ ( λ ) (cid:1) → (cid:0) L µ ( λ ) (cid:1) x . roof. The first isomorphism is H (cid:0) C n , L µ ( λ ) (cid:1) = H k (cid:16) C n , ( π ◦ m n ) ∗ ( ˙ t µ ) ∗ ( ˙ t µ ) ! B ( λ ) (cid:17) = H k (cid:16) ˙ T µ , ( ˙ t µ ) ! B ( λ ) (cid:17) . The second one requires the notion of a universally locally acyclic complex (see [9], sect. 5.1).Specifically, B ( λ ) is ( π ◦ m n ) -ULA ([42], proof of Proposition IV.3.4, or [43], Lemma 3.20), sothere is an isomorphism i ∗ B ( λ ) → i ! B ( λ )[2 n ] . Then H k (cid:16) ( ˙ T µ ) x , ( ˙ t ′ µ ) ! i ∗ B ( λ ) (cid:17) = H k (cid:16) ( ˙ T µ ) x , ( ˙ t ′ µ ) ! i ! B ( λ )[2 n ] (cid:17) = H k (cid:16) { x } , ( π ◦ m n ) ∗ ( ˙ t ′ µ ) ∗ ( ˙ t ′ µ ) ! i ! B ( λ )[2 n ] (cid:17) = H k (cid:16) { x } , ( π ◦ m n ) ∗ ( ˙ t ′ µ ) ∗ i ′ ! ( ˙ t µ ) ! B ( λ )[2 n ] (cid:17) = H k (cid:16) { x } , ( i ) ! ( π ◦ m n ) ∗ ( ˙ t µ ) ∗ ( ˙ t µ ) ! B ( λ )[2 n ] (cid:17) , the last step being proper base change. Now ( π ◦ m n ) ∗ ( ˙ t µ ) ∗ ( ˙ t µ ) ! B ( λ ) is the local system L µ ( λ ) shifted by − k , and therefore H k (cid:16) ( ˙ T µ ) x , ( ˙ t ′ µ ) ! i ∗ B ( λ ) (cid:17) = H (cid:16) { x } , ( i ) ! L µ ( λ )[2 n ] (cid:17) = H (cid:16) { x } , ( i ) ∗ L µ ( λ ) (cid:17) = (cid:0) L µ ( λ ) (cid:1) x as announced. (cid:3) By Proposition 5.4, the irreducible components of G r λ n ∩ ˙ T µ are all top-dimensional and can beindexed by G ( µ ,...,µ n ) ∈ Λ n µ + ··· + µ n = µ Z ( λ ) µ × · · · × Z ( λ n ) µ n ; (28)namely, to a tuple Z = ( Z , . . . , Z n ) is assigned the component X ( Z ) = Ψ ( Z ∝ · · · ∝ Z n ) ∩ G r λ n , the bar denoting closure in ˙ T µ . From now on, to lighten the writing, we will substitute Z ( λ ) µ for the cumbersome compound (28), using implicitly the bijection (2).The proof of Proposition 5.4 shows that for any x ∈ C n , the irreducible components of thefiber (cid:0) G r λ n ∩ ˙ T µ (cid:1) x have all the same dimension and can be indexed by Z ( λ ) µ . Let us lookmore closely at two particular cases. 53f x ∈ C n lies in the open locus U of points with pairwise different coordinates, then, underthe bijection ( G r n ) x ∼ = (Gr) n from Proposition 5.3, the irreducible components of (cid:0) G r λ n ∩ ˙ T µ (cid:1) x are identified with the sets X ( Z ) x ∼ = (cid:0) Z ∩ Gr λ (cid:1) × · · · × (cid:0) Z n ∩ Gr λ n (cid:1) (29)with Z = ( Z , . . . , Z n ) in Z ( λ ) µ .On the other hand, recalling that an element Z ∈ Z ( λ ) µ is a subset of Gr λ n , we may considerthe preimage Y ( Z ) of ∆ × (cid:0) Z ∩ Gr λ n (cid:1) under the isomorphism G r n (cid:12)(cid:12) ∆ ≃ −→ ∆ × Gr n . Then for any x ∈ ∆ , the irreducible components of the fiber (cid:0) G r λ n ∩ ˙ T µ (cid:1) x are the sets Y ( Z ) x for Z ∈ Z ( λ ) µ .Let us introduce a last piece of notation before stating the next theorem. In sect. 2.3, weexplained the construction of the MV basis of the µ -weight space of V ( λ ) ; this basis is inbijection with Z ( λ ) µ and we denote by h Z i the element indexed by Z . On the other hand,given Z = ( Z , . . . , Z n ) in Z ( λ ) × · · · × Z ( λ n ) , we can look at hh Z ii = h Z i ⊗ · · · ⊗ h Z n i ,another element in V ( λ ) . Theorem 5.8
Let ( Z ′ , Z ′′ ) ∈ ( Z ( λ ) µ ) . The coefficient a Z ′ , Z ′′ in the expansion (cid:10)(cid:10) Z ′′ (cid:11)(cid:11) = X Z ∈ Z ( λ ) µ a Z , Z ′′ (cid:10) Z (cid:11) is the multiplicity of Y ( Z ′ ) in the intersection product X ( Z ′′ ) · (cid:0) G r λ n (cid:1)(cid:12)(cid:12) ∆ computed in the ambientspace G r λ n .Proof. Taking into account Lemma 5.7, the diagram ( ♥ ) can be rewritten as follows. H (cid:0) C n , L µ ( λ ) (cid:1) ≃ / / ≃ (cid:15) (cid:15) H BMtop (cid:16) G r λ n ∩ ˙ T µ (cid:17) ( g ∗ u x ) ∩ (cid:15) (cid:15) (cid:0) L µ ( λ ) (cid:1) x ≃ / / H BMtop (cid:16)(cid:0) G r λ n ∩ ˙ T µ (cid:1) x (cid:17) The fundamental classes of the irreducible components of G r λ n ∩ ˙ T µ and (cid:0) G r λ n ∩ ˙ T µ (cid:1) x providebases of the two Borel-Moore homology groups, both indexed by Z ( λ ) µ . In these bases, theright vertical arrow can be regarded as a matrix, say Q x . This matrix can be computed byintersection theory: applying Theorem 19.2 in [15], we see that if x ∈ U (respectively, x ∈ ∆) ,then the entry in Q x at position ( Z ′ , Z ′′ ) is the multiplicity of X ( Z ′ ) x (respectively, Y ( Z ′ ) x )in the intersection product X ( Z ′′ ) · (cid:0) G r λ n (cid:1) x G r λ n . Because (29) holds steadily over the open set U , we seethat Q x U is just the identity matrix.According to the discussion at the end of sect. 5.3, the geometric Satake correspondence identi-fies V ( λ ) µ with each fiber of the local system L µ ( λ ) . The basis element h Z i is the fundamentalclass of X ( Z ) x when x ∈ ∆ , and the basis element hh Z ii is the fundamental class of X ( Z ) x when x ∈ U . Therefore, the coefficient a Z ′ , Z ′′ in the statement of the theorem is the entry atposition ( Z ′ , Z ′′ ) in the product Q x ∆ × ( Q x U ) − , for any choice of ( x ∆ , x U ) ∈ ∆ × U . (cid:3) In particular, the entries a Z ′ , Z ′′ of the transition matrix between our two bases are nonnegativeintegers. Proposition 5.9
In the setup of Theorem 5.8, the diagonal entry a Z ′′ , Z ′′ is equal to one.Proof. Write Z ′′ = ( Z , . . . , Z n ) in Z ( λ ) × · · · × Z ( λ n ) . By the slice theorem applied tothe quotient map G ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) → Gr (or, in this concrete situation, using Remark 15 andCorollary 5 in [17]), we can find, for each j ∈ { , . . . , n } , an affine variety U j and a map φ j : U j → G ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) such that u [ φ j ( u )] sends U j isomorphically to an open subset of Gr λ j which meets Z j .For x ∈ C and u ∈ U j , let φ j ( u ) | x denote the result of substituting z − x for z in φ j ( u ) . Wecan then define an open embedding φ as on the diagram C n × ( U × · · · × U n ) φ / / (cid:15) (cid:15) G r λ nπ ◦ m n (cid:15) (cid:15) C n C n by setting φ ( x , . . . , x n ; u , . . . , u n ) = (cid:16) x , . . . , x n ; h φ ( u ) | x , . . . , φ n ( u n ) | x n i(cid:17) . Since intersection multiplicities are of local nature, a Z ′′ , Z ′′ can be computed after restrictionto the image of φ , where the situation is that of a trivial bundle. (cid:3) It is possible to put coordinates on G r λ n and to effectively compute the intersection multiplicitiesmentioned in Theorem 5.8. In this section, we look at the case of the group G = SL . We55dopt the usual description Λ = ( Z ε ⊕ Z ε ⊕ Z ε ) / Z ( ε + ε + ε ) of the weight lattice, sothat V ( ε ) is the defining representation of G and V ( − ε ) is its dual.We consider the sequence of dominant weights λ = ( ε , − ε ) . The basic MV cycles are Z i =Gr ε ∩ T ε i and Z − i = Gr − ε ∩ T − ε i for i ∈ { , , } , and with this notation Z ( λ ) = (cid:8) ( Z i , Z − j ) (cid:12)(cid:12) ( i, j ) ∈ { , , } (cid:9) . To abbreviate, we set Z i, − j = ( Z i , Z − j ) . For weight reasons, hh Z i, − j ii = h Z i, − j i if i = j . Therest of the transition matrix between the two bases is given as follows. hh Z , − ii = h Z , − ihh Z , − ii = h Z , − i + h Z , − ihh Z , − ii = h Z , − i + h Z , − i From these relations, we get h Z , − i = hh Z , − ii − hh Z , − ii + hh Z , − ii . This allows to checkthat h Z , − i is G -invariant, which in truth is a consequence of the compatibility of the MVbasis of V ( λ ) with the isotypical filtration (Theorem 3.4).As an example, let us sketch out a computation which justifies that h Z , − i appears withcoefficient one in hh Z , − ii . We consider two charts on G r λ , both with C as domain: φ : ( x , x , a, b, c, d ) x , x ; z − x a b , c z − x d z − x , φ : ( x , x , a ′ , b ′ , c ′ , d ′ ) x , x ; a ′ z − x b ′ , z − x c ′
00 1 00 d ′ z − x . (The matrices here belong to the group PGL ( C [ z − x , z − x ] .) One easily computes thetransition map between these two charts: a ′ = 1 /a, b ′ = − b/a, c ′ = − a ( ac + bd + x − x ) , d ′ = − ad. In the chart φ , the cycle Y ( Z , − ) is defined by the equations a = b = x − x = 0 . Inthe chart φ , the cycle X ( Z , − ) is defined by the equations b ′ = c ′ = 0 . Thus, the ideals in R = C [ x , x , a, b, c, d ] of the subvarieties V = φ − (cid:0) Y ( Z , − ) (cid:1) and X = φ − (cid:0) X ( Z , − ) (cid:1) are respectively p = ( a, b, x − x ) and q = ( b, ac + x − x ) . q ⊆ p , we have V ⊆ X ; in fact, V is a subvariety of X of codimension one. The localring A = O V,X of X along V is the localization of R/ q at the ideal p / q . Observing that c is not in p , we see that its image in A is invertible, and then that x − x generates themaximal ideal of A . As a consequence, the order of vanishing of x − x along V (see [15],sect. 1.2) is equal to one. By definition, this is the multiplicity of Y ( Z , − ) in the intersectionproduct X ( Z , − ) · G r λ (cid:12)(cid:12) ∆ . A nice feature of the Beilinson-Drinfeld Grassmannian is its so-called factorizable structure (seefor instance [42], Proposition II.1.13). On the other side of the geometric Satake equivalence,this corresponds to associativity properties of partial tensor products.Let n = ( n , . . . , n r ) be a composition of n in r parts. We define the partial diagonal ∆ n = { ( x , . . . , x | {z } n times , . . . , x r , . . . , x r | {z } n r times ) | ( x , . . . , x r ) ∈ C r } . We write λ as a concatenation (cid:0) λ (1) , . . . , λ ( r ) (cid:1) , where each λ ( j ) belongs to (Λ + ) n j , and similarlywe write each Z ∈ Z ( λ ) µ as (cid:0) Z (1) , . . . , Z ( r ) (cid:1) with Z ( j ) ∈ Z ( λ ( j ) ) . Then V ( λ ) = V (cid:0) λ (1) (cid:1) ⊗ · · · ⊗ V (cid:0) λ ( r ) (cid:1) and (cid:10) Z ( j ) (cid:11) ∈ V (cid:0) λ ( j ) (cid:1) . Further, define X ( Z , n ) = Ψ ( Z ∝ · · · ∝ Z n ) (cid:12)(cid:12) ∆ n ∩ G r λ n where the bar means closure in (cid:0) ˙ T µ (cid:1)(cid:12)(cid:12) ∆ n . These X ( Z , n ) generalize the set X ( Z ) defined insect. 5.4, as the latter corresponds to the composition (1 , . . . , .Theorem 5.8 can then be extended to this context in a straightforward fashion, as demonstratedby the following statement. Proposition 5.10
Let ( Z ′ , Z ′′ ) ∈ ( Z ( λ ) µ ) . The coefficient b Z ′ , Z ′′ in the expansion (cid:10) Z ′′ (1) (cid:11) ⊗ · · · ⊗ (cid:10) Z ′′ ( r ) (cid:11) = X Z ∈ Z ( λ ) µ b Z , Z ′′ h Z i is the multiplicity of Y ( Z ′ ) in the intersection product X ( Z ′′ , n ) · (cid:0) G r λ n (cid:1)(cid:12)(cid:12) ∆ computed in theambient space G r λ n (cid:12)(cid:12) ∆ n . The proof does not require any new ingredient and is left to the reader.57 .7 Triangularity
In this section, we show that the transition matrix described in Theorem 5.8 is unitriangularwith respect to an adequate order on Z ( λ ) µ . Proposition 5.11
Let ( µ , . . . , µ n ) and ( ν , . . . , ν n ) in Λ n and let S be a stratum for theind-structure of G r n . If Ψ (cid:0) T ν ∝ · · · ∝ T ν n (cid:1) meets the closure of S ∩ Ψ (cid:0) T µ ∝ · · · ∝ T µ n (cid:1) , then ν ≥ µ , ν + ν ≥ µ + µ , . . . , ν + · · · + ν n ≥ µ + · · · + µ n . Proof.
Given a tuple ζ = ( ζ , . . . , ζ n ) in (Λ / Z Φ) n , we set G r n, ζ = G λ ∈ (Λ + ) n λ ∈ ζ , ..., λ n ∈ ζ n G r λ n . From equation (25), we deduce that each G r n, ζ is closed and connected in the ind-topology;as these subsets form a finite partition of the space G r n , they are its connected components.We easily verify that a subset of the form Ψ (cid:0) T µ ∝ · · · ∝ T µ n (cid:1) is contained in G r n, ζ if each ζ j is the coset of µ j modulo Z Φ . Therefore, a necessary condition for Ψ (cid:0) T ν ∝ · · · ∝ T ν n (cid:1) to meetthe closure of S ∩ Ψ (cid:0) T µ ∝ · · · ∝ T µ n (cid:1) is that µ j − ν j ∈ Z Φ for each j ∈ { , . . . , n } .Let λ ∨ ∈ Hom Z (Λ , Z ) be a dominant integral weight for the group G ∨ and let V be the finitedimensional irreducible representation of G ∨ of highest weight λ ∨ . Then G ∨ ( C ( z )) acts on V ⊗ C ( z ) . The standard lattice L = V ⊗ C [ z ] is left stable by G ∨ ( C [ z ]) .We choose a nonzero linear form p : V → C that vanishes on all weight subspaces of V but thehighest weight subspace. Extending the scalars, we regard p as a linear form V ⊗ C ( z ) → C ( z ) .For x = ( x , . . . , x n ) in C n , we set f x = ( z − x ) · · · ( z − x n ) . Let S be a stratum for theind-structure of G r n . There exists a positive integer k such that f k x L ⊆ β . . . β n ( L ) ⊆ f − k x L for each ( x , . . . , x n ; [ β , . . . , β n ]) ∈ S .Now we take ( µ , . . . , µ n ) ∈ Λ n and ( x , . . . , x n ; [ β , . . . , β n ]) in S ∩ Ψ (cid:0) T µ ∝ · · · ∝ T µ n (cid:1) . Then p ( β . . . β n ( L )) is the fractional ideal ( z − x ) h λ ∨ ,µ i · · · ( z − x n ) h λ ∨ ,µ n i C [ z ] , and therefore dim (cid:0) p ( β . . . β n ( L )) /f k x C [ z ] (cid:1) = kn − (cid:10) λ ∨ , µ + · · · + µ n (cid:11) .
58f the point ( x , . . . , x n ; [ β , . . . , β n ]) degenerates to ( y , . . . , y n ; [ γ , . . . , γ n ]) ∈ Ψ (cid:0) T ν ∝ · · · ∝ T ν n (cid:1) , then dim (cid:0) p ( γ . . . γ n ( L )) /f k y C [ z ] (cid:1) ≤ dim (cid:0) p ( β . . . β n ( L )) /f k x C [ z ] (cid:1) which translates to (cid:10) λ ∨ , ν + · · · + ν n (cid:11) ≥ (cid:10) λ ∨ , µ + · · · + µ n (cid:11) . This inequation holds for any dominant coweight λ ∨ , hence ν + · · · + ν n ≥ µ + · · · + µ n .This proves the last inequality among those announced in the statement. The other ones canbe shown in a similar way, by taking the image under the obvious truncation map G r n → G r j for each j ∈ { , . . . , n } . (cid:3) Corollary 5.12
Adopt the setup of Theorem 5.8. Let ( µ , . . . , µ n ) and ( ν , . . . , ν n ) in Λ n besuch that Z ′ ∈ Z ( λ ) ν × · · · × Z ( λ n ) ν n and Z ′′ ∈ Z ( λ ) µ × · · · × Z ( λ n ) µ n . A necessarycondition for a Z ′ , Z ′′ = 0 is that ν ≥ µ , ν + ν ≥ µ + µ , . . . , ν + · · · + ν n − ≥ µ + · · · + µ n − . We can obtain more stringent conditions regarding the transition matrix by looking at theassociativity properties from sect. 5.6. The sharpest result is obtained with a composition ( n , n ) of n in two parts. Accordingly, we write λ as a concatenation ( λ (1) , λ (2) ) and similarlywrite each Z ∈ Z ( λ ) as ( Z (1) , Z (2) ) . Here Z (1) is an element in Z ( λ ) × · · · × Z ( λ n ) , butowing to the bijection (2) it can also be regarded as a cycle in Gr λ (1) n . Theorem 5.13
Let ( Z ′ , Z ′′ ) ∈ ( Z ( λ ) µ ) . Consider the expansion (cid:10) Z ′′ (1) (cid:11) ⊗ (cid:10) Z ′′ (2) (cid:11) = X Z ∈ Z ( λ ) µ b Z , Z ′′ h Z i . If b Z ′ , Z ′′ = 0 , then either Z ′ = Z ′′ or Z ′ (1) ( Z ′′ (1) as cycles in Gr λ (1) n . In addition, b Z ′′ , Z ′′ = 1 .Proof. Let Z ′ = ( Z ′ , . . . , Z ′ n ) and Z ′′ = ( Z ′′ , . . . , Z ′′ n ) in Z ( λ ) µ .For j ∈ { , . . . , n } , let µ j be the weight such that Z ′′ j ∈ Z ( λ j ) µ j . Using the gallery modelsfrom [17] (or Theorem 4.6 and Proposition 4.8 above), we find a nonnegative integer d j and59onstruct a map φ j : C d j → N − , ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) such that (cid:8)(cid:2) φ j ( a ) z µ j (cid:3) (cid:12)(cid:12) a ∈ C d j (cid:9) is a densesubset of Z ′′ j . Then φ : ( x ; a , . . . , a n ) (cid:16) x ; h φ ( a ) | x ( z − x ) µ , . . . , φ n ( a n ) | x n ( z − x n ) µ n i(cid:17) maps C n × C d × · · · × C d n onto a dense subset of Ψ ( Z ′′ ∝ · · · ∝ Z ′′ n ) , where φ j ( a j ) | x j means theresult of substituting z − x j for z in φ j ( a j ) .Assume that b Z ′ , Z ′′ = 0 . By Proposition 5.10, Y ( Z ′ ) is contained in X ( Z ′′ , ( n , n )) , hence inthe closure of Ψ ( Z ′′ ∝ · · · ∝ Z ′′ n ) (cid:12)(cid:12) ∆ ( n ,n .Take a point in Z ′ (1) ∩ Gr λ (1) n , written as [ g , . . . , g n ] where each g j is in G ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) . Wecan complete this datum to get an element Γ = (cid:0) , . . . , (cid:2) g , . . . , g n (cid:3)(cid:1) of Y ( Z ′ ) . Working in the analytic topology for expositional simplicity, we see that Γ is the limitof a sequence ( φ ( x p ; a ,p , . . . , a n,p )) p ∈ N with x p ∈ ∆ ( n ,n ) and ( a ,p , . . . , a n,p ) ∈ C d ×· · ·× C d n .We write x p = ( x ,p , . . . , x ,p | {z } n times , x ,p , . . . , x ,p | {z } n times ) with of course lim p →∞ x ,p = lim p →∞ x ,p = 0 . (30)Then [ g , . . . , g n ] = lim p →∞ (cid:2) φ ( a ,p ) z µ , . . . , φ n ( a ,n ) z µ n (cid:3) | x ,p = lim p →∞ (cid:2) φ ( a ,p ) z µ , . . . , φ n ( a ,n ) z µ n (cid:3) is the limit of a sequence of points in Z ′′ (1) . Therefore Z ′ (1) ∩ Gr λ (1) n ⊆ Z ′′ (1) , whence the inclusion Z ′ (1) ⊆ Z ′′ (1) .In addition to b Z ′ , Z ′′ = 0 , assume that the latter inclusion is an equality. Then Z ′ (1) = Z ′′ (1) because these two MV cycles are irreducible components of the same Gr λ (1) n ∩ ( m n ) − ( T µ (1) ) ,with indeed µ (1) = µ + · · · + µ n . We regard Z ′ (2) and Z ′′ (2) as cycles in Gr λ (2) n . Take a point in Z ′ (2) ∩ Gr λ (2) n , written as [ g n +1 , . . . , g n ] where each g j is in G ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) . We can then lookat the element Γ = (cid:0) , . . . , (cid:2) z µ , . . . , z µ n , g n +1 , . . . , g n (cid:3)(cid:1) Y ( Z ′ ) . Again Γ is the limit of a sequence ( φ ( x p ; a ,p , . . . , a n,p )) p ∈ N with x p ∈ ∆ ( n ,n ) and ( a ,p , . . . , a n,p ) ∈ C d × · · · × C d n . We set B p = z − µ (1) φ ( a ,p ) z µ · · · φ n ( a n ,p ) z µ n . Writing again (30), we have L µ (1) = lim p →∞ (cid:2) z µ (1) B p (cid:3) | x ,p = lim p →∞ (cid:2) z µ (1) B p (cid:3) (31)and z µ (1) [ g n +1 , . . . , g n ] =lim p →∞ ( z µ (1) B p ) | x ,p (cid:2) φ n +1 ( a n +1 ,p ) z µ n , . . . , φ n ( a n,p ) z µ n (cid:3) | x ,p . (32)Let K be the kernel of the evaluation map N − , ∨ (cid:0) C (cid:2) z − (cid:3)(cid:1) → N − , ∨ ( C ) at z = ∞ . The multi-plication induces a bijection K × N − , ∨ ( C [ z ]) ≃ −→ N − , ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) . We decompose B p as a product B − ,p B + ,p according to this bijection. Using (31) and identifyingthe ind-variety T with K , we obtain that B − ,p tends to one when p goes to infinity. Insertingthis information in (32), we obtain [ g n +1 , . . . , g n ] = lim p →∞ B + ,p (cid:2) φ n +1 ( a n +1 ,p ) z µ n , . . . , φ n ( a n,p ) z µ n (cid:3) , so [ g n +1 , . . . , g n ] is the limit of a sequence of points in Z ′′ (2) . We conclude that Z ′ (2) ⊆ Z ′′ (2) ,and since these two cycles have the same dimension, that actually Z ′ (2) = Z ′′ (2) .To sum up: if b Z ′ , Z ′′ = 0 , then Z ′ (1) ⊆ Z ′′ (1) , and in case of equality Z ′ (1) = Z ′′ (1) , we additionallyhave Z ′ (2) = Z ′′ (2) . This proves the first statement in the theorem. The second one is proved inthe same manner as Proposition 5.9. (cid:3) Remark 5.14.
Using Theorem 5.13, one easily sharpens Corollary 5.12: with the notation ofthe latter, if a Z ′ , Z ′′ = 0 , then either Z ′ = Z ′′ or one of the displayed inequalities is strict. Theproof is left to the reader. Application to standard monomial theory.
Let λ ∈ Λ + and let ℓ ⊆ V ( λ ) ∗ be the line spanned by the highest weight vectors. The group G acts on the projective space P ( V ( λ ) ∗ ) ; let Q be the stabilizer of ℓ , a parabolic subgroup of G .The map g gℓ induces an embedding of the partial flag variety X = G/Q in P ( V ( λ ) ∗ ) . We61enote by L the pull-back of the line bundle O (1) by this embedding; then the homogeneouscoordinate ring of X is R λ = M m ≥ H (cid:0) X, L ⊗ m (cid:1) ; here H (cid:0) X, L ⊗ m (cid:1) is isomorphic to V ( mλ ) and the multiplication in R λ is given by the pro-jection onto the Cartan component V ( mλ ) ⊗ V ( nλ ) → V (( m + n ) λ ) . The algebra R λ is endowed with an MV basis, obtained by gathering the MV bases of thesummands V ( mλ ) .Each MV cycle Z ∈ Z ( λ ) defines a basis element h Z i ∈ V ( λ ) . Given an m -tuple Z =( Z , . . . , Z m ) of elements of Z ( λ ) , the product h Z i · · · h Z m i in the algebra R λ is the imageof hh Z ii = h Z i ⊗ · · · ⊗ h Z m i under the projection V ( λ ) ⊗ m → V ( mλ ) . This product is calledstandard if Z lies in the Cartan component of the crystal Z ( λ ) ⊗ m .Remark 3.5 implies that the MV basis element h Z i ∈ V ( λ ) ⊗ m goes, under the projection V ( λ ) ⊗ m → V ( mλ ) , either to an element in the MV basis of V ( mλ ) or to , depending onwhether Z lies or not in the Cartan component of Z ( λ ) ⊗ m .Using Corollary 5.12 and Remark 5.14, we can then endow, for each degree m , the Cartancomponent of Z ( λ ) ⊗ m with an order, so that the transition matrix expressing the standardmonomials in the MV basis of R λ is unitriangular. In particular, the standard monomials forma basis for the algebra R λ too, and straightening laws can be obtained from Theorem 5.8.The dual of the MV basis is compatible with the Demazure modules contained in V ( mλ ) ∗ ;this property is recorded as Remark 2.6 (ii) in [2] but the crux of the argument is due toKashiwara [27]. This implies that for any Schubert variety Y ⊆ X , the kernel of the restrictionmap M m ≥ H (cid:0) X, L ⊗ m (cid:1) → M m ≥ H (cid:0) Y, L ⊗ m (cid:1) is spanned by a subset of the MV basis of R λ . Therefore the homogeneous coordinate ring of Y is also endowed with an MV basis.These observations suggest that the MV basis could be a relevant tool for the study of thestandard monomial theory. Recall the notation set up in sects. 3.1–3.2. Given λ ∈ Λ + , we set λ ∗ = − w λ , where asusual w denotes the longest element in the Weyl group W . As is well known, there exists a62nique bijection σ : B ( λ ) → B ( λ ∗ ) which for each i ∈ I exchanges the actions of ˜ e i and ˜ f i .In our context, we will regard σ as a bijection Z ( λ ) → Z ( λ ∗ ) and may define it by means ofLemma 2.1 (e) in [33] and Theorem 4.9.Now let n ≥ and let λ = ( λ , . . . , λ n ) in (Λ + ) n . We set λ ∗ = ( λ ∗ n , . . . , λ ∗ ) and define abijection σ : Z ( λ ) × · · · × Z ( λ n ) → Z ( λ ∗ n ) × · · · × Z ( λ ∗ ) by σ ( Z , . . . , Z n ) = ( σ ( Z n ) , . . . , σ ( Z )) . (Using the same symbol σ to denote different bijectionsis certainly abusive, but adding extra indices to disambiguate would overload the notationwithout clear benefit.) The Cartesian products above are in fact tensor product of crystals,and here again σ exchanges the actions of ˜ e i and ˜ f i for each i ∈ I ([23], Theorem 2).Let µ ∈ Λ and choose ( Z ′ , Z ′′ ) ∈ ( Z ( λ ) µ ) ; we then obtain σ ( Z ′ ) and σ ( Z ′′ ) in Z ( λ ∗ ) − µ .Recall the notation introduced in Theorem 5.8 to denote the entries of the transition matrixbetween the two bases of V ( λ ) and adopt a similar notation as regards V ( λ ∗ ) . Conjecture 5.15.
The equality a Z ′ , Z ′′ = a σ ( Z ′ ) ,σ ( Z ′′ ) holds.According to [11], this conjecture is true in type A . Its general validity would have twointeresting consequences.Firstly, one could then strengthen Theorem 5.13; indeed b Z ′ , Z ′′ = 0 would imply not only Z ′ (1) ⊆ Z ′′ (1) , but also σ ( Z ′ (2) ) ⊆ σ ( Z ′′ (2) ) , restoring the symmetry between the two tensorfactors.Secondly, the MV basis of an irreducible representation V ( λ ) would then satisfy the analogueof [37], Proposition 21.1.2. In fact, one easily verifies that the MV basis enjoys this propertyif λ is minuscule or quasi-minuscule; our conjecture would allow to deduce the general case bytaking suitable tensor products, mimicking the strategy of proof from [40]. Let n ≥ and let λ ∈ (Λ + ) n . The MV basis of V ( λ ) is compatible with the isotypical filtration,hence provides a basis of the invariant subspace V ( λ ) G , called the Satake basis in [12]. In thissection we study two properties of this basis. 63 .1 Cyclic permutations Let us write λ = ( λ , . . . , λ n ) and consider the rotated sequence λ [1] = ( λ , . . . , λ n , λ ) . Thus, V ( λ ) = V ( λ ) ⊗ · · · ⊗ V ( λ n ) and V (cid:0) λ [1] (cid:1) = V ( λ ) ⊗ · · · ⊗ V ( λ n ) ⊗ V ( λ ) . The signed cyclic permutation x ⊗ · · · ⊗ x n ( − ρ ( λ ) x ⊗ · · · ⊗ x n ⊗ x , defines an isomorphism of G -modules R : V ( λ ) → V (cid:0) λ [1] (cid:1) . In particular, R induces a linearbijection between the invariant subspaces. Theorem 6.1
The signed cyclic permutation R maps the Satake basis of V ( λ ) G to the Satakebasis of V (cid:0) λ [1] (cid:1) G . Theorem 6.1 replicates a similar result for the dual canonical basis due to Lusztig ([37], 28.2.9),and our proof below mirrors Lusztig’s argument. It has been proved by Fontaine, Kamnitzerand Kuperberg in the case where all the weights λ j are minuscule ([12], Theorem 4.5). Thebijection induced by R between the two Satake bases has a nice interpretation, both in termsof crystals (see [13]) and in terms of cluster combinatorics (see [22], sect. 2.1.6).The rest of this section is devoted to the proof of Theorem 6.1.As in sect. 3, we denote by { α i | i ∈ I } the set of simple roots and choose simple root vectors e i and f i in the Lie algebra of G of weights ± α i such that [ e i , f i ] = − α ∨ i . The Weyl group W is generated by the simple reflections s i and contains a longest element w .Given λ ∈ Λ + and w ∈ W , we can pick a reduced word ( i , . . . , i ℓ ) of w and form the productof divided powers θ ( w, λ ) = f ( n ) i · · · f ( n ℓ ) i ℓ , where n j = h α ∨ i j , s i j +1 · · · s i ℓ λ i . This element does not depend on the choice of ( i , . . . , i ℓ ) ([37], Proposition 28.1.2), whichlegitimizes the notation. We note that θ ( w , λ ) acts on V ( λ ) by mapping highest weightvectors to lowest weight vectors.We set λ = λ , the first element in the sequence λ . With the notation of sect. 5.4, the highestand lowest weight elements in the MV basis of V ( λ ) are v λ = (cid:10) { L λ } (cid:11) and v w λ = D Gr λ E . v w λ = θ ( w , λ ) · v λ (see [2], Theorem 5.2 and Remark 2.10). We define ξ to be thelinear form on V ( λ ) such that h ξ, v λ i = 1 and that vanishes on all weight subspaces of weightdifferent from λ . Similarly, we define η to be the linear form on V ( λ ) such that h η, v w λ i = 1 and that vanishes on all weight subspaces of weight different from w λ .Let M be a representation of G . The assignment v ⊗ m ( − ρ ( λ ) m ⊗ v defines an isomor-phism P : V ( λ ) ⊗ M → M ⊗ V ( λ ) .We set λ ∗ = − w λ . Let M ◦ be the isotypical component of M corresponding to the highestweight λ ∗ , namely, the sum of all subrepresentations isomorphic to V ( λ ∗ ) . Given a weight µ ∈ Λ , we denote by M µ the corresponding weight subspace of M and set M ◦ µ = M ◦ ∩ M µ .Then M ◦ λ ∗ is the set of all vectors in M λ ∗ that are annihilated by all the root vectors e i and M ◦ w λ ∗ is the set of all vectors in M w λ ∗ that are annihilated by all the root vectors f i . Lemma 6.2
The following diagram commutes and consists of linear bijections. ( V ( λ ) ⊗ M ) G P / / η ⊗ id M (cid:15) (cid:15) ( M ⊗ V ( λ )) G id M ⊗ ξ (cid:15) (cid:15) M ◦ λ ∗ θ ( w ,λ ∗ ) / / M ◦ w λ ∗ (33) Proof.
By additivity, we can reduce to the case where M is a simple representation. If M isnot isomorphic to the dual of V ( λ ) , then all four spaces are zero and the statement is banal.We therefore assume that M ∼ = V ( λ ∗ ) ; in this case, all four spaces are one dimensional.Let m λ ∗ be a highest weight vector in M and set m w λ ∗ = θ ( w , λ ∗ ) · m λ ∗ . There exists aunique G -invariant bilinear form Φ : V ( λ ) × M → C such that Φ( v w λ , m λ ∗ ) = 1 . This form Φ is non-degenerate and a standard computation gives Φ( v λ , m w λ ∗ ) = ( − ρ ( λ ) .The assignment v ⊗ m Φ( v, ? ) m defines a G -equivariant isomorphism V ( λ ) ⊗ M → End( M ) .The preimage x of id M by this bijection spans the vector space ( V ( λ ) ⊗ M ) G . By construction, ( η ⊗ id M )( x ) = m λ ∗ and ( ξ ⊗ id M )( x ) = ( − ρ ( λ ) m w λ ∗ . Thus, both paths around the diagrammap x to m w λ ∗ . (cid:3) We take M = V ( λ ) ⊗ · · · ⊗ V ( λ n ) . We define M • to be the step in the isotypical filtrationof M where the component M ◦ is appended to smaller ones. There is a natural quotient map p : M • → M ◦ .We set M = Z ( λ ) × · · · × Z ( λ n ) . Using the notation introduced in sect. 5.4, the MV basis of M consists of elements h Z i for Z ∈ M . Let M • be the set of all Z ∈ M such that h Z i ∈ M • ;65ince MV bases are L -perfect, {h Z i | Z ∈ M • } is a basis of M • . Let M ◦ be the set of all Z ∈ M • such that h Z i / ∈ ker p ; then { p ( h Z i ) | Z ∈ M ◦ } is a basis of M ◦ . In consequence, eachweight subspace of M ◦ is endowed with a basis.As a crystal, M decomposes as the disjoint union (direct sum) of its connected components,and M ◦ is the union of the connected components of M that are isomorphic to Z ( λ ∗ ) . Foreach connected component C ⊆ M ◦ , the subspace of M ◦ spanned by B C = { p ( h Z i ) | Z ∈ C } is a subrepresentation isomorphic to V ( λ ∗ ) , and by Remark 3.5, B C identifies with the MVbasis of V ( λ ∗ ) . The action of θ ( w , λ ∗ ) therefore maps the highest weight element in B C tothe lowest element in B C . We conclude that the bottom horizontal arrow in (33) maps thebasis of M ◦ λ ∗ to the basis of M ◦ w λ ∗ .Each element in the MV basis of V (cid:0) λ [1] (cid:1) = M ⊗ V ( λ ) is of the form h Z i , with Z in Z (cid:0) λ [1] (cid:1) = M × Z ( λ ) . Let V ( λ ) = λ be the sum of all the weight subspaces of V ( λ ) other than the higherweight subspace. Theorem 5.13 implies that for each Z (1) ∈ M , (cid:10) Z (1) (cid:11) ⊗ (cid:10) { L λ } (cid:11) ≡ (cid:10)(cid:0) Z (1) , { L λ } (cid:1)(cid:11) (cid:0) mod M ⊗ V ( λ ) = λ (cid:1) . Thus, for Z (1) ∈ M and Z = (cid:0) Z (1) , { L λ } (cid:1) , we have (id M ⊗ ξ )( h Z i ) = (cid:10) Z (1) (cid:11) .As evidenced by the crystal structure on M ⊗ Z ( λ ) , the Satake basis of the space ( M ⊗ V ( λ )) G consists of the vectors h Z i for the pairs Z = (cid:0) Z (1) , { L λ } (cid:1) such that Z (1) ∈ M ◦ w λ ∗ . Noting that (cid:10) Z (1) (cid:11) ∈ M ◦ w λ ∗ for those Z (1) , we conclude that the right vertical arrow in (33) maps basiselements to basis elements.Similarly, the left vertical arrow in (33) maps the Satake basis of ( V ( λ ) ⊗ M ) G to the basisof M ◦ λ ∗ . Lemma 6.2 then concludes the proof of Theorem 6.1. Let ( n ′ , n ′′ ) be a composition of n in two parts. Correspondingly, we write λ ∈ (Λ + ) n as aconcatenation ( λ ′ , λ ′′ ) and view each element in Z ( λ ) as a pair ( Z ′ , Z ′′ ) ∈ Z ( λ ′ ) × Z ( λ ′′ ) .The following proposition implies that the Satake basis of the invariant subspace of V ( λ ) satisfies the second item in Khovanov and Kuperberg’s list of properties for the dual canonicalbasis (see the introduction of [30]). Proposition 6.3
Let ( Z ′ , Z ′′ ) ∈ Z ( λ ) . If (cid:10) Z ′ (cid:11) ∈ V ( λ ′ ) G , then (cid:10) Z ′ (cid:11) ⊗ (cid:10) Z ′′ (cid:11) = (cid:10) ( Z ′ , Z ′′ ) (cid:11) . Proof.
Let Z ′ ∈ Z ( λ ′ ) . Recall the map m n ′ : Gr n ′ → Gr defined in sect. 2.2 and the notation µ I from sect. 3.4 and set µ = µ I ( Z ′ ) . Then m n ′ ( Z ′ ) ⊆ Gr µ and (cid:10) Z ′ (cid:11) appear in the isotypicalfiltration of V ( λ ′ ) at the step where the component of type V ( µ ) is appended.66f (cid:10) Z ′ (cid:11) ∈ V ( λ ′ ) G , then µ = 0 , accordingly Gr µ = { L } , and as a result Z ′ ⊆ ( m n ′ ) − ( { L } ) ⊆ ( m n ′ ) − ( T ) . This implies that no MV cycle in Z ( λ ′ ) can be strictly contained in Z ′ . (Such a cycle wouldbe contained in ( m n ′ ) − ( T ) , so would be an irreducible component of Gr λ ′ n ′ ∩ ( m n ′ ) − ( T ) ,and would end up having dimension ρ ( | λ ′ | ) , the same as Z ′ .) The desired result now directlyfollows from Theorem 5.13. (cid:3) C [ N ] We adopt the notation set up in the preamble of sect. 3. Let N be the unipotent radical ofthe Borel subgroup B and let C [ N ] be the algebra of regular functions on N . At the expenseof an isogeny, which does not alter N , we can assume that G is simply-connected.For each dominant weight λ ∈ Λ + , we can choose a highest weight vector v λ in the represen-tation V ( λ ) and define the linear form v ∗ λ : V ( λ ) → C such that h v ∗ λ , v λ i = 1 et h v ∗ λ , v i = 0 for all weight vectors v of weight other than λ . This yields an embedding Ψ λ : v
7→ h v ∗ λ , ? v i of V ( λ ) into C [ N ] , where h v ∗ λ , ? v i stands for the function n
7→ h v ∗ λ , nv i . The MV bases ofthe representations V ( λ ) can be transported to C [ N ] through these maps Ψ λ , and they gluetogether to form a basis of C [ N ] , which we call the MV basis of C [ N ] (see [2]).The algebra C [ N ] comes with several remarkable bases: the MV basis, subject of our currentinvestigation, but also the dual canonical basis of Lusztig/upper global basis of Kashiwara,and (in simply laced type) the dual semicanonical basis. The theory of cluster algebras wasdeveloped in order to compute effectively these bases (or at least, the dual canonical basis).Concretely, the cluster structure of C [ N ] allows to define specific elements, called cluster mono-mials, which are linearly independent and easily amenable to calculations. It is known thatboth the dual canonical and the dual semicanonical bases contain all the cluster monomials[20, 25], but also that these bases differ (except when cluster monomials span C [ N ] ).The methods developed in sect. 5 allow to effectively compute products of elements of the MVbasis of C [ N ] . This allows us to prove that this basis contains quite a few cluster monomials(Proposition 7.2) and that it generally differs from both the dual canonical and the dualsemicanonical bases (Proposition 7.3). 67 .1 Cluster monomials As explained in [21], each reduced word ( i , . . . , i ℓ ) of the longest element w in the Weyl group W yields a standard seed of the cluster structure of C [ N ] . The main result of this section,Proposition 7.2, claims that sometimes the cluster monomials built from the variables of astandard seed belong to the MV basis of C [ N ] .Set t ∨ R = Hom Z (Λ , R ) and let C = { x ∈ t ∨ R | ∀ i ∈ I, h x, α i i > } be the Weyl chamber in t ∨ R .We consider the following condition about a reduced word ( i , . . . , i ℓ ) :(A) There exist x ∈ s i ( C ) , x ∈ ( s i s i )( C ) , . . . , x ℓ ∈ ( s i · · · s i ℓ )( C ) such that x k − x k +1 ∈ C for each k ∈ { , . . . , ℓ − } .For instance, choose ( x, y ) ∈ C in such a way that the straight line joining x to − y avoidsall the two-codimensional faces of the Weyl fan in t ∨ R . List in order the chambers successivelycrossed by this line: C , s i C , ( s i s i )( C ) , . . . The word ( i , i , . . . ) produced in this manner isthen reduced and obviously satisfies condition (A).Let Q ⊆ Λ be the root lattice. We denote by Q + the positive cone in Q with respect to thedominance order; in other words, Q + is the set of all linear combinations of the simple roots α i with non-negative integral coefficients. We set Q − = − Q + . Lemma 7.1
Let ( i , . . . , i ℓ ) be a reduced word, set w k = s i · · · s i k for k ∈ { , . . . , ℓ } , and let ( ν , . . . , ν ℓ ) ∈ w ( Q − ) × · · · × w ℓ ( Q − ) . Assume that ν + · · · + ν k ∈ Q + for all k ∈ { , . . . , ℓ − } ,that ν + · · · + ν ℓ = 0 , and that ( i , . . . , i ℓ ) satisfies condition (A). Then ν = · · · = ν ℓ = 0 .Proof. We set µ = 0 and µ k = ν + · · · + ν k for k ∈ { , . . . , ℓ } . We pick x , . . . , x ℓ as statedin condition (A). Then ℓ X k =1 h x k , ν k i = ℓ X k =1 h x k , µ k − µ k − i = ℓ − X k =1 h x k − x k +1 , µ k i . From x k ∈ w k ( C ) and ν k ∈ w k ( Q − ) , we deduce that h x k , ν k i ≤ for each k ∈ { , . . . , ℓ } . Onthe other hand, from x k − x k +1 ∈ C and µ k ∈ Q + , we deduce that h x k − x k +1 , µ k i ≥ for each k ∈ { , . . . , ℓ − } . We conclude that each h x k , ν k i is indeed zero, which implies ν k = 0 . (cid:3) In sect. 6.1, we defined, for each ( λ, w ) ∈ Λ + × W , a product θ ( λ, w ) of divided powers of theroot vectors f i . We can then set v wλ = θ ( λ, w ) · v λ ; this is a vector of weight wλ in V ( λ ) .We define ∆ λ,wλ = Ψ λ ( v wλ ) , usually called a flag minor if λ is minuscule. We denote by { ̟ i | i ∈ I } the set of fundamental weights. 68 roposition 7.2 Let ( i , . . . , i ℓ ) be a reduced word and define x k = ∆ ̟ ik ,s i ··· s ik ̟ ik for each k ∈ { , . . . , ℓ } . If ( i , . . . , i ℓ ) satisfies condition (A), then any monomial in x , . . . , x ℓ belongsto the MV basis of C [ N ] .Proof. We choose λ = ( λ , . . . , λ ℓ ) in (Λ + ) ℓ . For k ∈ { , . . . , ℓ } , we set w k = s i · · · s i k .The extremal weight vector v w k λ k ∈ V ( λ k ) belongs to the MV basis ([2], Remark 2.10 andTheorem 5.2), so v w k λ k = h Z k i where Z k is the cycle Gr λ k ∩ T w k λ k . We set µ = w λ + · · · + w ℓ λ ℓ and Z = ( Z , . . . , Z ℓ ) . We adopt the convention of sect. 5.4 and regard Z as an elementof Z ( λ ) µ ; then hh Z ii = v w λ ⊗ · · · ⊗ v w ℓ λ ℓ .Let us expand this element on the MV basis of V ( λ ) . As in Theorem 5.8, we write hh Z ii = X Z ′ ∈ Z ( λ ) µ a Z ′ , Z (cid:10) Z ′ (cid:11) . (34)Suppose Z ′ ∈ Z ( λ ) µ satisfies a Z ′ , Z = 0 . Let ( ν , . . . , ν ℓ ) ∈ Λ ℓ be the tuple of weights suchthat Z ′ ∈ Z ( λ ) ν × · · · × Z ( λ ℓ ) ν ℓ . For each k ∈ { , . . . , ℓ } , we have Z ( λ k ) ν k = ∅ , so w − k ν k is a weight of V ( λ k ) , whence ( ν k − w k λ k ) ∈ w k ( Q − ) . From ν + · · · + ν ℓ = µ , we deducethat ( ν − w λ ) + · · · + ( ν ℓ − w ℓ λ ℓ ) = 0 . And by Corollary 5.12, we get ( ν − w λ ) + · · · + ( ν k − w k λ k ) ∈ Q + for each k ∈ { , . . . , ℓ − } . Then, assuming that ( i , . . . , i ℓ ) satisfies condition (A) and applyingLemma 7.1, we obtain ν k = w k λ k for each k ∈ { , . . . , ℓ − } . In other words, none of theinequalities given in Corollary 5.12 is strict. By Remark 5.14, this forces Z ′ = Z . Thus, theexpansion (34) contains a single term, namely h Z i .Set λ = λ + · · · + λ ℓ and let p : V ( λ ) → V ( λ ) be the unique morphism that maps v λ ⊗· · ·⊗ v λ n to v λ . Noting that p is the quotient map to the top factor in the isotypical filtration of V ( λ ) and applying Remark 3.5, we obtain that p ( h Z i ) belongs to the MV basis of V ( λ ) . From theequality v w λ ⊗ · · · ⊗ v w ℓ λ ℓ = h Z i , we deduce that ∆ λ ,w λ · · · ∆ λ ℓ ,w ℓ λ ℓ = (cid:10) v ∗ λ , p (cid:0) ? h Z i (cid:1)(cid:11) = Ψ λ (cid:0) p (cid:0) h Z i (cid:1)(cid:1) belongs to the MV basis of C [ N ] . The claim in the proposition is the particular case whereeach λ k is a multiple of ̟ i k . (cid:3) D In [29], Kashiwara and Saito found an example in type A where the singular support of asimple perverse sheaf related to the canonical basis is not irreducible. Looking again at this69ituation, Geiß, Leclerc and Schröer [19] computed the dual canonical and dual semicanonicalelements and found that they were different. They also observed that a similar situation occursin type D . In [2], Kamnitzer, Knutson and the first author observed that in both situations A and D , the MV basis is a third basis, different from the two other ones.Let us have a closer look at the D case. As usual, we label the vertices of the Dynkin diagramfrom to , with for the central node. Our three bases are indexed by the crystal B ( ∞ ) :given b ∈ B ( ∞ ) , we denote the corresponding dual semicanonical basis element by C ( b ) , thedual canonical basis element by C ′ ( b ) , and the MV basis element by C ′′ ( b ) . Calling b thehighest weight element in B ( ∞ ) , we set b = (cid:0) ˜ f (cid:0) ˜ f ˜ f ˜ f (cid:1) ˜ f (cid:1) b and b = (cid:0) ˜ f (cid:1) (cid:0) ˜ f ˜ f ˜ f (cid:1) (cid:0) ˜ f (cid:1) b . Proposition 7.3
The basis elements are related by the equations C ( b ) = C ′′ ( b ) + 2 C ( b ) and C ′ ( b ) = C ′′ ( b ) + C ( b ) . The proof is given in [2], sect. 2.7, except for one justification left to the present paper. Wehere fill the gap.The fundamental weight ̟ is the highest root α + 2 α + α + α . The crystal of the repre-sentation V ( ̟ ) (the adjoint representation) is pictured below. Highest weights are towardsthe left, vertices are represented as keys pq with p , q in { , , , , , , , } , and operators ˜ f , ˜ f , ˜ f and ˜ f are indicated by dashed, solid, dotted and dash-dotted arrows, respectively.
12 13 141423 132424 12233434 22334444 21324343 314242 414132 31 21
If we endow the weight lattice Λ with its usual basis ( ε , ε , ε , ε ) , then the weight of theelement pq is simply ε p + ε q , with the convention that ε ı = − ε i for i ∈ { , , , } . The crystalcontains four elements of weight zero, namely , , and .70e set λ = ( ̟ , ̟ ) and look at the tensor square V ( λ ) = V ( ̟ ) ⊗ . As in sect. 5.4, its MVbasis consists of symbols h Z i , where Z = ( Z , Z ) is a pair in Z ( ̟ ) × Z ( ̟ ) . In addition, V ( λ ) is endowed with the tensor product basis. In the matter of notation, we simply usethe keys pq to denote MV cycles, taking advantage that Z ( ̟ ) is isomorphic to the crystalpictured above.We claim that D E ⊗ D E = 2 D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E + D(cid:16) , (cid:17)E . (35)Let p : V ( ̟ ) ⊗ → V (2 ̟ ) be the unique morphism that maps v ̟ ⊗ v ̟ to v ̟ . Applying Ψ ̟ ◦ p to the equality (35), we obtain the equation C ′′ ( b ) C ′′ ( b ) = 2 C ′′ ( b ) + X i =2 C ′′ ( b i ) + C ′′ ( b ) asserted without proof in [2]. Thus, establishing (35) will complete the proof of Proposition 7.3.Actually, an inspection of the proof in loc. cit. reveals that it is enough to justify that thecoefficient in front of D(cid:16) , (cid:17)E is strictly larger than one.We will use Theorem 5.8 to prove this fact. Here the group G ∨ is SO . For ( i, j ) ∈ { , . . . , } , wedenote by E i,j the matrix of size × with zeros everywhere except for a one at position ( i, j ) .For each coroot α ∨ ∈ Φ ∨ , we define a subgroup x α ∨ : C → G ∨ by the following formulas, where I is the identity matrix, a ∈ C , and i , j are elements in { , , , } such that i < j . x ( ε i − ε j ) ∨ ( a ) = I + a ( E i,j − E − j, − i ) x ( ε i + ε j ) ∨ ( a ) = I + a ( E i, − j − E j, − i ) x ( ε j − ε i ) ∨ ( a ) = I + a ( E − i, − j − E j,i ) x ( − ε i − ε j ) ∨ ( a ) = I + a ( E − i,j − E − j,i ) For each root α , we define a map χ α : C → G ∨ (cid:0) C (cid:2) z, z − (cid:3)(cid:1) by the formula χ α ( a )( z ) = Y k =1 x β ∨ k ( a k ) ! x α ∨ ( a + za ) z α where a stands for the tuple ( a , . . . , a ) ∈ C and where β ∨ , . . . , β ∨ are the coroots β ∨ suchthat h β ∨ , α i = 1 . We specify the enumeration in our cases of interest as follows.71 β ∨ β ∨ β ∨ β ∨ β ∨ β ∨ β ∨ β ∨ ε + ε ( ε − ε ) ∨ ( ε − ε ) ∨ ( ε + ε ) ∨ ( ε + ε ) ∨ ( ε − ε ) ∨ ( ε − ε ) ∨ ( ε + ε ) ∨ ( ε + ε ) ∨ − ε − ε ( − ε − ε ) ∨ ( − ε − ε ) ∨ ( ε − ε ) ∨ ( ε − ε ) ∨ ( − ε − ε ) ∨ ( − ε − ε ) ∨ ( ε − ε ) ∨ ( ε − ε ) ∨ ε − ε ( ε − ε ) ∨ ( ε − ε ) ∨ ( ε + ε ) ∨ ( ε + ε ) ∨ ( − ε − ε ) ∨ ( − ε − ε ) ∨ ( ε − ε ) ∨ ( ε − ε ) ∨ ε − ε ( ε − ε ) ∨ ( ε − ε ) ∨ ( ε + ε ) ∨ ( ε + ε ) ∨ ( − ε − ε ) ∨ ( − ε − ε ) ∨ ( ε − ε ) ∨ ( ε − ε ) ∨ We now define two charts on G r λ , both with C as domain: φ : ( x , x , a , b ) (cid:0) x , x ; (cid:2) χ ε + ε ( a )( z − x ) , χ − ε − ε ( b )( z − x ) (cid:3)(cid:1) , φ : ( x , x , a ′ , b ′ ) (cid:0) x , x ; (cid:2) χ ε − ε ( a ′ )( z − x ) , χ ε − ε ( b ′ )( z − x ) (cid:3)(cid:1) . One can then compute the transition map between these two charts. (The calculations wereactually carried out with the help of the computer algebra system
Singular [10].) One findsthe variables a ′ , . . . , b ′ as rational functions in x − x , a , . . . , b . We denote by f thel.c.m. of the denominators.Recall the notation used in sect. 5.4. In the chart φ , the cycle Y (cid:16) , (cid:17) is defined by theequations a = · · · = a = x − x = 0 , so the ideal in R = C [ x , x , a , . . . , a , b , . . . , b ] of V = φ − (cid:16) Y (cid:16) , (cid:17)(cid:17) is p = ( a , . . . , a , x − x ) . In the chart φ , the cycle X (cid:16) , (cid:17) is defined by the equations a ′ = a ′ = a ′ = a ′ = a ′ = a ′ = b ′ = b ′ = b ′ = b ′ = 0 . Since the zero locus of f contains the locus where the transitionmap between the charts is not defined, the ideal q of the subvariety X = φ − (cid:16) X (cid:16) , (cid:17)(cid:17) is the preimage in R of the ideal q f = ( a ′ , a ′ , a ′ , a ′ , a ′ , a ′ , b ′ , b ′ , b ′ , b ′ ) of the localized ring R f . Singular gives the following expression: q = ( a a + a a , a a − a a , a a + a a , a a − a a , a a + a a , a a − a a ,a a − a a , a a − a a , a a + a a , a , a , a b + a b + a b + a b ,a b + a b + a b + a b , a b + a b + a b + a b − ( x − x ) , a b − a b + a b − a b ,a b + a b + a b + a b , a b − a b + a b − a b , a b − a b + a b − a b ) .
72e observe that q ⊆ p , hence V ⊆ X .Let a and x be two indeterminates. Let B be the field C ( x, b , . . . , b ) . Extract the last sevenequations from q and remove the term x − x in the third one: we then deal with seven linearequations with coefficients in B in the eight variables a , . . . , a . This system has a non-zerosolution ( c , . . . , c ) ∈ B . We can then define an algebra morphism u : R/ q → B [ a ] / ( a ) by u ( x ) = u ( x ) = x, u ( b i ) = b i for i ∈ { , . . . , } ,u ( a ) = u ( a ) = 0 , u ( a i ) = c i a for i ∈ { , . . . , } . The ring B [ a ] / ( a ) is local with maximal ideal ( a ) and the preimage of this ideal by u is theideal p / q of R/ q .Let A be the localization of R/ q at p / q . Then u extends to an algebra morphism u : A → B [ a ] / ( a ) . By construction, the kernel of u contains x − x but not all a , . . . , a . Therefore x − x does not generate the maximal ideal of A . Since A is the local ring O V,X of X along V ,this means that the order of vanishing of x − x along V is larger than one. In other words,the multiplicity of Y (cid:16) , (cid:17) in the intersection product X (cid:16) , (cid:17) · G r λ (cid:12)(cid:12) ∆ is larger than one.Applying Theorem 5.8, we conclude that in (35) the coefficient in front of D(cid:16) , (cid:17)E is strictlylarger than one. References [1] P. Baumann, S. Gaussent,
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Peter Littelmann, Universität zu Köln, Mathematisches Institut, Weyertal 86–90, 50931 Köln,Germany. [email protected]@math.uni-koeln.de