aa r X i v : . [ m a t h . R T ] A p r THE HARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS
YURI BAZLOV
Abstract.
We study an analogue of the Harish-Chandra homomorphism where the universalenveloping algebra U ( g ) is replaced by the Clifford algebra, C ℓ ( g ), of a semisimple Lie algebra g .Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphismbetween the subalgebra of g -invariants in C ℓ ( g ) and the Clifford algebra C ℓ ( h ) of the Cartansubalgebra of g . Second, the Cartan subalgebra h is identified, via this isomorphism, with thegraded space of the so-called primitive skew-symmetric invariants of g . The grading leads to adistinguished orthogonal basis of h , which turns out to be induced from the Langlands dual Liealgebra g ∨ via the action of its principal three-dimensional subalgebra. This settles a conjectureof Kostant. Introduction
Introduced by Harish-Chandra more than half a century ago, the Harish-Chandra homomorphismis of utmost significance in representation theory of semisimple Lie groups and algebras; charactertheory is one of the areas where it plays a key role. Recall that, given a complex semisimple Liealgebra g and its Cartan subalgebra h , the Harish-Chandra homomorphism is a one-to-one algebramap between the centre Z ( g ) of the universal enveloping algebra U ( g ) of g and the algebra S ( h ) W of (translated) Weyl group W invariant polynomial functions on the space h ∗ . Because W is afinite reflection group, by the well-known Chevalley-Shephard-Todd theorem S ( h ) W is a polynomialalgebra. The Harish-Chandra map thus establishes the polynomiality of the algebra Z ( g ) andidentifies the characters of Z ( g ) with W -orbits in h ∗ .The subject of the present paper is a natural analogue of the Harish-Chandra homomorphismwhere U ( g ) is replaced with the Clifford algebra, C ℓ ( g ), of a semisimple Lie algebra g . This analogueis based on a remarkably easy algebraic construction, which underlines the “classical” Harish-Chan-dra homomorphism but applies to a class of algebras much wider than that of universal envelopingalgebras U ( g ). Let A be an associative algebra that factorises as A − ⊗ A ⊗ A + , where A ± and A are subalgebras in A and the tensor product is realised by the multiplication in A ; we stress that A − , A and A + need not commute in A . Some further conditions on this factorisation guaranteethat there exists a projection (in general, not an algebra map) pr = ε − ⊗ id ⊗ ε + of A onto itssubalgebra A . (General details are discussed in [5].)In the case A = U ( g ), the standard triangular decomposition g = n − ⊕ h ⊕ n + of a semisimpleLie algebra g gives rise to a factorisation as above with A ± = U ( n ± ) and A = U ( h ) = S ( h ) thepolynomial algebra. The Harish-Chandra map is obtained by restricting pr to Z ( g ) = U ( g ) g , the ad g -invariants in U ( g ). (This method also works for the quantised universal enveloping algebra U ~ ( g ) [25].) The same approach is used in [26] to give a completely algebraic construction of thehomomorphism D ( g ) g → D ( h ) W , also due to Harish-Chandra (here D ( · ) stands for polynomialdifferential operators).The Clifford algebra C ℓ ( g ) factorises as a product of three of its subalgebras, C ℓ ( n − ), C ℓ ( h ) and C ℓ ( n + ), therefore admitting a Harish-Chandra map Φ : C ℓ ( g ) → C ℓ ( h ). Studying the map Φ involves structural theory of the Clifford algebra C ℓ ( g ), based mostly on results of Kostant. In the presentpaper, we focus on the restriction of the Harish-Chandra map Φ to the subalgebra J = C ℓ ( g ) g of g -invariants, which in the Clifford algebra plays a role similar to that of the centre Z ( g ) = U ( g ) g in U ( g ). Indeed, Kostant’s “separation of variables” result [19] states that C ℓ ( g ) = E ⊗ J is a freemodule over the algebra J , and J is further described as a Clifford algebra C ℓ ( P ) of a remarkablespace P of the so-called primitive invariants.Our first main result, Theorem 4.1, shows that Φ restricts to an isomorphism J ∼ −→ C ℓ ( h ).Comparing this to the universal enveloping algebra, it is worth noting that the action of the Weylgroup W on C ℓ ( h ) is conspicuously absent from the picture. We then refine the isomorphism resultby showing that the Harish-Chandra map Φ identifies the space P of primitive invariants with theCartan subalgebra h of g .The above immediately raises the next question: the space P is naturally graded, via its inclusionin the exterior algebra V g and its identification with primitive homology classes in the homologyof g ; what is the grading induced on the Cartan subalgebra by the Harish-Chandra map? It turnsout that the answer involves the so-called Langlands dual Lie algebra g ∨ of g . This is a complexsemisimple Lie algebra with a root system dual to that of g . There is a canonical copy of sl inside g ∨ ,and its adjoint action on g ∨ splits g ∨ into a direct sum of sl -submodules (“strings”). Intersectionsof these “strings” with the Cartan subalgebra h , which is viewed as shared between g and g ∨ , definethe graded components of h . This is established in Theorem 5.5 which is the second and final mainresult of the paper. In particular, Theorem 5.5 confirms a conjecture made by Kostant [20].When g is a simple Lie algebra, the graded components in h typically are one-dimensional, givingrise to a distinguished basis of h . (The only deviation from this occurs in a simple Lie algebra withDynkin diagram of type D on an even number of nodes.) This basis is orthogonal with respect tothe Killing form and contains ρ , half the sum of positive roots of g ; we refer to this basis of h as the(Langlands dual) principal basis of the Cartan subalgebra. Acknowledgments.
Section 4 of the present paper is based on a revision of work done in my PhDthesis [3], which was completed with the guidance of Anthony Joseph. I am grateful to BertramKostant for conversations on the subject of this paper; having seen partial results of [3], he suggesteda conjectural link to principal TDS bases, proved in the present paper. I thank Anton Alekseevfor providing helpful ideas for the proof of Theorem 5.5. These useful ideas and calculations arealso made clear in a recent preprint [23] by R. P. Rohr. I am grateful to Victor Ginzburg andEckhard Meinrenken for their comments on earlier versions of this work and to David Kazhdan fora stimulating discussion.
Contents
Introduction 11. The classical Chevalley projection and Harish-Chandra map 32. Clifford algebras 53. The algebras of invariants and Kostant’s ρ -decomposition of C ℓ ( g ) 94. Φ is an isomorphism between C ℓ ( g ) g and C ℓ ( h ) 115. The principal basis of the Cartan subalgebra 14References 20 ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 3 The classical Chevalley projection and Harish-Chandra map
We start by recalling the Chevalley projection map and the Harish-Chandra map associated witha semisimple Lie algebra g . This will serve as an introduction to the subsequent treatment of theanalogues of these for skew-symmetric tensors.1.1. Invariant symmetric tensors and the Chevalley projection.
Let g be a semisimple Liealgebra of rank r over the complex field C . Let a Cartan subalgebra h and a Borel subalgebra b of g , such that h ⊂ b , be fixed. This partitions the root system of g into positive and negative parts,and gives rise to the triangular decomposition g = n − ⊕ h ⊕ n + of g , where n − and n + are subspaces of g spanned by root vectors corresponding to negative andpositive roots, respectively. By S ( g ) we denote the algebra of symmetric tensors over g , which isthe same as the algebra of polynomial functions on the space g ∗ . It is graded by degree: S ( g ) = L ∞ n =0 S n ( g ), and we identify g with S ( g ). The action of g on S ( g ) is extended from the adjointaction of g on g = S ( g ) by derivations of degree 0; we denote the action of x ∈ g by ad x ∈ End S ( g ).We refer to the set J S = S ( g ) g = { f ∈ S ( g ) | ( ad x ) f = 0 ∀ x ∈ g } as the space of symmetric g -invariants. Note that J S is a graded subalgebra of S ( g ). It is obviousthat the algebra S ( g ) has triangular factorisation, S ( g ) ∼ = S ( n − ) ⊗ S ( h ) ⊗ S ( n + ) , into three subalgebras generated by n − , h , n + , respectively. Denote by ε the character of an algebraof polynomials given by the evaluation of a polynomial at zero. The homomorphismΨ = ε ⊗ id ⊗ ε : S ( g ) → S ( h )of commutative algebras is what is typically called the Chevalley projection map. By a classical resultof Chevalley, see [10, Theorem 7.3.7], the restriction of Ψ to J S is a graded algebra isomorphismΨ : J S ∼ −→ S ( h ) W . Here S ( h ) W denotes symmetric tensors over h invariant under the action of the Weyl group W of g (this action is extended from h to S ( h )).1.2. The classical Harish-Chandra map.
Let U ( g ) be the universal enveloping algebra of g , and J U = U ( g ) g be its centre, which is the ring of invariants of the adjoint representation of g in U ( g ). ThePoincar´e-Birkhoff-Witt symmetrisation map β : S ( g ) → U ( g )is a g -module isomorphism between S ( g ) and U ( g ), where β ( x x . . . x n ) for x i ∈ g is defined as n ! times the sum of x π (1) x π (2) . . . x π ( n ) ∈ U ( g ) over all permutations π in n letters. In particular, β identifies J S and J U as linear spaces. Note that β is not an algebra isomorphism between J S and J U . (An algebra isomorphism J S ∼ −→ J U , known as the Duflo map, was explicitly constructed in[11].) The Poincar´e-Birkhoff-Witt theorem for U ( g ) implies the triangular factorisation U ( g ) = U ( n − ) ⊗ U ( h ) ⊗ U ( n + ) YURI BAZLOV of the algebra U ( g ) into the subalgebras generated by n − , h , n + , respectively. Let ε − : U ( n − ) → C be the algebra homomorphism defined by ε − ( x ) = 0 for x ∈ n − , and let ε + : U ( n + ) → C be definedsimilarly. We will refer to the mapΨ = ε − ⊗ id ⊗ ε + : U ( g ) → U ( h )as the Harish-Chandra map, slightly abusing the terminology. As h is an Abelian Lie algebra, U ( h )is identified with the polynomial algebra S ( h ). The map Ψ is not an algebra homomorphism, butits restriction to the subalgebra U ( g ) h = { f ∈ U ( g ) | ( ad h ) f = 0 ∀ h ∈ h } is. Further restricting Ψto J U which is a subalgebra of U ( g ) h , one obtains the isomorphismΨ : J U ∼ −→ S ( h ) W · between the centre of U ( g ) and the ring S ( h ) W · of symmetric tensors over h invariant under theshifted action of W . (The shifted action of w ∈ W is defined on λ ∈ h ∗ by w.λ = w ( λ + ρ ) − ρ where ρ ∈ h ∗ is the half sum of positive roots of g , and hence on S ( h ) which is viewed as the algebra ofpolynomial functions on h ∗ .) This isomorphism is due to Harish-Chandra; see [15], [10, Theorem7.4.5].1.3. An analogue of the Chevalley projection for V g . We are going to consider V g , theexterior algebra of g , as a “skew-symmetric analogue” of S ( g ). Furthermore, the universal envelopingalgebra U ( g ) — a deformation of S ( g ) — will be replaced by C ℓ ( g ), the Clifford algebra of g , whichis a deformation of V g . We will now discuss the analogue of the Chevalley projection Ψ (and later,of the Harish-Chandra map Ψ) in the skew-symmetric situation.The action of g on the finite-dimensional algebra V g = L dim g n =0 V n g is an extension of the adjointaction of g as a derivation of degree 0. Let us denote the action of x ∈ g by θ ( x ) ∈ End( V g ); thatis, θ ( x ) y = [ x, y ] where y ∈ ∧ g = g , and θ ( x )( u ∧ v ) = ( θ ( x ) u ) ∧ v + u ∧ θ ( x ) v for u, v ∈ V g . Thesubspace J = ( V g ) g = { u ∈ V g | θ ( x ) u = 0 ∀ x ∈ g } , of θ ( g ) invariants (the invariant skew-symmetric tensors over g ) is a graded ∧ -subalgebra of V g .The triangular factorisation V g = V n − ⊗ V h ⊗ V n + and the “augmentation maps” ε ± : V n ± → C which are algebra homomorphisms uniquely definedby ε ± ( n ± ) = 0, give rise to a degree-preserving projection mapΦ = ε − ⊗ id ⊗ ε + : V g → V h . Let us consider the restriction of Φ to J . In contrast to the Chevalley projection map for symmetrictensors, this restriction fails to be one-to-one: one hasΦ ( J ) = V h = C . Indeed, one shows that the Φ -image of J must lie in the W -invariants in V h . However, the fixedpoints of W in V h are just the one-dimensional space V h ; see [13, Section 5.1].Although the injectivity of the Chevalley projection Φ on V g fails so miserably, passing to itscounterpart Φ (a Clifford algebra version of the Harish-Chandra map) rectifies the situation, as wewill discover in due course. ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 5 Clifford algebras
In this Section, we recall some basics on Clifford algebras associated to quadratic forms oncomplex vector spaces, define the algebra C ℓ ( g ), and introduce the Clifford algebra version of theHarish-Chandra map.2.1. Identification of C ℓ ( V ) with V V . Let V be a finite-dimensional vector space over C , equip-ped with a symmetric bilinear form ( , ). The Clifford algebra C ℓ ( V ) of V is the quotient of thefull tensor algebra T ( V ) modulo the two-sided ideal generated by { x ⊗ x − ( x, x ) | x ∈ V } . We willdenote the Clifford product of u, v ∈ C ℓ ( V ) by u · v . An isomorphic image of the space V = T ( V )is contained in C ℓ ( V ); one has xy + yx = 2( x, y ) for x, y ∈ V .A convenient point of view that we adhere to in the present paper is that the Clifford algebra C ℓ ( V ) has the same underlying linear space as V V , but the Clifford product is a deformation of theexterior product. Explicitly, following [19], for x ∈ V let the operator ι ( x ) : V V → V V be definedby ι ( x ) y = ( x, y ) where y ∈ V . Extend ι ( x ) to a superderivation of V V of degree −
1, i.e., by therule ι ( x )( u ∧ v ) = ( ι ( x ) u ) ∧ v + ( − | u | u ∧ ι ( x ) v, u ∈ V | u | V, v ∈ V V. We refer to ι ( x ) : V V → V V as the contraction operator associated to x ∈ V . Now define theoperator γ ( x ) ∈ End( V V ) by γ ( x ) u = x ∧ u + ι ( x ) u , u ∈ V g . The superderivation property of ι ( x )and the fact that ι ( x ) = 0 and x ∧ x = 0 imply that γ ( x ) is multiplication by the scalar ( x, x ).Therefore, γ extends to a homomorphism γ : C ℓ ( V ) → End( V V ). The linear map σ : C ℓ ( V ) → V V, σ ( u ) = γ ( u )1 , is bijective, cf. [8, Theorem II.1.6]. It is this map σ that is used to identify the underlying linearspace of C ℓ ( V ) with V V . Remark . The symmetric algebra S ( V ) and the exterior algebra V V of a vector space V arethe simplest examples of the so-called Nichols algebras (terminology introduced by Andruskiewitschand Schneider). Nichols algebras are Hopf algebras in a braided category; see [22, 27, 2, 4] forbackground. In particular, Nichols algebra B ( V ) of a braided space V (a space equipped with aninvertible operator c ∈ End( V ⊗ V ) satisfying the quantum Yang-Baxter equation) has a set ofbraided derivations ∂ ξ , indexed by ξ ∈ V ∗ , that satisfy the braided Leibniz rule inferred from thebraiding c . One takes the braiding c ( x ⊗ y ) = y ⊗ x , respectively c ( x ⊗ y ) = − y ⊗ x , to obtain theNichols algebra S ( V ), respectively V V , of V . Via the map V → V ∗ given by the form ( , ), thepartial derivative ∂∂x and the contraction operator ι ( x ), x ∈ V , are braided derivations of S ( V ) and V V , respectively.We also remark that the inverse to the map σ : C ℓ ( V ) → V V can be written, in a completelydifferent fashion, as the skew-symmetrisation map σ − = β ∧ : V V → C ℓ ( V ) , β ∧ ( x ∧ · · · ∧ x n ) = 1 n ! X π (sgn π ) x π (1) x π (2) . . . x π ( n ) , where x i ∈ V and the sum on the right is over all permutations π of the indices 1 , . . . , n .2.2. Algebras C ℓ ~ ( V ) . To emphasise our earlier point that the Clifford product on V V is a de-formation of the wedge product, and for later use in calculations, we introduce a complex-valueddeformation parameter ~ . To each value of ~ we associate a bilinear form ( x, y ) ~ := ~ · ( x, y ) on V .Denote by C ℓ ~ ( V ) the Clifford algebra of the form ( , ) ~ . Observe that, for ~ = 0, the algebra YURI BAZLOV C ℓ ~ ( V ) is isomorphic to the original Clifford algebra C ℓ ( V ) = C ℓ ~ ( V ) | ~ =1 , because the linear map V → V , x ~ / x , extends to an algebra isomorphism C ℓ ~ ( V ) → C ℓ ( V ). On the other hand, C ℓ ~ ( V ) | ~ =0 coincides with the exterior algebra V V and is not isomorphic to C ℓ ( V ) unless the form( , ) is identically zero.This construction gives rise to a family of Clifford products {· ~ | ~ ∈ C } on the space V V . Theway the product a · ~ b depends on ~ is described in Lemma 2.2. If a ∈ V i V , b ∈ V j V are homogeneous elements in the exterior algebra of V , thereexist u i + j − s ∈ V i + j − s V for s = 1 , , . . . , ⌊ i + j ⌋ such that a · ~ b = a ∧ b + ~ u i + j − + ~ u i + j − + · · · = a ∧ b + X ≤ s ≤ ( i + j ) / ~ s u i + j − s . Proof.
The statement is proved by induction in i , the degree of a . If i = 0, put u j − s = 0 forall s . If i = 1 and a = x ∈ V , then a · ~ b = x ∧ b + ~ ι ( x ) b . In this case, put u j − = ι ( x ) b and u j − s = 0 for all s > i ≥
2, one may assume that a = x ∧ a ′ for some x ∈ V and a ′ ∈ V i − V .Put a ′′ = ~ ι ( x ) a ′ so that the element a ′′ is homogeneous of degree i − V V . Then a = x · ~ a ′ − a ′′ ,hence a · ~ b = x · ~ ( a ′ · ~ b ) − a ′′ · ~ b . By the induction hypothesis, the Clifford products a ′ · ~ b and a ′′ · ~ b have required expansions in V V . It only remains to apply the i = 1 case to the product x · ~ ( a ′ · ~ b ) and to collect the terms, which gives the required expansion for a · ~ b . (cid:3) The superalgebra structure on C ℓ ( V ) . Clearly, the Clifford product on the exterior algebra V V does not respect the grading on V V , and C ℓ ( V ) is not a graded algebra. It is, however, easyto see (and is apparent from Lemma 2.2) that C ℓ ( V ) is still a superalgebra: C ℓ ( V ) = C ℓ ¯0 ( V ) ⊕ C ℓ ¯1 ( V ) , with C ℓ i ( V ) = P n ≥ V n + i V for i = ¯0 , ¯1 (residues modulo 2). For later use, we observe the factthat the contraction operators ι ( x ) : V V → V V are superderivations with respect to the Cliffordmultiplication: Lemma 2.3.
For any i ∈ { ¯0 , ¯1 } , x ∈ V , u ∈ C ℓ i ( V ) and v ∈ C ℓ ( V ) , ι ( x )( u · v ) = ( ι ( x ) u ) · v + ( − i u · ι ( x ) v. Proof.
We have to show that ι ( x ) supercommutes with γ ( u ) : V V → V V , the operator of the leftClifford multiplication by u . Since γ ( u ) is in the subalgebra of End( V V ) generated by γ ( y ), y ∈ V ,it is enough to show that ι ( x ) supercommutes with γ ( y ). Write γ ( y ) = ( y ∧ · ) + ι ( y ). Now, ι ( x )supercommutes with the operator y ∧ · of the left exterior multiplication by y , because ι ( x ) is asuperderivation of the wedge product. Finally, to show that ι ( x ) supercommutes with ι ( y ), observethat, by a general fact about superderivations, the supercommutator ι ( x ) ι ( y ) + ι ( y ) ι ( x ) must be aneven superderivation of the wedge product; but it obviously vanishes on V , hence is identically zeroon V V . (cid:3) The Clifford algebra C ℓ ( g ) . We are interested in the case when V = g is a semisimple Liealgebra. We fix ( , ) to be a non-degenerate ad -invariant symmetric bilinear form on g . For example,( , ) may be the Killing form, or be proportional to the Killing form with a non-zero coefficient. (If g is simple, there are no other options.) We denote by C ℓ ( g ) the Clifford algebra of g with respectto the form ( , ). ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 7
Recall that by θ ( g ) is denoted the adjoint action of g ∈ g on the exterior algebra V g ; one has θ ( g )( x ∧ u ) − x ∧ θ ( g ) u = [ g, x ] ∧ u for all g, x ∈ g and u ∈ V g . Furthermore, it is easy to seethat the ad -invariance of the form ( , ) implies θ ( g ) ι ( x ) − ι ( x ) θ ( g ) = ι ([ g, x ]). Thus, θ ( g ) is an(even) derivation of the Clifford product. It immediately follows that g -invariants J = ( V g ) g forma Clifford subalgebra in C ℓ ( g ). Recall that J is also a wedge-subalgebra in V g . We will elaborateon these two non-isomorphic algebra structures on J in the next Section.2.5. The Harish-Chandra map Φ for C ℓ ( g ) . The central object of the paper is the followinganalogue of the Harish-Chandra map, defined for the Clifford algebra C ℓ ( g ). Observe that C ℓ ( g ), likeall the algebras considered so far, factorises into its subalgebras, generated by the direct summands n − , h and n + in the triangular decomposition of g . These subalgebras are themselves Cliffordalgebras that correspond to the restriction of the form ( , ) on the respective subspaces of g : C ℓ ( g ) = C ℓ ( n − ) ⊗ C ℓ ( h ) ⊗ C ℓ ( n + ) , where the tensor product is realised by the Clifford multiplication (note that the tensorands do notcommute with respect to Clifford multiplication). The subalgebras C ℓ ( n ± ) are supercommutativeand are isomorphic to the exterior algebras V n ± , because the restriction of ( , ) to n − (respectivelyto n + ) is necessarily zero. The restriction of ( , ) to the Cartan subalgebra h is non-degenerate, thus C ℓ ( h ) is a simple algebra or a direct sum of two simple algebras, much like the bigger algebra C ℓ ( g ).In line with all the previous definitions of Harish-Chandra type maps, introduce the Harish-Chan-dra map for C ℓ ( g ) by Φ = ε − ⊗ id ⊗ ε + : C ℓ ( g ) → C ℓ ( h ) , where ε ± : C ℓ ( n ± ) = V n ± → C are the augmentation maps as above. Similar to the U ( g ) situation,the Harish-Chandra map Φ is not a homomorphism of algebras, but its restriction to h -invariantsis: Lemma 2.4.
The restriction of Φ to the subalgebra C ℓ ( g ) h = { u ∈ C ℓ ( g ) | θ ( h ) u = 0 } is asuperalgebra homomorphism between C ℓ ( g ) h and C ℓ ( h ) .Proof. First of all, Φ is a map of superspaces: the triangular factorisation C ℓ ( g ) = C ℓ ( n − ) ⊗ C ℓ ( h ) ⊗ C ℓ ( n + ) is compatible with the superspace structure on all tensorands, and moreover, the maps ε ± : C ℓ ( n ± ) → C are superspace maps (where C is a one-dimensional even space), and Φ is definedas ε − ⊗ id ⊗ ε + in this triangular factorisation.Furthermore, write L = n − C ℓ ( g ) n + ⊂ C ℓ ( g ). Then C ℓ ( g ) h ⊆ ⊗ C ℓ ( h ) ⊗ ⊕ L . The Lemmafollows immediately from the fact that L · C ℓ ( g ), C ℓ ( g ) · L are in the kernel of Φ. (cid:3) The r-matrix formula for Φ . Let us now express the Harish-Chandra map Φ : C ℓ ( g ) → C ℓ ( h )in terms of the projection Φ : V g → V h . For i = 1 , . . . , n , let x i (respectively y i ) be the positive(respectively negative) root vectors in g , normalized so that ( x i , y i ) = 1. Denote r = n X i =1 x i ∧ y i ∈ V g . The formula for r is a standard way to write a classical skew-symmetric r-matrix of g . Now introducethe operator ι ( r ) ∈ End( V g ) , ι ( r ) = n X i =1 ι ( x i ) ι ( y i ) , of degree − V g . This operator is used in the following YURI BAZLOV
Proposition 2.5.
Modulo the identification of the spaces C ℓ ( g ) and V g , respectively C ℓ ( h ) and V h , Φ( u ) = Φ ( e ι ( r ) u ) for any u ∈ V g .Proof. The algebra V h is viewed as an exterior and Clifford subalgebra of V g . For any u ∈ V h onehas Φ( u ) = Φ ( u ) = u , and ι ( r ) u = 0 so that e ι ( r ) u = u . Thus, both sides of the equation agree on u ∈ C ℓ ( h ) ⊂ C ℓ ( g ).Observe, as in the proof of Lemma 2.4, that C ℓ ( g ) h ⊂ C ℓ ( h ) + n − · C ℓ ( g ). Let us show thatΦ( u ) = Φ ( e ι ( r ) u ) = 0 when u ∈ n − · C ℓ ( g ). Of course, Φ( u ) = 0 simply by definition of the mapΦ. Now, we may assume that u ∈ y j · C ℓ ( g ) for some j between 1 and n ; but since ι ( r ) does notdepend on a particular ordering of positive roots, we may assume j to be 1, i.e., u = γ ( y ) u ′ forsome u ′ ∈ V g . Here γ ( y ) = y ∧ · + ι ( y ) is the operator of left Clifford multiplication by y as in 2.1.Note that the operators ι ( x i ) ι ( y i ), i = 1 , . . . , n , pairwise commute and square to zero. This followsfrom the fact that ι ( x ) and ι ( y ) anticommute for all x, y ∈ g , see the proof of Lemma 2.3. Hence, wemay write e ι ( r ) as Q ni =1 e ι ( x i ) ι ( y i ) = Q ni =1 (1+ ι ( x i ) ι ( y i )). Furthermore, because ( x i , y ) = ( y i , y ) = 0for i = 1, it follows from Lemma 2.3 that ι ( x i ) commutes with γ ( y ) for i = 1. Thus,Φ ( e ι ( r ) u ) = Φ (cid:0) (1 + ι ( x ) ι ( y )) γ ( y ) u ′′ (cid:1) for some u ′′ ∈ V g . It remains to note that(1 + ι ( x ) ι ( y )) z = γ ( x ) γ ( y ) z − x ∧ y ∧ z and that, obviously, Φ ( x ∧ y ∧ z ) = 0 for all z ∈ V g . We are left withΦ ( e ι ( r ) u ) = Φ ( γ ( x ) γ ( y ) u ′′ )which is zero since y = ( y , y ) = 0 in the Clifford algebra C ℓ ( g ).We have shown that both sides of the required equation agree when u ∈ C ℓ ( g ) h . Now supposethat u is an eigenvector of non-zero weight for the adjoint action of h on V g . Since the maps Φ, Φ and ι ( r ) preserve the weight with respect to the h -action, Φ( u ) and Φ ( e ι ( r ) u ) must have, in V h , thesame non-zero weight as u . Since the adjoint action of h on V h is trivial, the latter is only possibleif Φ( u ) = Φ ( e ι ( r ) u ) = 0. The Proposition is proved. (cid:3) Remark . The map p TG ◦ T in [1, 3.1] coincides with Φ ◦ e ι ( r ) (in our notation). Proposition 2.5thus implies that the map p TG ◦ T from [1] is the same as our Harish-Chandra map Φ. Remark . Recall that for each non-zero value of the deformation parameter ~ we can equip V g with the structure of Clifford algebra C ℓ ~ ( g ). The latter Clifford algebra is built with respect tothe bilinear form ( , ) ~ = ~ · ( , ) on g which is ad -invariant and non-degenerate. In particular, theHarish-Chandra map Φ ~ : C ℓ ~ ( g ) → C ℓ ~ ( h )is defined. Let us apply Proposition 2.5 to the Clifford algebra C ℓ ~ ( g ).Namely, if x i , y i was a positive/negative root vector pair normalised by ( x i , y i ) = 1 , then x i , ~ − y i will be such pair for the form ( , ) ~ . It follows that the classical r-matrix of g correspondingto the new form ( , ) ~ is given by r ~ = ~ − r . Furthermore, the contraction operators on V g withrespect to the bilinear form ( , ) ~ are given by ι ~ ( x ) = ~ ι ( x ). Hence we have a new operator ι ~ ( r ~ ) := n X i =1 ι ~ ( x i ) ι ~ ( ~ − y i ) = ~ ι ( r ) , ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 9 which leads to the following corollary of Proposition 2.5:
Corollary . Φ ~ ( u ) = Φ ( e ~ ι ( r ) u ) = Φ ( u ) + ~ Φ ( ι ( r ) u ) + ~ Φ ( ι ( r ) u ) + . . . . (cid:3) The map Φ ~ is thus given as a deformation of the Chevalley projection Φ . We emphasise thatΦ ~ is injective on the space J of invariants for ~ = 0 while Φ = Φ ~ | ~ =0 is not.3. The algebras of invariants and Kostant’s ρ -decomposition of C ℓ ( g )This section contains the information on the structure of the algebras J S = S ( g ) g , J = C ℓ ( g ) g and C ℓ ( g ) which will be used in the proof of our main results. We will recall several theorems ofKostant from [19] where it is assumed that ( , ) is the Killing form on g ; however, they are easilyseen to hold when ( , ) is any non-degenerate ad -invariant form.3.1. Generators of the algebra of symmetric invariants.
Recall from Section 1 that theChevalley projection map establishes an algebra homomorphism between J S = S ( g ) g and the al-gebra S ( h ) W of Weyl group invariants in the polynomial algebra S ( h ). From the invariant theoryof reflection groups [24, 9] it follows that J S has r = rank ( g ) algebraically independent homoge-neous generators f , f , . . . , f r . These are defined up to multiplication by non-zero constants andmodulo ( J + S ) , where J + S = J S ∩ ⊕ n> S n ( g ). We will denote by P S the linear span of some chosen f , f , . . . , f r . The space P S is, in general, not uniquely defined and depends on the choice of the f i .Define the positive integers m , . . . , m r by deg f i = m i + 1. The numbers m i are independent(up to reordering) of the choice of a particular set of the f i and are called the exponents of g .3.2. Primitive skew-symmetric invariants.
It turns out that there is a skew symmetric coun-terpart, P , of the space P S ⊂ J S . First of all, extend the ad -invariant form ( , ) from g to the wholeof V g in a standard way: V m g is orthogonal to V n g unless m = n , and( x ∧ · · · ∧ x n , y ∧ · · · ∧ y n ) = det(( x i , y j )) ni,j =1 where x i , y i ∈ g . The space P ⊂ V g of primitive alternating invariants is defined as the ( , )-ortho-complement of J + ∧ J + in J + , where J + is the augmentation ideal J ∩ ( P m> V m g ). By a theoremof Koszul (see [19, Theorem 26]), the restriction of ( , ) to J and to J + is non-singular. Hence P isa graded subspace of J . Moreover, the dimension of P is equal to the rank r of g , and one actuallyknows the degrees where the graded components of P are located: P = span { p , p , . . . , p r } , p i ∈ ( V m i +1 g ) g , where m i , as before, are the exponents of g .Under a natural bijection between J and the homology H ∗ ( g ) of g , the elements of P correspondto what is known as primitive homology classes (see [19, 4.3]).Now it turns out that the space J of invariants, with the product induced from V g , is itself anexterior algebra. The Hopf-Koszul-Samelson theorem (see [19, 4.3] which refers to [21, Theorem10.2]) asserts that J = V P, meaning that the map ζ : V P → J , which is an algebra homomorphism extending the inclusion map P ֒ → J (the elements of P anticommute in J , being of odd degree), is an isomorphism. The subalgebra J ⊂ C ℓ ( g ) . Even more surprising is the result, due to Kostant, that thesubalgebra J = C ℓ ( g ) g of the Clifford algebra C ℓ ( g ), is itself a Clifford algebra, generated by P asprimitive generators.Of course, the assertion J = C ℓ ( P ) has a chance to be valid only if a primitive tensor p ∈ P ,being Clifford-squared, yields a constant. And this is indeed true; as shown in [19, Theorem B],primitive skew-symmetric invariants behave under Clifford multiplication as if they were elementsof degree 1: p · p = ( α ( p ) , p ) for p ∈ P ,where α is defined as multiplication by the constant ( − m on P ∩ V m +1 g . A further result ofKostant [19, Theorem 35] asserts that the map ζ C ℓ : C ℓ ( P ) → J , which extends, as an algebrahomomorphism, the inclusion map P ֒ → J , is an isomorphism of algebras. Here C ℓ ( P ) is theClifford algebra of the space P equipped with the non-degenerate bilinear form ( p, q ) = ( α ( p ) , q ).For later use, we record this here as theorem. Theorem 3.1 (Kostant) . In the above notation, the diagram V P ζ −−−−→ ∼ J β ∧ ,P y y β ∧ , g C ℓ ( P ) ζ C ℓ −−−−→ ∼ J commutes. Here β ∧ ,P : V P → C ℓ ( P ) and β ∧ , g : V g → C ℓ ( g ) are skew-symmetrisation maps for therespective exterior algebras. The Chevalley transgression map.
We briefly recall the useful transgression map, following[19, Section 6]. Let d : g → V g , dx = 12 X e a ∧ [ e a , x ]be the coboundary map for g . (Its extension to V g as a derivation of degree 1 is the coboundary inthe standard Koszul complex for g .) Here { e a } , { e a } is any pair of dual bases of g with respect tothe form ( , ). Now introduce the algebra homomorphism s : S ( g ) → V even g , s ( x x . . . x n ) = dx ∧ dx ∧ · · · ∧ dx n between S ( g ) and the commutative subalgebra V even g = P n V n g of V g .Denote by ι S ( x ) f the directional derivative of f ∈ S ( g ) with respect to x ∈ g (attention: g is identified with its dual space g ∗ via the form ( , )). In other words, ι S ( x ) is the derivation of S ( g ) of degree − y in S ( g ) = g , one has ι S ( x ) y = ( x, y ) ∈ S ( g ). The Chevalleytransgression map may now be defined by the formula t ( f ) = ( m !) (2 m + 1)! X a e a ∧ s ( ι S ( e a ) f )due to Kostant [19, Theorem 64]. This maps symmetric tensors of degree m + 1 to alternatingtensors of degree 2 m + 1. By a result of Chevalley (see [19, Theorem 66]), for any choice of thespace P S of primitive symmetric invariants, t : P S → P is a linear isomorphism. (One observes that t vanishes on ( J + S ) .) One can choose f i ∈ P S ∩ S m i +1 ( g ) so that t ( f i ) = p i . ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 11
The ρ -decomposition of C ℓ ( g ) . Kostant’s “separation of variables” result for the Cliffordalgebra [19] asserts that C ℓ ( g ) is a free module over its subalgebra J . There is a subalgebra E ⊂ C ℓ ( g ), which is in fact the Clifford centraliser of J in C ℓ ( g ), so that the Clifford algebra factorisesas C ℓ ( g ) = E ⊗ J, where ⊗ is realised by Clifford multiplication. Moreover, E can also be described as follows. For x ∈ g , denote δ ( x ) = 14 X a e a · [ e a , x ] ∈ C ℓ ( g ) , where, as usual, { e a } , { e a } are a pair of dual bases of g . It is easy to show that in fact, identifyingthe spaces V g and C ℓ ( g ), one has δ ( x ) = dx where dx ∈ V g is the coboundary of x introducedearlier. The equation δ ([ x, y ]) = δ ( x ) δ ( y ) − δ ( y ) δ ( x ) holds in the Clifford algebra C ℓ ( g ); see [19,Proposition 28]. Therefore, δ extends to a homomorphism δ : U ( g ) → C ℓ ( g )of associative algebras. One has E = δ ( U ( g )).Now, let ρ be the half sum of positive roots of g , and let V ρ denote the irreducible g -modulewith highest weight ρ . Kostant identifies E , as an algebra and a g -module, with the matrix algebraEnd V ρ , which is why the factorisation C ℓ ( g ) = E ⊗ J is referred to as the ρ -decomposition of C ℓ ( g ).Indeed, let x , . . . , x n be an ordering of positive root vectors in g , and let µ + = x · · · · · x n in C ℓ ( g ).Let z ∈ U ( g ) act on the subspace Eµ + of C ℓ ( g ) via left multiplication by δ ( z ). This makes Eµ + into a U ( g )-module isomorphic to V ρ , with highest weight vector µ + . The action of E on V ρ ∼ = Eµ + by left multiplication induces a homomorphism E → End V ρ . It turns out to be an isomorphism[19, Theorem 40]. 4. Φ is an isomorphism between C ℓ ( g ) g and C ℓ ( h )4.1. The first main theorem.
In this section we prove our first main result about the Clifford al-gebra analogue, Φ : C ℓ ( g ) → C ℓ ( h ), of the Harish-Chandra map. We will use the notation introducedin previous sections. Theorem 4.1.
The restriction of the Harish-Chandra map Φ to the g -invariants in C ℓ ( g ) is asuperalgebra isomorphism Φ : C ℓ ( g ) g → C ℓ ( h ) .Proof. We use Kostant’s ρ -decomposition, C ℓ ( g ) = E ⊗ J , of the Clifford algebra C ℓ ( g ). Let usdenote by E h the subalgebra { u ∈ E | θ ( h ) u = 0 } of C ℓ ( g ) h . Then we have the tensor factorisation C ℓ ( g ) h = E h ⊗ J. Recall from the previous Section that E is the image of U ( g ) under the algebra map δ : U ( g ) → C ℓ ( g ).As δ is also a g -module map, E h = δ ( U ( g ) h ), where U ( g ) h is ad h -invariants in U ( g ). Choose a basisof U ( g ) h consisting of monomials y a . . . y a k h b . . . h b l x c . . . x c m , where y a (resp. x c ) are negative(resp. positive) root vectors in g , h b are vectors in the Cartan subalgebra, and the product is, ofcourse, in U ( g ). To see what the δ -image of such a monomial can be, we use the following Lemma 4.2. ( i ) For any h ∈ h , Φ( δ ( h )) is equal to the constant ρ ( h ) , where ρ ∈ h ∗ is half the sumof positive roots of g . ( ii ) For any x ∈ n + ⊂ g , δ ( x ) is in C ℓ ( g ) · n + . Proof of the lemma.
Choose a pair of dual bases of g in the following special way. Let x , . . . , x n be positive root vectors in g , corresponding to the positive roots β , . . . , β n and constituting thebasis of n + . Let y , . . . , y n be negative root vectors so that ( x i , y i ) = 1, and let h , . . . , h r besome basis of h , orthonormal with respect to ( , ). The bases x , . . . , x n ; h , . . . , h r ; y , . . . , y n and y , . . . , y n ; h , . . . , h r ; x , . . . , x n of g are dual with respect to the ad -invariant non-degenerate form( , ). By definition of δ and using the fact that h is an Abelian Lie subalgebra of g , one calculates δ ( h ) = 14 n X i =1 ( x i · [ y i , h ] + y i · [ x i , h ]) = 14 X i β i ( h )( x i y i − y i x i ) . Observe that x i y i − y i x i = 2 − y i x i in the Clifford algebra, and Φ( y i x i ) = 0 because, by definition ofΦ, the kernel of Φ contains n − · C ℓ ( g ) and C ℓ ( g ) · n + . Thus one obtains Φ( δ ( h )) = P i β i ( h ) · ρ ( h ),establishing part ( i ) of the Lemma.Now for x ∈ n + , calculation of δ ( x ) with respect to the same special pair of dual bases of g willyield an expression with the following terms: x i · [ y i , x ], y i · [ x i , x ] and h j · [ h j , x ]. The latter twoclearly belong to C ℓ ( g ) n + . Rewrite x i · [ y i , x ] as − [ y i , x ] · x i +2( x i , [ y i , x ]). Here − [ y i , x ] · x i is again in C ℓ ( g ) n + , and ( x i , [ y i , x ]) = ([ x i , y i ] , x ) = 0 because [ x i , y i ] ∈ h and x ∈ n + . Thus δ ( x ) ∈ C ℓ ( g ) · n + .The Lemma is proved. (cid:3) Remark . The proof of Lemma 4.2 is similar to [19, Proposition 37, Lemma 38 and Theorem 39].These statements lead to a Clifford algebra realisation of the representation with highest weight ρ of a semisimple Lie algebra (Chevalley-Kostant construction). This construction can be generalisedto central extensions of the corresponding loop algebra and in particular for c sl . See the paper [16]by Joseph. One may realise the basic modules of c sl , which is done by Greenstein and Joseph in[14] and has no analogue in the semisimple case. In the infinite dimensional case, the ordering ofthe factors in the expression for δ ( h ) becomes crucial.We now continue the proof of Theorem 4.1. Consider a typical monomial y a . . . y a k h b . . . h b l x c . . . x c m in U ( g ) h as above. If m >
0, the δ -image of this monomial lies in C ℓ ( g ) δ ( x c m ), which isin C ℓ ( g ) n + by Lemma 4.2. By definition of Φ, C ℓ ( g ) n + lies in the kernel of Φ, thus the Φ ◦ δ -imageof the monomial is zero.If m = 0, then k = 0 because the monomial must have weight zero with respect to the adjointaction of h on U ( g ). The δ -image of the monomial h b . . . h b l is δ ( h b ) . . . δ ( h b l ). By Lemma 4.2 onehas Φ( δ ( h b ) . . . δ ( h b l )) = ρ ( h b ) . . . ρ ( h b l ) ∈ C . Thus, we have shown thatΦ( E h ) = C . Because Φ : C ℓ ( g ) h → C ℓ ( h ) is a superalgebra homomorphism (by Lemma 2.4) and is surjective(coincides with the identity map on C ℓ ( h ) ⊂ C ℓ ( g ) h ), we have C ℓ ( h ) = Φ( E h )Φ( J ). It follows that C ℓ ( h ) = Φ( J ), that is, Φ : J → C ℓ ( h ) is surjective, hence bijective by comparison of dimensions(both are 2 r ). Theorem 4.1 is proved. (cid:3) A formula for Φ ~ ◦ δ . Looking at the proofs of Lemma 4.2 and Theorem 4.1, we conclude thatthe calculations which have been made lead to a formula for the map Φ ◦ δ : U ( g ) → C . Namely,it is apparent that Φ( δ ( u )) = Ψ( u )( ρ ), where Ψ : U ( g ) S ( h ) is the classical Harish-Chandra mapintroduced in 1.2, and · ( ρ ) denotes the evaluation of an element of S ( h ), viewed as a polynomialfunction on the space h ∗ , at the point ρ ∈ h ∗ . For later purposes we will need a slightly more generalversion of this formula, which is nevertheless established by a completely analogous calculation.Recall the “deformed” Harish-Chandra map Φ ~ : C ℓ ( g ) → C ℓ ( h ), introduced in Remark 2.7. ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 13
Lemma 4.4.
The map Φ ~ ◦ δ : U ( g ) → C is given by Φ ~ ( δ ( u )) = Ψ( u )( ~ ρ ) . (cid:3) identifies P and h . In the proof of Theorem 4.1 we relied on the result, due to Kostant,that C ℓ ( g ) factorises as E ⊗ J . We are going to obtain more information about the Harish-Chandramap Φ : C ℓ ( g ) g → C ℓ ( h ) using Kostant’s description of J as the Clifford algebra C ℓ ( P ), where P isthe space of primitive g -invariants in C ℓ ( g ) as in the previous Section. It is one of the key results of[19] that ι ( x ) p ∈ E for p ∈ P, x ∈ g ;see [19, Theorem E]. From this, we deduce our Proposition 4.5.
The restriction of the Harish-Chandra map Φ to the space P ⊂ J of primitiveinvariants is a bijective linear map between P and the Cartan subalgebra h .Proof. Take a primitive invariant p ∈ P . For any h ∈ h , one has ι ( h ) p ∈ E by the above result ofKostant. Therefore, it follows from Lemma 4.2 that Φ( ι ( h ) p ) ∈ C . More precisely, Φ( ι ( h ) p ) is inthe one-dimensional subspace C · V h of V h . Let us now use Lemma 4.6.
For any h ∈ h and u ∈ C ℓ ( g ) h , one has ι ( h )Φ( u ) = Φ( ι ( h ) u ) .Proof of Lemma 4.6. We have already observed that C ℓ ( g ) h decomposes as a direct sum C ℓ ( h ) ⊕ C ℓ ( g ) n + ∩ C ℓ ( g ) h . Because ι ( h ) is a superderivation of C ℓ ( g ) (Lemma 2.3) and ι ( h ) n + = 0, ι ( h )preserves the subspaces C ℓ ( h ) and C ℓ ( g ) n + ∩ C ℓ ( g ) h of C ℓ ( g ), hence commutes with the projectiononto C ℓ ( h ). (cid:3) Let Φ( p ) be some element q ∈ V h . Applying Lemma 4.6 we establish that ι ( h ) q ∈ V h for all h ∈ h . The restriction of the form ( , ) to h is non-degenerate, therefore there exists h ∈ h = V h such that ι ( h ) h = ι ( h ) q for all h ∈ h . The intersection of kernels of all contraction operators ι ( h )in the exterior algebra V h is its zero degree part, V h = C : this fact can be checked directly and isa particular case of a Nichols algebra property (see Remark 2.1 above and [4, Criterion 3.2]). Thus, q = h + h for some h ∈ V h .But Φ is a map of superspaces by Lemma 2.4, and a primitive invariant p ∈ P is odd. Therefore,the even component h of q is zero, thus Φ( p ) ∈ h . Hence Φ( P ) ⊂ h , and by injectivity of Φ on C ℓ ( g ) g (Theorem 4.1) and comparison of dimensions, Φ( P ) = h . Proposition 4.5 is proved. (cid:3) induces an isomorphism between J and V h . We finish this Section with an observationthat the Harish-Chandra map Φ respects not only Clifford but also exterior multiplication on thespace of g -invariants in V g . Corollary 4.7.
The map
Φ : ( V g ) g → V h is an algebra isomorphism.Remark . The Corollary asserts that the restriction Φ to J respects the wedge multiplication.This is not trivial, because it is not readily seen from the construction of Φ why Φ( a ∧ b ) = Φ( a ) ∧ Φ( b )when a, b ∈ J . Recall that in terms of the wedge product, the map Φ is given by the r-matrix formula(Proposition 2.5), and note that in general, Φ( a ∧ b ) = Φ( a ) ∧ Φ( b ) for a, b ∈ ( V g ) h . Proof of Corollary 4.7.
We need to show that the wedge product in ( V g ) g is respected by the map,fully written as σ h ◦ Φ ◦ σ − g , where σ h : C ℓ ( h ) → V h and σ g : C ℓ ( g ) → V g are maps identifying theunderlying linear space of a Clifford algebra with the corresponding exterior algebra; see 2.1. But by Theorem 3.1 it is enough to show that σ h ◦ Φ ◦ σ − P respects the wedge product in V P , where P isthe space of primitive invariants in V g . Let p , . . . , p r be a basis of P such that p i are homogeneousin V g and are pairwise orthogonal with respect to the form ( , ) extended to V g . Then p i are alsoorthogonal with respect to the form ( α ( · ) , · ), which gives rise to the Clifford algebra C ℓ ( P ) as in 3.3.It follows that the p i pairwise anticommute in C ℓ ( P ), hence also in C ℓ ( g ).For each subset I = { i < · · · < i k } of { , . . . , r } , denote p I = p i ∧ · · · ∧ p i k . The 2 r elements p I form a basis of V P . By orthogonality of the p i , p I = σ P ( p i p i . . . p i k ); applying Φ gives h i h i . . . h i k ∈ C ℓ ( h ), where h i = Φ( p i ). By Proposition 4.5, h , . . . , h r form a basis of the Cartansubalgebra h . Moreover, this basis is orthogonal with respect to the restriction of the form ( , ) on h , because the h i pairwise anticommute in the Clifford algebra C ℓ ( h ) being the images of the p i .Therefore, σ h ( h i h i . . . h i k ) = h i ∧ h i ∧ · · · ∧ h i k , which may be denoted by h I .Thus, modulo the appropriate identifications, the map Φ : ( V g ) g → V h is given on the basis { p I | I ⊆ { , . . . , r }} of ( V g ) g by Φ( p I ) = h I . It manifestly follows that Φ is an isomorphism ofexterior algebras. (cid:3) The principal basis of the Cartan subalgebra
In the previous Section we established that the Harish-Chandra map Φ : C ℓ ( g ) → C ℓ ( h ) restrictsto a bijective linear map between the space P of primitive g -invariants in C ℓ ( g ) and the Cartansubalgebra h of g . Recall that P is a graded subspace of the exterior algebra V g of g . The linearbijection Φ : P → h thus induces a grading on the Cartan subalgebra h . We will now show thatthis grading coincides with one arising in a different context — from the decomposition of g ∨ , theLanglands dual Lie algebra of g , as a module over its principal three-dimensional simple subalgebra.5.1. Principal three-dimensional simple subalgebras.
In this subsection, we briefly recallknown facts about principal three-dimensional simple subalgebras (principal TDS) of g . Our stan-dard reference for this is [17, § r be the rank of a semisimple Lie algebra g . A nilpotent element e of g is called principalnilpotent (or regular nilpotent), if the centraliser g e of e in g is of minimal possible dimension,namely dim g e = r . By the well-known Jacobson-Morozov theorem, any non-zero nilpotent element e ∈ g can be included in an sl -triple ( e, h, f ) of elements of g , i.e., a triple that fulfils the relations[ h, e ] = 2 e , [ h, f ] = − f , [ e, f ] = h . Such an sl -triple is called principal, if e is principal nilpotent.The linear span of a principal sl -triple in g is referred to as a principal three-dimensional simplesubalgebra (principal TDS) of g .The following characterisation of principal TDS is due to Kostant: Lemma 5.1. If a is a principal TDS of g , then g , viewed as an a -module via the adjoint action, isa direct sum of precisely r simple a -modules. (cid:3) If a subalgebra a ∼ = sl of g is not principal, then g is a direct sum of strictly more than r simple a -modules.All principal TDS g are conjugate (with respect to the action of the adjoint group of g ). Wewill be interested in one particular principal TDS a of g , discovered independently by Dynkin andde Siebenthal. To define a , fix a Cartan subalgebra h of g and recall that our chosen ad -invariantform ( , ), restricted to h , is non-degenerate, hence one may identify h with its dual space h ∗ . Inparticular, the root system of g becomes a subset of h ; a root α and the corresponding coroot α ∨ are related via α ∨ = 2 α/ ( α, α ). ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 15
Choose a set of simple roots α , . . . , α r ∈ h . Denote by ρ ∨ the element of h defined by thecondition ( α i , ρ ∨ ) = 1, i = 1 , . . . , r ; this element is half the sum of positive coroots of g . Let e i be root vectors corresponding to the roots α i , and f i be root vectors corresponding to roots − α i normalised so that ( e i , f i ) = 1. Observe that [ e i , f i ] = α i ∈ h . Indeed, for arbitrary h ∈ h one has([ e i , f i ] , h ) = ( e i , [ f i , h ]) = ( e i , α i ( h ) f i ) = ( α i , h ). Put e = r X i =1 e i , h = 2 ρ ∨ , f = r X i =1 c i f i , where c i are the coefficients in the expansion 2 ρ ∨ = P ri =1 c i α i . It is not difficult to show that( e , h , f ) is an sl -triple (using the fact that ( α i , ρ ∨ ) = 2 for all i , that [ e i , f j ] = 0 for i = j , andthat [ e i , f i ] = α i ). Moreover, e is a principal nilpotent element of g . The principal TDS a of g isthe linear span of the triple ( e , h , f ).5.2. The principal grading on h . Recall that under the adjoint action of a principal TDS a , theLie algebra g breaks down into a direct sum of r = rank g simple a -modules. The dimensions of thesesimple modules were determined by Kostant in [17]: writing V d for the d -dimensional sl -module,one has g ∼ = V m +1 ⊕ . . . ⊕ V m r +1 , where m i , i = 1 , , . . . , r , are the exponents of g ; see 3.1. Each sl -module V m i +1 , being of odddimension, has a one-dimensional zero weight subspace V m i +1 . We now turn to the case a = a , thedistinguished principal TDS introduced above. Fix an isomorphism between g and ⊕ i V m i +1 andregard each V m i +1 as an a -submodule of g . Clearly, ⊕ i V m i +1 is the centraliser, g h , of h = 2 ρ ∨ in g . Since all root vectors in g are eigenvectors of ad h corresponding to non-zero eigenvalues, itfollows that g h = h , the Cartan subalgebra of g (i.e., h is a regular semisimple element of g ). Onehas the direct sum decomposition h = V m +1 ⊕ . . . ⊕ V m r +1 . This decomposition of h is not canonical, because there is some freedom in choosing the isomorphismbetween g and ⊕ i V m i +1 . However, for each d , the V d -primary component ⊕{ V m i +1 : 2 m i + 1 = d } is canonically defined. There is thus a grading h = ⊕ d h d , h d = h ∩ ⊕{ V m i +1 : 2 m i + 1 = d } on the Cartan subalgebra h , which only depends on the choice of a set of simple roots of h in g (i.e., the choice of a Borel subalgebra b ⊃ h ). Note that the element ρ ∨ ∈ h and the grading donot depend on a particular non-degenerate ad -invariant form ( , ). We refer to this grading as theprincipal grading on h . Remark . There is another grading on g associated to any sl -triple ( e, h, f ) in g , namely theeigenspace decomposition of ad h . Such gradings are referred to as Dynkin gradings in [12]. Notethat the Dynkin grading arising from ( e , h , f ) has h as the degree zero subspace and breaksdown each of the V m i +1 into a direct sum of 1-dimensional graded subspaces, so it is, in a sense,transversal to the principal grading. Elementary properties of the principal grading.
Let us observe a few straightforwardproperties of the principal grading on h . Lemma 5.3.
Let h = ⊕ d h d be the principal grading on the Cartan subalgebra h . ( a ) Non-zero homogeneous components h d occur only in degrees d = 2 m + 1 , where m is anexponent of g . The least such degree is . The dimension of h m +1 is the number of times m occursas an exponent of g . ( b ) dim h is the number of connected components in the Dynkin diagram of g . ( c ) The element ρ ∨ of h belongs to h . ( d ) Homogeneous components of h of different degrees are orthogonal with respect to any ad -in-variant form ( , ) on g .Proof. ( a ) is apparent from the definition of the principal grading. To establish ( b ), one observesthat 1 occurs as an exponent of g as many times as is the number of summands in the decompositionof g into a direct sum of simple Lie algebras, which is the number of connected components of theDynkin diagram of g . (A simple Lie algebra has only one exponent equal to 1, and taking the directsum of semisimple Lie algebras corresponds to taking the union of the multisets of exponents.) Notethat a ⊂ g is a three-dimensional a -submodule of g , hence a ∩ h = C ρ ∨ ⊂ h and ( c ) follows.To prove ( d ), take h, k ∈ h such that h ∈ V m +1 ⊂ g and k ∈ V n +1 ⊂ g , with m < n two distinctexponents of g . By the representation theory of sl , there is a highest weight vector x ∈ V n +1 for a , such that k = ( ad f ) n x . Then ( h, k ) = (( ad f ) n x, k ) = ( − n ( x, ( ad f ) n k ). But ( ad f ) n k = 0,thus h , k are orthogonal. (cid:3) The principal basis of h in a simple Lie algebra g . When g is simple, we can say moreabout the principal grading on h . Lemma 5.4.
Let h be the Cartan subalgebra of a simple Lie algebra g . If g is not of type D r with r even, then all non-zero homogeneous components of the principal grading h = ⊕ d h d areone-dimensional. If g is of type D r with r even, then dim h d = 1 for d = r − , and dim h r − = 2 .Proof. According to [6, Table IV], simple Lie algebras of types other than D r with even r havedistinct exponents; in type D r with even r all exponents have multiplicity 1 except r/ − a ). (cid:3) Thus, in all simple Lie algebras apart from so (2 r ), r even, a choice, b ⊃ h , of a Borel and a Cartansubalgebra gives rise to a distinguished basis h , h , . . . , h r of h . The vector h i is determined, up toa scalar factor, by the condition h m i +1 = C h i . This basis is orthogonal with respect to the Killingform, and h = ρ ∨ . We call h , . . . , h r the principal basis of h .For g ∼ = so (2 r ), r − h r − of h . We still call any basis, obtained by this procedure, a principal basis of h .5.5. The grading on h induced from g ∨ . Recall that we have identified the Cartan subalgebra h with its dual space h ∗ via the non-degenerate ad -invariant form ( , ), so that both the roots andthe coroots of g lie in h . Consider g ∨ , the Langlands dual Lie algebra to g ; that is, the complexsemisimple Lie algebra with a root system dual to that of g . The roots of g ∨ are the coroots of g ,so we will assume that g ∨ and g share the same Cartan subalgebra h ∨ = h . ARISH-CHANDRA ISOMORPHISM FOR CLIFFORD ALGEBRAS 17
The above definition of principal grading applies to g ∨ . One has, therefore, a grading h = ⊕ d h ∨ d on the Cartan subalgebra, which is the principal grading induced from g ∨ . This grading dependson a choice of simple coroots α ∨ , . . . , α ∨ r which is the same as a choice α , . . . , α r of simple roots of g . If g is a simple Lie algebra, g ∨ is also simple, so in that case h has a dual principal basis inducedfrom g ∨ .5.6. Main result.
We are now ready to state and prove the final main result of the paper, Theo-rem 5.5 which also solves a conjecture of Kostant. Once again, recall that P ⊂ ( V g ) g is the spaceof primitive skew-symmetric invariants. It is graded by degree in V g . The space P is also viewed asa subspace of C ℓ ( g ) g . We already know that under the Harish-Chandra map Φ : C ℓ ( g ) g ∼ −→ C ℓ ( h ),the space P is isomorphically mapped onto h . Theorem 5.5.
The grading on the Cartan subalgebra h of g , induced from the grading on P bydegree via the map Φ : P ∼ −→ h , coincides with the principal grading h = ⊕ d h ∨ d on h in the Langlandsdual Lie algebra g ∨ .Proof. We are going to use a result due to Chevalley, cited in 3.4, that the primitive skew-symmetricinvariants are transgressive; i.e., if f , . . . , f r are independent homogeneous generators of S ( g ) g ,then t ( f ) , . . . , t ( f r ) is a basis of P . Let m ≤ m ≤ · · · ≤ m r be the exponents of g so that f i ∈ S m i +1 ( g ). Denote p i = t ( f i ), thus p i ∈ V m i +1 g . Furthermore, let z , . . . , z r be any basis ofthe Cartan subalgebra h orthonormal with respect to the non-degenerate ad -invariant form ( , ). AsΦ( p i ) is an element of h by Proposition 4.5, we may writeΦ( p i ) = r X j =1 (Φ( p i ) , z j ) z j = r X j =1 ι ( z j )Φ( p i ) · z j = r X j =1 Φ( ι ( z j ) p i ) · z j = r X j =1 Φ( ι ( z j ) t ( f i )) · z j , where we used Lemma 4.6 to exchange the maps Φ and ι ( z j ). We now observe that further tobeing two examples of braided derivations (remark 2.1), the operators ι S ( z ) : S m +1 ( g ) → S m ( g )and ι ( z ) : V m +1 g → V m g are intertwined (up to a scalar factor) by the Chevalley transgressionmap t . One has ι ( z ) t ( f ) = ( m !) (2 m )! s ( ι S ( z ) f ) for all f ∈ J S ∩ S m +1 ( g ) , z ∈ g by [19, Theorem 73], where s : S ( g ) → V g is the homomorphism introduced in 3.4. HenceΦ( p i ) = c r X j =1 Φ( s ( ι S ( z j ) f i )) · z j for some non-zero constant c .In the next calculation, we are going to use results from [19] which are obtained for the case ofsimple g . We thus assume g to be simple until further notice.Let us now assume that the independent homogeneous generators f , . . . , f r of the algebra S ( g ) g of symmetric invariants are chosen in a particular way, so as to span the orthogonal complementto ( J + S ) in J + S (compare with 3.1). Here orthogonality is with respect to a natural extension of the form ( , ) from g onto S ( g ): identify S ( g ) with S ( g ∗ ) which is acting on S ( g ) by differentialoperators with constant coefficients, i.e., to f ∈ S ( g ) there corresponds ∂ f ∈ S ( g ∗ ); now put ( f, g )to be the evaluation of the polynomial function ∂ f g at zero. Such choice of primitive symmetricinvariants is due to Dynkin. Denote by P D the span of f , . . . , f r where deg f i = m i + 1, and let( P D ) ( m ) = P D ∩ S ≤ m ( g ).The space P D is especially useful for us because of the following property [19, Theorem 87]: thereexist u , . . . , u r ∈ P D , such that u i − f i ∈ ( P D ) ( m i ) and δ u i ( z ) = 12 m i s ( ι S ( z ) f i )for any z ∈ g . Here δ u : g → C ℓ ( g ) , δ u ( z ) = δ ( β ( ι S ( z ) u ))is a g -equivariant map from g to C ℓ ( g ), defined in [19, 6.12]; we remind the reader that β : S ( g ) → U ( g ) is the PBW symmetrisation map. (It is clear from the definition of the map δ u that the imageof δ u is in the subalgebra E = Im δ of C ℓ ( g ).) In fact, the u i are not homogeneous, and are of theform u i = f i + P k
Let b , . . . , b r be algebraically independent homogeneous generators of the algebra S ( g ) g of symmetric invariants of a simple Lie algebra g such that deg b k = m k + 1 . Put h k = ι S ( ρ ) m k b k .If the h k are non-zero and pairwise orthogonal with respect to the Killing form, then h . . . , h r area principal basis of h induced from the Langlands dual Lie algebra g ∨ .Proof. Let W ⊂ GL ( h ) be the Weyl group of g , which is the same as the Weyl group of g ∨ . TheLanglands dual principal basis of h is orthogonal with respect to the Killing form on g ∨ , which inrestriction to h is the unique, up to a scalar factor, W -invariant form on h . Thus, the Langlandsdual principal basis is orthogonal with respect to the Killing form on g . It is therefore sufficient toshow that h k is in the kernel of ( ad e ∨ ) m k +1 in the algebra g ∨ , where ( e ∨ , ρ, f ∨ ) is the canonicalprincipal sl -triple of g ∨ . Here m k is the k th exponent of g ∨ , which is the same as the k th exponentof g .Putting n = m k + 1 and b = b k , it is enough to prove that for b ∈ S n ( g ) g , the element h = ι S ( ρ ) n − b of h is in the kernel of ( ad e ∨ ) n in the algebra g ∨ . Let ¯ b = Ψ ( b ) denote the image of b under the Chevalley projection map Ψ : S ( g ) g → S ( h ) W . Clearly, h = ι S ( ρ ) n − ¯ b . Now regard h as the Cartan subalgebra of g ∨ , and let Ψ ∨ : S ( g ∨ ) g ∨ → S ( h ) W be the Chevalley projection for g ∨ .Put b ∨ = (Ψ ∨ ) − (¯ b ) so that b ∨ is the image of b under the algebra isomorphism(Ψ ∨ ) − Ψ : S ( g ) g ∼ −→ S ( g ∨ ) g ∨ , and write h = ι S ( ρ ) n − b ∨ . Using the formula ( ad x ) ι S ( y ) = ι S ([ x, y ]) + ι S ( y ) ad x where x, y ∈ g ∨ and both sides are operators on S ( g ∨ ), we can now write( ad e ∨ ) n ι S ( ρ ) n − b ∨ = X n ! d ! . . . d n ! ι S (( ad e ∨ ) d ρ ) . . . ι S (( ad e ∨ ) d n − ρ )( ad e ∨ ) d n b ∨ . The sum on the right is over all n tuples d , . . . , d n of non-negative integers, such that d + · · · + d n = n . Observe that for each such n tuple, either d i ≥ i < n so that ( ad e ∨ ) d i ρ = 0(remembering the key sl relation ( ad e ∨ ) ρ = 0); or else d n ≥
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