AN EXPLICIT DETERMINATION OF THE SPRINGER MORPHISM SEAN ROGERS 7 Abstract. Let G be a simply connected semisimple algebraic groups over C and let ρ : 1 G → GL(Vλ) be an irreducible representation of G of highest weight λ. Suppose that ρ 0 hasfinite kernel.Springerdefinedanadjoint-invariantregularmapwithZariskidense image 2 from the group to the Lie algebra, θλ : G → g, which depends on λ [BP,§9]. By a lemma n in [Kum] θλ takes the maximal torus to its Lie algebra t. Thus, for a given simple group G a n J and an irreducible representation Vλ, one may write θλ(t) = ci(t)αˇi, where we take the i=1 6 P simple coroots{αˇi} as a basis for t. We givea complete determination for these coefficients ] ci(t) for any simple group G as a sum over the weights of the torus action on Vλ. T R 1. Introduction . h t LetGbeaconnectedreductivealgebraicgroupoverCwithBorelsubgroupB andmaximal a m torus T ⊂ B of rank n with character group X∗(T). Let P be a standard paraoblic subgroup [ with Levi subgroup L containing T. Let W (resp. WL) be the Weyl group of G (resp. L). Let V be an irreducible almost faithful representation of G with highest weight λ, i.e. λ is 1 λ v a dominant integral weight and the corresponding map ρ : G → Aut(V ) has finite kernel. λ λ 8 Then, Springer defined an adjoint-invariant regular map with Zariski dense image from the 3 group to its Lie algebra, θ : G → g, which depends on λ (Sect. 2.1). 5 λ 1 In recent work by Kumar [Kum], the Springer morphism is used in a crucial way to ex- 0 tend the classical result relating the polynomial representation ring of the general linear . 1 group GL and the singular cohomology ring H∗(Gr(r,n)) of the Grassmanian of r-planes 0 r 7 in Cn to the Levi subgroups of any reductive group G and the cohomology of the cor- 1 responding flag varieties G/P. Computing θ | is integral to this process. By a lemma λ T v: in [Kum], θ takes the maximal torus T to its Lie algebra t, thus inducing a C-algebra λ Xi homomorphism (θλ|T)∗ : C[t] → C[T] between the corresponding affine coordinate rings. r The Springer morphism is adjoint invariant and thus (θλ|T)∗ takes C[t]WL to C[T]WL. One a can then define the λ − polynomial subring RepC (L) to be the image of C[t]WL un- λ−poly der (θ | )∗ (as RepC(L) ≃ C[T]WL). This leads to a surjective C-algebra homomorphism λ T ξP : RepC (L) → H∗(G/P,C), as in [Kum]. The aim of this work is to compute θ | in λ λ−poly λ T a uniform way for all simple algebraic groups G and any dominant integral weight λ. As θ | maps T into t, we have that for a given simple group G and an irreducible λ T representation V , one may write λ n θ (t) = c (λ)αˇ λ i i Xi=1 , where we take the simple coroots {αˇ} as a basis for t. We give a complete determination i for these coefficients c (t) for any simple, simply-connected algebraic group G as a sum over i Date: October 2016. 1 2 SEAN ROGERS the weights of the torus action on V . For a given representation V , let Λ be the set of λ λ λ weights appearing in the weight space decomposition of V = V , listed with multiplicity. λ µ Let ω1,...,ωn be the fundamental weights in t∗, and consider Lthe weights µ ∈ Λλ written in the fundamental weight basis, i.e. µ = (µ ,...,µ ) = µ ω +... +µ ω . Let eµ(t) ∈ X∗(T) 1 n 1 1 n n be the corresponding character of T. Then we find (Sect. 3) that, Theorem 1. The coefficients c (t) are determined by the following set of equations. i µ1 ·eµ(t) c1(t) µP∈Λλ  c (t) .  2  .. = S(G,λ) .. ,   . µP∈Λλµn ·eµ(t) cn(t) where S(G,λ) = { µ µ } . i j ij µP∈Λλ Our main result (Sect. 4) determines that Theorem 2. The above matrix 1 S(G,λ) := { µ µ } = ( µ2)S , i j ij 2 i µX∈Λλ µX∈Λλ where S is a symmetrization of the Cartan matrix A for G, and µ is the coordinate of the i fundamental weight corresponding to a long root (or in the simply-laced case any root). In particular, for the simply-laced groups S(G,λ) = (1 µ2)A. The determination of 2 1 µP∈Λλ S(G,λ) relies on the fact that Λ is invariant under the action of the Weyl group W, and λ moreover that if σ ∈ W then dim(V ) = dim(V ). µ σ.µ 2. Preliminaries Let G be a simply-connected semi-simple algebraic group over C, with Lie algebra g = t⊕ g of rank n, and fixed base of simple roots ∆ = {α }. Take the set of simple co-roots α j α L n ∆ˇ = {αˇj} as a basis for the Cartan subalgebra t ⊂ g. Then tZ = Zαˇj is the co-root jL=1 n lattice. Further, the weight lattice is t∗ = Zω , where ω ∈ t∗ is the ith fundamental Z i i i=1 L weight of g defined by ω (αˇ ) = δ . Then the maximal torus T ⊂ G (with Lie algebra t) can i j ij be identified with T = HomZ(t∗Z,C∗) as in [Kum2]. Finally, let W be the Weyl group of G, generated by the simple reflections s . So for µ ∈ t∗, s (µ) = µ−µ(αˇ)α . i i i i Let V be the irreducible representation of G with highest weight λ. Then V has weight λ λ space decomposition V = V λ µ M where V = {v ∈ V | t.v = ((µ ω + ... + µ ω )(t))v ∀v ∈ V } is the weight space µ1,µ2,...,µn λ 1 1 n n λ with weight µ = µ ω +...+µ ω . 1 1 n n So for t ∈ T and v ∈ V we have that the action of t on v is given by µ1,µ2,...,µn t.v = t(µ ,...,µ )v = eµ(t)v 1 n AN EXPLICIT DETERMINATION OF THE SPRINGER MORPHISM 3 where (µ ,...µ ) = µ ω +...+µ ω . Additionally αˇ ∈ t acts on v by 1 n 1 1 n n j αˇ .v = (µ ω +...+µ ω )(αˇ )v = µ v. j 1 1 n n j j 2.1. Springer Morphism. For a given almost faithful irreducible representation V of G λ we define the Springer morphism as in [BP] θ : G → g λ given by G ❚❚❚❚❚❚// ❚A❚u❚❚t❚(❚Vθ❚λ❚(❚λ❚)❚)❚❚⊂❚❚❚E❚❚n** d(cid:15)(cid:15) π(V(λ)) = g⊕g⊥ g where g sits canonically inside End(V ) via the derivative dρ , the orthogonal complement λ λ g⊥ is taken via the adjoint invariant form < A,B >= tr(AB) on End(V ), and π is the λ projection onto the g component. Note, that since π ◦ dρ is the the identity map, θ is λ λ a local diffeomorphism at 1. Since the decomposition End(V ) = g ⊕g⊥ is G-stable, θ is λ λ invariant under conjugation in G. Importantly, θ restricts to θ : T 7→ t. [Kum] λ λ|T 3. General Case Let V be a d dimensional almost faithful irreducible representation of G of highest weight λ λ. Let Λ = {(µi,...,µi)}d be an enumeration of the set of weights considered with their λ 1 n i=1 multiplicity that appear in the weight space decomposition of V (so µi is the coordinate of λ j the jth fundamental weight for the ith weight in the decomposition) Then we can take a basis of weight vectors {v }d on which the torus T and each simple co-root acts diagonally. µi1,...,µin i=1 Thus, ρ (t) = diag{eµ1(t),...,eµd(t)} ∈ Aut(V ) λ λ and for a simple co-root αˇ we have that j dρ (αˇ ) = diag{µ1,...,µd} ∈ End(V ). λ j j j λ To take the projection we calculate dρ (g)⊥ ∈ End(V ) with respect to the symmetric λ λ bilinear form tr(AB). So letting X = (x ) be a d×d matrix in End(V ) we have that for ij λ any co-root αˇ ∈ t we require that j d tr(dρ (αˇ )·X) = 0 =⇒ µix = 0 λ j j ii Xi=1 in order for X ∈ dρ (g)⊥. λ So µix = µix = ... = µix = 0. Now to project ρ (t) onto dρ (t) we write ρ 1 ii 2 ii n ii λ λ λ as a sPuΛλm PΛλ PΛλ n ρ (t) = c (t)dρ (αˇ )+X(t). λ j λ j Xj=1 4 SEAN ROGERS where c : T 7→ C is a function that depends on λ, and X(t) ∈ dρ (g)⊥. It follows then j λ that θ (t) = c (t)αˇ λ j j X So we aim to solve for the coefficients c (t). Note that for the root space g , we have j α that g .V ⊂ V . Thus, dρ (e ) for e ∈ g will only have off diagonal entries, and as α µ µ+α λ α α α such the condition tr(dρ (e )·X) = 0 will only add constraints to the off diagonal entries λ α of X ∈ dρ (g)⊥. As the action of t and αˇ are both diagonal, by comparing coordinates we λ j have the following set of d equations eµ1(t) = c (t)µ1 +...+c (t)µ1 +x 1 1 n n 11 eµ2(t) = c (t)µ2 +...+c (t)µ2 +x 1 1 n n 22 . . . eµd(t) = c (t)µd +...+c (t)µd +x . 1 1 n n dd d This can be reduced to n equations by utilizing the fact that µix = 0, as follows. j ii i=1 P Multiply each equation above by µi and sum (then repeat with µi,...,µi) 1 2 n d d d d µie(µi1,...,µin)(t) = (µi)2c (t)+ µiµic (t)+...+ µiµic (t) 1 1 1 1 2 2 1 n n Xi=1 Xi=1 Xi=1 Xi=1 . . . d d d d µie(µi1,...,µin) = µiµic (t)+ µiµic (t)+...+ (µi)2c (t) n 1 n 1 2 n 2 n n Xi=1 Xi=1 Xi=1 Xi=1 More cleanly this can be written as µ1 ·eµ(t) c1(t) PΛλ  c (t) .  2  .. = S(G,λ) ..   . PΛλ µn ·eµ(t) cn(t) where µ ·µ µ ·µ ... µ ·µ 1 1 1 2 1 n PΛλ PΛλ PΛλ  µ ·µ µ ·µ ... µ ·µ 1 2 2 2 2 n S(G,λ) := PΛλ PΛλ PΛλ   .. .. ..   . . .     µ1 ·µn ... µn−1 ·µn µn ·µn   PΛλ PΛλ PΛλ  Then, we have that c1(t) µ1eµ(t) c (t) PΛλ   2  . .. = S−1(G,λ) .. .   cn(t) PΛλ µneµ(t) AN EXPLICIT DETERMINATION OF THE SPRINGER MORPHISM 5 In the next section we calculate the matrix S(G,λ) for the classical and exceptional simple algebraic groups. In the following sections, we continue the notation Λ = {(µ ,...µ )| µ ω +...+µ ω is a weight ofV } λ 1 n 1 1 n n λ counted with multiplicity. 4. Main Result Our main result will be calculating the matrix S(G,λ) as defined in section 3, for the simple algebraic groups. We use the convention that the Cartan matrix associated to the root system of g is A = (A ), where A = α (αˇ). Then A is a change-of-basis matrix ij ij i j for t∗ between the fundamental weights and the simple roots. Furthermore, A satisfies the following properties • For diagonal entries A = 2 ii • For non-diagonal entries A ≤ 0 ij • A = 0 iff A = 0 ij ji • A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix. Let D be the diagonal matrix defined by D = δij(α ,α ), where if we realize the root system ij 2 i j R associated to g as a set of vectors in a Euclidean space E, then (·,·) is the standard inner product. In this framework we can write A = α (αˇ) = 2(αi,αj) Then, writing A = DS, we ij i j (αj,αj) find that the matrix S has coordinate entries given by 4(α ,α ) i j S = ij (α ,α )(α ,α ) i i j j and is clearly symmetric. (·,·) is an invariant bilinear form on t∗, normalized so that so that (α ,α ) = 2 where α is i i i the highest root. Note that under this formulation, if G is of simply-laced type then D is the identity matrix and S is the Cartan matrix. We find that in general for a given simple group G that S(G,λ) is a multiple of S. Before stating our result precisely we fix the following notation. If α is any long simple root (for the simply laced case α can be any simple root), j j consider the corresponding fundamental weight ω . Let x (λ) := µ2, where µ is the jth j j j j µP∈Λλ coordinate of the weight µ ∈ Λ in the fundamental weight basis. λ Proposition 4.1. Let G be a simple algebraic group. Let S(G,λ) be defined as in section 3. Set x (λ) := µ2 for a long root α . This is independent of the choice of long root α . j j j j µP∈Λλ Let S be a symmetrization of the Cartan matrix as above. Then S(G,λ) is a multiple of S. More precisely, 1 S(G,λ) = x (λ)·S j 2 Proof. The proof will rely on the fact that the set of weights Λ of V is invariant under the λ λ action of the Weyl Group on t∗, i.e. for w ∈ W, w.Λ = Λ . The following Lemma is true λ λ for all simple groups. The following two lemmas are sufficient to prove the simply-laced case but also hold for the non-simply laced cases. 6 SEAN ROGERS Lemma 4.2. For a given simple group G, if the Cartan matrix entry A = 0, i.e the nodes ij representing the simple roots α and α are not connected on the associated Dynkin diagram, i j then µ ·µ = 0, i j µX∈Λλ where µ = (µ ,...,µ ). 1 n Proof. Consider the simple reflection s acting on a weight µ = (µ ,...µ ) ∈ Λ . Then i 1 n λ s (µ) = (µ ,...µ )−((µ ,...µ )(αˇ))(α ) i 1 n 1 n i i Where (µ ,...µ )(αˇ) = (µ ω + ...µ ω )(αˇ) = µ . Using the Cartan matrix to write the 1 n i 1 1 n n i i simple roots α in the fundamental weight basis gives α = (A ,...,A ). Then the above i i i,1 i,n reflection yields s (µ) = (µ ,...µ )−µ (A ,...,A ) = (µ −µ A ,...,µ −µ A ) i 1 n i i,1 i,n 1 i i1 n i in Now note that A = 2 and A = 0. So the ith coordinate of s (µ) is [s (µ)] = µ −µ A = ii ij i i i i i ii −µ and the jth coordinate of s (µ) is [s (µ)] = µ −µ A = µ . Thus we find that i i i j j i ij j µ µ = µ µ = [s (µ)] ·[s (µ)] = −µ µ , i j i j i i i j i j µX∈Λλ si(Xµ)∈Λλ µX∈Λλ µX∈Λλ by invariance of Λ under s . Thus, the result follows. (cid:3) λ i Lemma 4.3. If simple roots α and α of G are connected via the Dynkin diagram and have i j the same length then (µ )2 = (µ )2. i j µX∈Λλ µX∈Λλ Furthermore, 1 µ ·µ = − µ ·µ i j i i 2 µX∈Λλ µX∈Λλ Proof. Let α and α be roots of the same length whose corresponding nodes on the Dynkin i j diagram are connected. So A = A = −1. Then as above with µ = (µ ,...µ ) ∈ Λ , we ij ji 1 n λ have that s (µ) = (µ −µ A ,...,µ −µ A ). Now consider i 1 i i1 n i in s s (µ) = ((µ −µ A )−(µ −µ A )A ,...,(µ −µ A )−(µ −µ A )A ) j i 1 i i1 j i ij j1 n i in j i ij jn Thus, [s s (µ)] = (µ −µ A )−(µ −µ A )A = −µ −(µ +µ )(−1) = µ . Thus, j i i i i ii j i ij ji i j i j µ ·µ = [s s (µ)] ·[s s (µ)] = µ ·µ i i j i i j i i j j X X X Λλ Λλ Λλ The second part of the lemma follows from the fact that [s (µ)] = µ −µ A with A = −1. i j j i ij ij It follows that µ2 = [s (µ)]2 = (µ +µ )2 j i j j i X X X Λλ Λλ Λλ Thus, µ ·µ = −2 µ ·µ (cid:3) i i i j PΛλ PΛλ AN EXPLICIT DETERMINATION OF THE SPRINGER MORPHISM 7 With the above results we see that for groups of simply-laced type that µ eµ(t) 1 c (t) 1. 2 µP∈Λλ .   ..  = A−1 .. µ2 cn(t) µP∈Λλ i  µneµ(t) µP∈Λλ  The inverses of the Cartan matrices for the simply laced root systems are in the Appendix. 4.1. Non-simply laced groups. Recall that the roots systems of simple groups of type B ,C ,G ,F contain long and short simple roots. Our convention will be the same as in n n 2 4 Bourbaki [Bo]. That is, for B that α ,...,α are the long roots and α is short, for C n 1 n−1 n n that α ,...α are short and α is long, for G that α is short and α is long, and for F 1 n−1 n 2 1 2 4 that the first and second are long and that the third and fourth are short. 4.1.1. G of type B,C or F. Proposition 4.1.1. Let G be a rank n simple group of types B , C , or F . For any long n n 4 root α , set x = µ2. If α is a short root, then µ2 = 2x, where x is defined in §4. If i i j j PΛλ µP∈Λλ either or both of α and α are short, then µ µ = −x i j i j µP∈Λλ Proof. Note that if α and α are both long roots, connected via the Dynkin diagram, then i j A = A = −1 So the same argument as in Lemma 4.3 shows that ij ji µ2 = µ2, i j X X Λλ Λλ and that µ µ = −1 µ2. The same is true for the short roots as A = A = −1 i j 2 i ij ji PΛλ PΛλ for connected short roots. So we need to show that if α and α are short and long i j roots respectively and connected via the Dynkin diagram, then µ2 = 2x, and that i PΛλ µ µ = −x. To show this we first note that A = −1 and A = −2 and then com- i j ij ji PΛλ pare [s (µ)] ,[s (µ)] ,[s (µ)] and [s (µ)] . Note that [s (µ)] = −µ and s (µ ) = −µ as i i j j j i i j i i i j j j before. Also, [s (µ)] = µ −µ A = µ +µ and [s (µ)] = µ −µ A = µ +2µ . Thus, we i j j i i,j j i j i i j ji i j have that µ µ = [s (µ)] ·[s (µ)] = (µ +2µ )(−µ ) = −µ µ −2µ2 i j j i j j i j j i j j X X X X Λλ Λλ Λλ Λλ Thus µ µ = − µ2 = −x. Applying, s to µ gives i j j i PΛλ PΛλ µ µ = [s (µ)] ·[s (µ)] = −µ µ −µ2 i j i i i j i j i X X X Λλ Λλ Λλ Thus, µ2 = 2x (cid:3) i PΛλ 8 SEAN ROGERS So it follows that with x = µ2, where α is a long root, then j j PΛλ 2 −1 4 −2 −1 2 −1 −2 4 −2     x −1 ... x −2 ... S(B ,λ) =  ,S(C ,λ) =   n n 2  2 −1  2  4 −2       −1 2 −2  −2 4 −2      −2 4   −2 2      2 −1 0 0 x −1 2 −2 0  S(F ,λ) = 4 2 0 −2 4 −2    0 0 −2 4    We give inverses of these matrices in the appendix. 4.1.2. G of type G . Let α be the short root, and α the long root of G . 2 1 2 2 Proposition 4.1.2. µ2 = −2 µ µ = 3 µ2 1 1 2 2 PΛλ PΛλ PΛλ 2 −1 Proof. Let µ = (µ ,µ ) ∈ Λ . Then since A = , we find that s (µ) = (−µ ,µ + 1 2 λ (cid:18)−3 2 (cid:19) 1 1 1 µ ) and that s (µ) = (µ +3µ ,−µ ). So, 2 2 1 2 2 µ2 = (µ +3µ )2 1 1 2 X X Λλ Λλ from which it follows that µ µ = −3 µ2. Additionally, we have that 1 2 2 2 PΛλ PΛλ µ2 = (µ +µ )2 2 1 2 X X Λλ Λλ from which we can see that µ2 = −2 µ µ = 3 µ2. Thus, 1 1 2 2 PΛλ PΛλ PΛλ 1 6 −3 S(G ,λ) = µ2 2 2 2(cid:18)−3 2 (cid:19) X Λλ (cid:3) In particular, we can solve for c (t) and c (t) as 1 2 µ eµ(t) 1 c (t) 1 = (S(G ,λ)−1PΛλ  (cid:18)c (t)(cid:19) 2 µ eµ(t) 2 2 PΛλ  2 3 then, letting x = µ2 we have that S−1(G,λ) = 2 . Thus, 2 3x (cid:18)3 6(cid:19) PΛλ 2 c (t,λ) = (2µ +3µ )eµ(t) 1 1 2 3x X Λλ AN EXPLICIT DETERMINATION OF THE SPRINGER MORPHISM 9 2 c (t,λ) = (3µ +6µ )eµ(t) 2 1 2 3x X Λλ (cid:3) . 5. Example(G = C ,Defining Reresentation) n 0 I Consider G = Sp(2n,C)={A ∈ GL(2n)|M = AtMA} where M = n where I is (cid:18)−In 0(cid:19) n the n×n identity matrix, and sp(2n,C)={X ∈ gl(2n)|XtM +MX = 0}. Let λ = ω ,the defining representation. Then we have that Λ ={±ω and ±(ω − ω ) 1 λ 1 i i+1 for 1 ≤ i ≤ n − 1}. So, x = µ2 = 2. Let T = diag{t ,...,t ,t−1,...,t−1}. The simple n 1 n 1 n PΛλ roots are α = ǫ −ǫ for 1 ≤ i ≤ n−1 and α = 2ǫ . The simple coroots in t are then i i i+1 n n αˇ = E − E − E + E for 1 ≤ 1 ≤ n − 1 and αˇ =E − E where E is the i i i+1 n+i n+i+1 n n 2n i diagonal matrix with a 1 in the ith slot and 0’s elsewhere [FH]. In the orthogonal basis for t, ω = ǫ +...+ǫ . Thus, the character eµ(t) is given by eµ(t) = tµ1+...µn ·tµ2+...+µn ·...·tµn. i 1 i 1 2 n Then, we have that 1 1 1 ... 1 t1 −t−11 −t2 +t−21 c1(.t) 1 1 2 2 ... 2 t2 −t−21 −. t3 +t−31   ..  = 1 2 3 ... 3 .. 2    c (t) ... ... ... ... ...t −t−1 −t +t−1  n    n−1 n−1 n n  1 2 3 ... n t −t−1    n n  which gives t −t−1 c (t) 1 1 1 . . 1  ..   ..  = 2 t −t−1 c (t)  n−1 n−1   n  t −t−1 +...+t −t−1  1 1 n n  Thus, t −t−1 t −t−1 t −t−1 t −t−1 θ (t) = c (t)αˇ +...+c (t)αˇ = diag( 1 1 ,..., n n ,− 1 1 ,...,− n n ). λ 1 1 n n 2 2 2 2 Note that this is equivalent to the Cayley transform as in §6 of [Kum]. Similiar results hold for θ (t) for the standard maximal tori of SO(2n,C) and SO(2n,C). ω1 Appendix A. inverse of the cartan matrices and their symmetrizations S The the inverses of the Cartan matrices for A ,D ,E ,E ,E respectively have the form n n 6 7 8 (as in [Rosenfeld])) n n−1 n−2 ... 3 2 1 n−1 2(n−1) 2(n−3) ... 6 4 2   1 n−2 2(n−2) 3(n−2) ... 9 6 3 , n+1  ... ... ... ... ... ... ...     2 4 6 ... (2n−2) 2(n−1) n−1    1 2 3 ... n−2 n−1 n    10 SEAN ROGERS 1 1 1 ... 1 1 1 2 2 1 2 2 ... 2 1 1   1 2 3 ... 3 3 3 2 2 ... ... ... ... ... ... ...    1 2 3 ... n−2 n−2 n−2  2 2  1 1 3 ... n−2 n n−2 2 2 2 4 4  1 1 3 ... n−2 n−2 n  2 2 2 4 4  4 5 7 10 8 6 4 2 2 2 3 4 3 2 1 4 1 5 2 4 2 5 8 10 15 12 9 6 3 3 3 3 3 2 2 4 6 9 3 3   1 2 2 3 2 1  2 2 2 7 10 14 20 16 12 8 4 3 4 6 8 6 4 2 5 2 10 4 8 4 10 15 20 30 24 18 12 6 3 3 3 3,4 6 8 12 9 6 3,  2 3 4 6 4 2   8 12 16 24 20 15 10 5   3 9 6 9 15 5 5   4 2 8 4 10 5  2 2 2 6 9 12 18 15 12 8 4 3 3 3 3 2 3 4 6 5 4 2   2 1 4 2 5 4   4 6 8 12 10 8 6 3 3 3 3 3 1 3 2 3 5 2 3    2 2 2 2 3 4 6 5 4 3 2   The inverse of the matrix S for types C ,B ,G ,F have the form n n 2 4 2 2 2 ... 2 1 1 1 1 ... 1 2 4 4 ... 4 2 2 3 2 1 1 2 2 ... 2   1 1 2 4 6 ... 6 3 2 1 3 6 4 2 1 2 3 ... 3 , , 3 , 2   2 ... ... ... ... ... ...  (cid:18)1 2(cid:19) 2 4 3 3 ... ... ... ... ...    2   2 4 6 ... 2(n−1) n−1 1 2 3 1 1 2 3 ... n    2    1 2 3 ... n−1 2    References [BR] P. Bardsley and R.W. Richardson, E´tale slices for algebraic transformationgroups in characteristic p, Proc. London Math. Soc. 51 (1985), 295–317. [Bo] N. Bourbaki, Groupes et Alg`ebres de Lie, Chap. 4–6, Masson, Paris,1981. [Kum1] S.Kumar,RepresentationringofLeviSubgroupsversuscohomologyringofflavarieties, Mathema- tische Annalen, 366(2016), 395–415. [Kum2] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Progress in Math- ematics, vol. 204, Birkh¨auser, 2002. [FH] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer, 1991. [Ro] Boris Rosenfeld, Geometry of Lie Groups, Kluwer Academic Publishers, 1993.