Dimension of zero weight space: An algebro-geometric approach
aa r X i v : . [ m a t h . R T ] A p r DIMENSION OF ZERO WEIGHT SPACE: ANALGEBRO-GEOMETRIC APPROACH
SHRAWAN KUMAR AND DIPENDRA PRASAD
1. I
NTRODUCTION
Let G be a connected, adjoint, simple algebraic group over the complexnumbers C with a maximal torus T and a Borel subgroup B ⊃ T . The studyof zero weight spaces in irreducible representations of G has been a topicof considerable interest; there are many works which study the zero weightspace as a representation space for the Weyl group. In this paper, we studythe variation on the dimension of the zero weight space as the irreduciblerepresentation varies over the set of dominant integral weights for T whichare lattice points in a certain polyhedral cone.The theorem proved here asserts that the zero weight spaces have dimen-sions which are piecewise polynomial functions on the polyhedral cone ofdominant integral weights. The precise statement of the theorem is givenbelow.Let Λ = Λ ( T ) be the character group of T and let Λ + ⊂ Λ (resp. Λ ++ )be the semigroup of dominant (resp. dominant regular) weights. Then, bytaking derivatives, we can identify Λ with Q , where Q is the root lattice(since G is an adjoint group). For λ ∈ Λ + , let V ( λ ) be the irreducible G -module with highest weight λ . Let µ : Λ + → Z + be the function: µ = dim V ( λ ) , where V ( λ ) is the 0-weight space of V ( λ ).Let Γ = Γ G ⊂ Q be the sublattice as in Theorem (3.1).Also, let Λ ( R ) : = Λ ⊗ Z R and let Λ ++ ( R ) be the cone inside Λ ( R ) gen-erated by Λ ++ . Let C , . . . , C N ⊂ Λ ++ ( R ) be the chambers (i.e., the GITclasses in Λ ++ ( R ) of maximal dimension: equal to the dimension of Λ ( R ),with respect to the T -action) (see Section 2).For any w ∈ W and 1 ≤ i ≤ ℓ , define the hyperplane H w , i : = { λ ∈ Λ ( R ) : λ ( wx i ) = } , where W is the Weyl group of G and { x , . . . , x ℓ } is the basis of t dual tothe basis of t ∗ given by the simple roots. Then, by virtue of Corollary (3.6), C , . . . , C N are the connected components of Λ ++ ( R ) \ (cid:0) ∪ w ∈ W , ≤ i ≤ ℓ H w , i (cid:1) . With this notation, we have the following main result of our paper (cf.Theorem (4.1)).
Theorem (1.1).
Let µ = µ + Γ be a coset of Γ in Q. Then, for any GIT classC k , ≤ k ≤ N, there exists a polynomial f µ, k : Λ ( R ) → R with rationalcoefficients of degree ≤ dim C X − ℓ , such that (1) f µ, k ( λ ) = µ ( λ ) , for all λ ∈ ¯ C k ∩ µ, where C k is the closure of C k inside Λ ( R ) and X is the full flag variety G / B.Further, f Γ , k has constant term 1. The proof of the above theorem relies on Geometric Invariant Theory(GIT). Specifically, we realize the function µ restricted to C k ∩ Λ as anEuler-Poincar´e characteristic of a reflexive sheaf on a certain GIT quotient(depending on C k ) of X = G / B via the maximal torus T . Then, one canuse the Riemann-Roch theorem for singular varieties to calculate this Euler-Poincar´e characteristic. From this calculation, we conclude that the function µ restricted to C k ∩ ( µ + Γ ) is a polynomial function. The result on descentof the homogeneous line bundles on X to the GIT quotient plays a crucialrole (cf. Lemma (3.7)).We end the paper by determining these piecewise polynomials for thegroups of type A and B (in Section 5) and A (in Section 6), all of whichwe do via some well-known branching laws.The results of the paper can easily be extended to show the piecewisepolynomial behavior of the dimension of any weight space (of a fixed weight µ ) in any finite dimensional irreducible representation V ( λ ).By a similar proof, we can also obtain a piecewise polynomial behavior ofthe dimension of H -invariant subspace in any finite dimensional irreduciblerepresentation V ( λ ) of G , where H ⊂ G is a reductive subgroup. However,the results in this general case are not as precise (cf. Remark (4.2)).It should be mentioned that Meinrenken-Sjamaar [MS] have obtained aresult similar to our above result Theorem (1.1) (also in the generality of H -invariants) by using techniques from Symplectic Geometry. But, theirresult in the case of T -invariants is less precise than our Theorem (1.1).The example of PGL suggests that the part of our theorem describing thedomains of validity of these piecewise polynomial functions is not optimal.Moreover, our theorem says nothing about the explicit nature of these poly-nomials. So this work should be taken only as a step towards the eventualgoal of describing the variation of the dimension of the zero weight spaceas the irreducible representation varies over the set of dominant integralweights for T . IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 3
Acknowledgements:
We are grateful to M. Brion, N. Ressayre, C. Telemanand M. Vergne for some very helpful correspondences. We especially thankVinay Wagh without whose help the computations on GL given in Section6 could not have been done. The first author was supported by the NSFgrant no. DMS-1201310. 2. N OTATION
Let G be a connected, adjoint, semisimple algebraic group over the com-plex numbers C . Fix a Borel subgroup B and a maximal torus T ⊂ B . Wedenote their Lie algebras by the corresponding Gothic characters: g , b and t respectively. Let R + ⊂ t ∗ be the set of positive roots (i.e., the roots of B )and let ∆ = { α , . . . , α ℓ } ⊂ R + be the set of simple roots. Let Q = ℓ L i = Z α i be the root lattice. Then, the group of characters Λ of T can be identifiedwith Q (since G is adjoint) by taking the derivative. We will often makethis identification. Let Λ + (resp. Λ ++ ) be the semigroup of dominant (resp.dominant regular) weights, i.e., Λ + : = { λ ∈ Λ : λ ( α ∨ i ) ∈ Z + , for all the simple coroots α ∨ i } , and Λ ++ : = { λ ∈ Λ + : λ ( α ∨ i ) ≥ α ∨ i } . Then, Λ + bijectively parameterizes the isomorphism classes of finite dimen-sional irreducible G -modules. For λ ∈ Λ + , let V ( λ ) be the correspondingirreducible G -module (with highest weight λ ).Let W : = N ( T ) / T be the Weyl group of G , where N ( T ) is the normalizerof T in G . Let Λ ( R ) : = Λ ⊗ Z R and let Λ + ( R ) (resp. Λ ++ ( R )) be the coneinside Λ ( R ) generated by Λ + (resp. Λ ++ ). Any element λ ∈ Λ ( R ) canuniquely be written as(2) λ = ℓ X i = z i ω i , z i ∈ R , where ω i ∈ Λ + ( R ) is the i -th fundamental weight: ω i ( α ∨ j ) = δ i , j . Then, Λ + ( R ) = ⊕ ℓ i = R ≥ ω i , Λ ++ ( R ) = ⊕ ℓ i = R > ω i , where R ≥ (resp. R > ) is the set of non-negative (resp. strictly positive) realnumbers. We will denote any λ ∈ Λ ( R ) in the coordinates z λ = ( z i ) ≤ i ≤ ℓ asin (2). SHRAWAN KUMAR AND DIPENDRA PRASAD
A function f : S ⊂ Λ + → Q defined on a subset S of Λ + is called a polynomial function if there exists a polynomial ˆ f ( z ) ∈ Q [ z i ] ≤ i ≤ ℓ such that f ( λ ) = ˆ f ( z λ ), for all λ ∈ S .For any λ ∈ Λ , we have the G -equivariant line bundle L ( λ ) on X : = G / B associated to the principal B -bundle G → G / B via the character λ − of B ,i.e., L ( λ ) = G B × C − λ → G / B , where C − λ denotes the one dimensional T -module with weight − λ . (Ob-serve that for any λ ∈ Λ , the T -module structure on C − λ extends to a B -module structure). The line bundle L ( λ ) is ample if and only if λ ∈ Λ ++ .Following Dolgachev-Hu [DH], λ , µ ∈ Λ ++ ( R ) are said to be GIT equiv-alent if X ss ( λ ) = X ss ( µ ), where X ss ( λ ) denotes the set of semistable pointsin X with respect to the element λ ∈ Λ ++ ( R ). Recall that if λ ∈ Λ ++ ( Q ) : = ⊕ ℓ i = Q > ω i , then X ss ( λ ) is the set of T -semistable points of X with respectto the T -equivariant line bundle L ( d λ ), for any positive integer d such that d λ ∈ Λ ++ . Definition (2.1).
By a rational polyhedral cone C in Λ ++ ( R ), one meansa subset of Λ ++ ( R ) defined by a finite number of linear inequalities withrational coefficients.For a R -linear form f on Λ ( R ) which is non-negative on C , the set ofpoints c ∈ C such that f ( c ) = face of C .By [DH] or [R, Proposition 7], any GIT equivalence class in Λ ++ ( R ) isthe relative interior of a rational polyhedral cone in Λ ++ ( R ) and moreoverthere are only finitely many GIT classes (cf. [DH, Theorem 1.3.9] or [R,Theorem 3]). Let C , . . . , C N be the GIT classes of maximal dimension, i.e.,of dimension equal to that of Λ ( R ). These are called chambers . Let X T ( C k )denote the GIT quotient X ss ( λ ) // T for any λ ∈ C k .Since for any λ ∈ Λ + , the irreducible module V ( λ ) has its zero weightspace V ( λ ) nonzero, we have X ss ( λ ) , ∅ for any λ ∈ Λ ++ ( R ).Let t + : = { x ∈ t : α i ( x ) ≥
0, for all the simple roots α i } be the dominantchamber. Clearly,(3) t + = ℓ M i = R + x i , where { x i } is the basis of t dual to the basis of t ∗ consisting of the simpleroots, i.e.,(4) α i ( x j ) = δ i , j . IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 5
3. D
ESCENT OF LINE BUNDLES TO
GIT
QUOTIENTS ANDDETERMINATION OF CHAMBERS
There exists the largest lattice Γ ⊂ Q such that for any λ ∈ Λ ++ ∩ Γ , thehomogeneous line bundle L ( λ ) descends as a line bundle b L ( λ ) on the GITquotient X T ( λ ). In fact, Γ is determined precisely in [Ku, Theorem 3.10] forany simple G , which we recall below. Theorem (3.1).
For any simple G,
Γ = Γ G is the following lattice (followingthe indexing in [B, Planche I-IX] ). (1) G of type A ℓ ( ℓ ≥
1) : Q (2) G of type B ℓ ( ℓ ≥
3) : 2 Q (3) G of type C ℓ ( ℓ ≥
2) : Z α + · · · + Z α ℓ − + Z α ℓ (4) G of type D : { n α + n α + n α + n α : n i ∈ Z and n + n + n is even } (5) G of type D ℓ ( ℓ ≥
5) : { n α + n α + · · · + n ℓ − α ℓ − + n ℓ − α ℓ − + n ℓ α ℓ : n i ∈ Z and n ℓ − + n ℓ is even } (6) G of type G : Z α + Z α (7) G of type F : Z α + Z α + Z α + Z α (8) G of type E : e Λ (9) G of type E : e Λ (10) G of type E : Q,where e Λ is the lattice generated by the fundamental weights. Definition (3.2).
Let S be any connected reductive algebraic group actingon a projective variety X and let L be an S -equivariant line bundle on X .Let O ( S ) be the set of all one parameter subgroups (for short OPS) in S .Take any x ∈ X and δ ∈ O ( S ). Then, X being projective, the morphism δ x : G m → X given by t δ ( t ) x extends to a morphism e δ x : A → X .Following Mumford, define a number µ L ( x , δ ) as follows: Let x o ∈ X bethe point e δ x (0). Since x o is G m -invariant via δ , the fiber of L over x o is a G m -module; in particular, it is given by a character of G m . This integer isdefined as µ L ( x , δ ).Let V be a finite dimensional representation of S and let i : X ֒ → P ( V )be an S -equivariant embedding. Take L : = i ∗ ( O (1)). Let λ ∈ O ( S ) and let { e , . . . , e n } be a basis of V consisting of eigenvectors, i.e., λ ( t ) · e l = t λ l e l ,for l = , . . . , n . For any x ∈ X , write i ( x ) = [ P nl = x l e l ]. Then, it is easy tosee that, we have ([MFK, Proposition 2.3, page 51])(5) µ L ( x , λ ) = max l : x l , ( − λ l ) . We record the following standard properties of µ L ( x , δ ) (cf. [MFK, Chap.2, § SHRAWAN KUMAR AND DIPENDRA PRASAD
Proposition (3.3).
For any x ∈ X and δ ∈ O ( S ) , we have the following (forany S -equivariant line bundles L , L , L ): (a) µ L ⊗ L ( x , δ ) = µ L ( x , δ ) + µ L ( x , δ ) . (b) If µ L ( x , δ ) = , then any element of H ( X , L ) S which does not vanishat x does not vanish at lim t → δ ( t ) x as well. (c) For any projective S -variety X ′ together with an S -equivariant mor-phism f : X ′ → X and any x ′ ∈ X ′ , we have µ f ∗ L ( x ′ , δ ) = µ L ( f ( x ′ ) , δ ) . (d) (Hilbert-Mumford criterion) Assume that L is ample. Then, x ∈ X issemistable (resp. stable) (with respect to L ) if and only if µ L ( x , δ ) ≥ (resp. µ L ( x , δ ) > ), for all non-constant δ ∈ O ( S ) . Lemma (3.4).
For any λ ∈ Λ ++ , the set X s ( λ ) of stable points (in X ss ( λ ) ) isnonempty.Proof. Consider the embedding i λ : X ֒ → P ( V ( λ )) , gB [ gv λ ] , where v λ is a highest weight vector in V ( λ ). Then, the line bundle O (1) over P ( V ( λ )) restricts to the line bundle L ( λ ) on X via i λ (as can be easily seen).Consider the open subset U λ ⊂ X defined by U λ = { gB ∈ X : gv λ has anonzero component in each of the weight spaces V ( λ ) w λ of weight w λ , forall w ∈ W } .Since V ( λ ) is an irreducible G -module, it is easy to see that U λ is nonempty.We claim that(6) U λ ⊂ X s ( λ ) . By the Hilbert-Mumford criterion (cf. Proposition (3.3) (d)), it sufficesto prove that for any gB ∈ U λ , the Mumford index(7) µ L ( λ ) ( gB , σ ) > , for any nonconstant one parameter subgroup σ : G m → T . Express gv λ = X µ ∈ X ( T ) v µ , as a sum of weight vectors. Let ˙ σ be the derivative of σ considered as anelement of t . Then, by the identity (5), µ L ( λ ) ( gB , σ ) = max µ ∈ X ( T ): v µ , {− µ ( ˙ σ ) }≥ max w ∈ W { λ ( − w ˙ σ ) } , since gB ∈ U λ . (8)Choose w ′ ∈ W such that − w ′ ˙ σ ∈ t + . Since σ is nonconstant, − w ′ ˙ σ , λ ( − w ′ ˙ σ ) > IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 7
To prove this, first observe that any fundamental weight ω j belongs to ⊕ ℓ i = Q > α i . (One could check this case by case for any simple group from[B, Planche I-IX]. Alternatively, one can give a uniform proof as well.)Thus, by the decomposition (3), since − ω ′ ˙ σ , ∈ t + , we get (9). In partic-ular, by (8), µ L ( λ ) ( gB , σ ) >
0, proving (7). This proves the lemma. (cid:3)
Proposition (3.5).
For λ ∈ Λ ++ , X s ( λ ) , X ss ( λ ) if and only if there existsw ∈ W and x j such that λ ( wx j ) = , where x i ∈ t is defined by (4) .Proof. Assume first that X s ( λ ) , X ss ( λ ). Take x ∈ X ss ( λ ) \ X s ( λ ). Then,by the Mumford criterion Proposition (3.3) (d), there exists a non-constantone parameter subgroup δ in T such that µ L ( λ ) ( x , δ ) =
0. Since both of X s ( λ ) and X ss ( λ ) are N ( T )-stable under the left multiplication on X by N ( T )(by loc. cit.), we can assume that δ is G -dominant, i.e., the derivative ˙ δ ∈ t + . Thus, by Proposition (3.3) (b), x o : = lim t → δ ( t ) x ∈ X ss ( λ ), since x issemistable. Let G δ be the fixed point subgroup of G under the conjugationaction by δ . Then, G δ is a (connected) Levi subgroup of G . Let W G δ bethe set of minimal length coset representatives in the cosets W / W G δ , where W G δ ⊂ W is the Weyl group of G δ . The fixed point set of X under the leftmultiplication by δ is given by X δ = ⊔ v ∈ W G δ G δ v − B / B . Let w ∈ W G δ besuch that x o ∈ G δ w − B / B . Thus, by [Ku, Lemma 3.4],(10) w − λ ∈ X α i ∈ ∆ ( G δ ) Z α i , where ∆ ( G δ ) ⊂ ∆ is the set of simple roots of G δ . Since δ is non-constant, G δ is a proper Levi subgroup. Take α j ∈ ∆ \ ∆ ( G δ ). Then, by (10), λ ( wx j ) = . Conversely, assume that(11) λ ( wx j ) = , for some w ∈ W and some x j . For any 1 ≤ i ≤ ℓ , let L i be the Levisubgroup containing T such that ∆ ( L i ) = ∆ \ { α i } . By the assumption (11), w − λ ∈ P α i ∈ ∆ ( L j ) Z α i . Moreover, we can choose w ∈ W L j and hence w − λ is a dominant weight for L j . In particular, v w − λ is a highest weight vectorfor L j , where v w − λ is a nonzero vector of (extremal) weight w − λ in V ( λ ).(To prove this, observe that | w − λ + α i | > | λ | for any α i ∈ ∆ ( L j ), and hence w − λ + α i can not be a weight of V ( λ ).) Thus, the L j -submodule V L j ( w − λ ) of V ( λ ) generated by v w − λ is an irreducible L j -module. By [Ku, Lemma 3.1],applied to the L j -module V L j ( w − λ ), we get that V L j ( w − λ ) contains the zeroweight space. Hence, by [Ku, Lemma 3.4], there exists a g ∈ L j such that gw − B ∈ X ss ( λ ). Define the one parameter subgroup δ j : = Exp( zx j ). Then, µ L ( λ ) ( gw − B , δ j ) = µ L ( λ ) ( w − B , δ j ), since g fixes δ j . But, µ L ( λ ) ( w − B , δ j ) = gw − B < X s ( λ ) by Proposition(3.3) (d). (cid:3) SHRAWAN KUMAR AND DIPENDRA PRASAD
For any w ∈ W and 1 ≤ i ≤ ℓ , define the hyperplane H w , i : = { λ ∈ Λ ( R ) : λ ( wx i ) = } . Decompose into connected components: Λ ++ ( R ) \ (cid:0) ∪ w ∈ W , ≤ i ≤ ℓ H w , i (cid:1) = ⊔ Nk = C k . The following corollary follows immediately from Proposition (3.5) and[DH, Theorems 3.3.2 and 3.4.2].
Corollary (3.6).
With the notation as above, { C , . . . , C N } are precisely theGIT classes of maximal dimension (equal to dim t ). Lemma (3.7).
For any GIT class C k (of maximal dimension) and any λ ∈ Γ ,the line bundle L ( λ ) descends as a line bundle on the GIT quotient X T ( C k ) .We denote this line bundle by b L C k ( λ ) .Proof. By Theorem (3.1), for any λ ∈ Λ ++ ∩ Γ , the line bundle L ( λ ) on X descends to a line bundle on X T ( λ ). Hence, for any λ ∈ Γ ∩ C k , the linebundle L ( λ ) descends to a line bundle b L C k ( λ ) on X T ( C k ).Let Z ( Γ ∩ C k ) denote the subgroup of Γ generated by the semigroup Γ ∩ C k .For any λ = λ − λ ∈ Z ( Γ ∩ C k ) (for λ , λ ∈ Γ ∩ C k ), define b L C k ( λ ) = b L C k ( λ ) ⊗ b L C k ( λ ) ∗ . We now show that b L C k ( λ ) is well defined, i.e., it does not depend uponthe choice of the decomposition λ = λ − λ as above. Take another decom-position λ = λ ′ − λ ′ , with λ ′ , λ ′ ∈ Γ ∩ C k . Thus, λ + λ ′ = λ ′ + λ ∈ Γ ∩ C k (since Γ ∩ C k is a semigroup). In particular, b L C k ( λ + λ ′ ) ≃ b L C k ( λ ′ + λ ).But, from the uniqueness of b L C k ( λ ) (cf. [T, § b L C k ( λ + λ ′ ) ≃ b L C k ( λ ′ ) ⊗ b L C k ( λ ). This proves the assertion that L C k ( λ ) is well defined.Observe that, by definition, C k is an open convex cone in Λ ( R ). We nextclaim that(12) Z ( Γ ∩ C k ) = Γ . Take a Z -basis { γ , . . . , γ ℓ } of Γ and let d : = max i || γ i || , with respect to anorm || · || on Λ ( R ). Take a ‘large enough’ γ ∈ Γ ∩ C k such that the closedball B ( γ, d ) of radius d centered at γ is contained in C k . Then, for any1 ≤ i ≤ ℓ , γ + γ i ∈ B ( γ, d ) and hence γ , γ + γ i ∈ Γ ∩ C k for any i . Thus, each γ i ∈ Z ( Γ ∩ C k ) and hence Γ = Z ( Γ ∩ C k ), proving the assertion (12). Thus,the lemma is proved. (cid:3) IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 9
4. T
HE MAIN RESULT AND ITS PROOF
Let µ : Λ + → Z + be the function: µ = dim V ( λ ) , where V ( λ ) is the0-weight space of V ( λ ). Following the notation from Sections 2 and 3, thefollowing is our main result. Theorem (4.1).
Let G be a connected, adjoint, simple algebraic group. Let µ = µ + Γ be a coset of Γ in Q, where Γ is as in Theorem (3.1). Then, for anyGIT class C k (of maximal dimension), ≤ k ≤ N, there exists a polynomialf µ, k : Λ ( R ) → R with rational coefficients of degree ≤ dim C X − ℓ , such that (13) f µ, k ( λ ) = µ ( λ ) , for all λ ∈ ¯ C k ∩ µ, where C k is the closure of C k inside Λ ( R ) . Further, f Γ , k has constant term 1.Proof. By the Borel-Weil theorem, for any λ ∈ Λ + ,(14) µ ( λ ) = dim (cid:16) H ( X , L ( λ )) T (cid:17) , since dim( V ( λ ) ) = dim (( V ( λ ) ∗ ) ) . Moreover, by the Borel-Weil-Bott theorem, for λ ∈ Λ + ,(15) H p ( X , L ( λ )) = , for all p > . We first prove the theorem for λ ∈ C k ∩ µ :Take λ ∈ C k ∩ µ . Let π : X ss ( C k ) → X T ( C k ) be the standard quotientmap. For any T -equivariant sheaf S on X ss ( C k ), define the T -invariant directimage sheaf π T ∗ ( S ) as the sheaf on X T ( C k ) with sections U Γ ( π − ( U ) , S ) T .Then, by Lemma (3.7), and the projection formula for π T ∗ ,(16) π T ∗ ( L ( λ )) ≃ π T ∗ ( L ( µ )) ⊗ b L C k ( λ − µ ) . By [T, Remark 3.3(i)] and (15), we get H p (cid:16) X T ( C k ) , π T ∗ ( L ( λ )) (cid:17) ≃ H ( X , L ( λ )) T , for p = = , otherwise . (17)Thus, for λ ∈ C k ∩ µ , by (14),(18) µ ( λ ) = χ (cid:16) X T ( C k ) , π T ∗ ( L ( λ )) (cid:17) , where for any projective variety Y and a coherent sheaf S on Y , we definethe Euler-Poincar´e characteristic χ ( Y , S ) : = X i ≥ ( − i dim H i ( Y , S ) . Now, take a basis (as a Z -module) { γ , . . . , γ ℓ } of the lattice Γ ⊂ Λ ( R ). Then,for any λ = µ + P ℓ i = a i γ i ∈ ¯ µ , with a i ∈ Z , we have by (16),(19) π T ∗ ( L ( λ )) ≃ π T ∗ ( L ( µ )) ⊗ b L C k ( X a i γ i ) . Thus, by the Riemann-Roch theorem for singular varieties (cf. [F, Theorem18.3]) applied to the sheaf π T ∗ ( L ( λ )), we get for any λ = µ + P a i γ i ∈ ¯ µ , χ (cid:16) X T ( C k ) , π T ∗ ( L ( λ )) (cid:17) = X n ≥ Z X T ( C k ) ( a c ( γ ) + · · · + a ℓ c ( γ ℓ )) n n ! ∩ τ (cid:16) π T ∗ ( L ( µ )) (cid:17) , (20)where τ ( π T ∗ ( L ( µ ))) is a certain class in the chow group A ∗ ( X T ( C k )) ⊗ Z Q and c ( γ i ) is the first Chern class of the line bundle b L C k ( γ i ). Combining (18) and(20), we get that for any λ ∈ C k ∩ ¯ µ , µ ( λ ) is a polynomial f ¯ µ, k with rationalcoefficients in the variables { a i } : λ = µ + ℓ P i = a i γ i .Since X s ( C k ) , ∅ by Lemma (3.4), dim ( X T ( C k )) = dim X − ℓ . Thus, deg f ¯ µ, k ≤ dim X − ℓ. This proves the theorem for λ ∈ C k ∩ µ .We now come to the proof of the theorem for any λ ∈ ¯ C k ∩ ¯ µ :Let P = P λ ⊃ B be the unique parabolic subgroup such that the linebundle L ( λ ) descends as an ample line bundle (denoted L P ( λ )) on X P : = G / P via the standard projection q : G / B → G / P . Fix µ ∈ C k ∩ Λ . By [T, § q : G / B → G / P , we get that q ∗ ( L P ( λ )) is adapted to thestratification on X induced from q ∗ ( L P ( λ )) + ǫ L ( λ ), for any small rational ǫ > λ ∈ ¯ C k ∩ ¯ µ )(21) µ ( λ ) = χ (cid:16) X T ( C k ) , π T ∗ q ∗ ( L P ( λ )) (cid:17) = χ (cid:16) X T ( C k ) , π T ∗ ( L ( λ )) (cid:17) . Hence, the identity (18) is established for any λ ∈ ¯ C k ∩ ¯ µ . Thus, by theabove proof, µ ( λ ) = f ¯ µ, k , where f ¯ µ, k is the polynomial given above.By the formula (20), the constant term of f Γ , k is equal to χ (cid:16) X T ( C k ) , π T ∗ ( L (0)) (cid:17) , which is 1 by the identity (21), since µ (0) =
1. This completes the proofof the theorem. (cid:3)
Remark (4.2). (a) By a similar proof, we can obtain a piecewise polynomialbehavior of the dimension of any weight space (of a fixed weight µ ) inany finite dimensional irreducible representation V ( λ ), by considering theGIT theory associated to the T -equivariant line bundle L ( λ ) twisted by thecharacter µ − .(b) By a similar proof, we can also obtain a piecewise polynomial be-havior of the dimension of H -invariant subspace in any finite dimensionalirreducible representation V ( λ ) of G , where H ⊂ G is a reductive subgroup.In this case, we will need to apply the GIT theory to the line bundle L ( λ )itself but with respect to the group H . However, in this general case, we do IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 11 not have a precise description of the lattice Γ as in Theorem (3.1), nor dowe have an explicit description of the GIT classes of maximal dimension asin Corollary (3.6).(c) As pointed out by Kapil Paranjape, we can obtain the polynomial be-haviour of χ (cid:16) X T ( C k ) , π T ∗ ( L ( λ )) (cid:17) as in the above proof (by using the Riemann-Roch theorem) more simply by applying Snapper’s theorem (cf. [K, The-orem in Section 1]). However, the use of Riemann-Roch theorem gives amore precise result. 5. E XAMPLES OF A AND B In this section, we calculate the dimension of the T -invariant subspacein an irreducible representation of the rank 2 groups G of types A and B .In these cases, we can do the calculation via certain well-known branchinglaws to certain subgroups. But lacking any such general branching laws, wehave not been able to handle G .We recall that irreducible representations of GL n + ( C ) are parametrizedby their highest weights, which is an ( n + λ ≥ λ ≥ · · · ≥ λ n ≥ λ n + . It is a well-known theorem that an irreducible representation of GL n + ( C )when restricted to GL n ( C ) decomposes as a sum of irreducible representa-tions with highest weights ( µ ≥ µ ≥ · · · ≥ µ n ) with λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ n ≥ µ n ≥ λ n + , and that these representations of GL n ( C ) appear with multiplicity exactlyone (cf. [GW, Theorem 8.1.1]).Note that for an irreducible representation of GL n + ( C ) to have a nonzerozero weight space, it is necessary (and sufficient) for it to have trivial centralcharacter. For determining the zero weight space of a representation ofGL n + ( C ) with trivial central character, it suffices to restrict it to GL n ( C )and consider those summands which have zero weight spaces for GL n ( C ),and then to add these zero weight spaces of GL n ( C ).We calculate the dimension of the zero weight space of an irreduciblerepresentation of GL ( C ) by restricting the representation to GL ( C ), andnoting that an irreducible representation of GL ( C ) parametrized by ( µ ≥ µ ) has a nonzero weight space if and only if µ + µ = Lemma (5.1).
An irreducible representation of GL ( C ) with highest weight ( λ ≥ λ ≥ λ ) , and with trivial central character, i.e., λ + λ + λ = , haszero weight space of dimension (1) λ − λ + , if λ ≥ , and (2) λ − λ + , if λ ≤ . We next recall that irreducible representations of SO n + ( C ) are parametrizedby their highest weights, which is an n -tuple of integers with λ ≥ λ ≥ · · · ≥ λ n ≥ . Similarly, the irreducible representations of SO n ( C ) are parametrized bytheir highest weights, which is an n -tuple of integers with λ ≥ λ ≥ · · · ≥ | λ n | . It is a well-known theorem that an irreducible representation of SO n + ( C )when restricted to SO n ( C ) decomposes as a sum of irreducible representa-tions with highest weights ( µ ≥ µ ≥ · · · ≥ | µ n | ) with λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ n ≥ | µ n | , and that these representations of SO n ( C ) appear with multiplicity exactlyone (cf. [GW, Theorem 8.1.3]).We use this branching law from SO ( C ) to SO ( C ) to calculate the dimen-sion of the zero weight space in an irreducible representation of SO ( C ). Forthis, we again note that the zero weight space in a SO ( C )-representation iscaptured by those subrepresentations of SO ( C ) which have nonzero zeroweight space. Further, note that SO ( C ) being the quotient of SU ( C ) × SU ( C ) by the diagonal central element ±
1, an irreducible representation ofSO ( C ) has nonzero zero weight space if and only if its central character istrivial, and in this case the zero weight space is 1 dimensional. We also needto use the fact that the irreducible representation of SO n ( C ) parametrizedby ( λ , λ , · · · , λ n ) has trivial central character if and only if λ + λ + · · · + λ n is an even integer.With these preliminaries, we leave the details of the straightforward proofof the following lemma to the reader. Lemma (5.2).
An irreducible representation of SO ( C ) with highest weight λ ≥ λ ≥ has zero weight space of dimension (1) ( λ − λ ) · λ + λ + λ + , if λ + λ is an even integer. (2) ( λ − λ ) · λ + λ + λ + , if λ + λ is an odd integer. IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 13
6. T HE E XAMPLE OF
PGL In this section, we compute the dimension of the zero weight space ofany irreducible representation of G = PGL ( C ). Theorem (6.1).
For an irreducible representation of GL ( C ) with highestweight ( λ ≥ λ ≥ λ ≥ λ ) , and with trivial central character, i.e., λ + λ + λ + λ = , the dimension d ( λ , λ , λ , λ ) of the zero weight space is givenas a piecewise polynomial in the domain λ ≥ λ ≥ λ ≥ λ as follows: (1) λ ≤ , where it is given by the polynomialp ( λ , λ , λ , λ ) =
12 ( λ − λ + λ − λ + λ − λ + . (2) λ ≥ , where it is given by the polynomialp ( λ , λ , λ , λ ) =
12 ( λ − λ + λ − λ + λ − λ + . (3) λ > , λ < , λ + λ ≥ , where it is given by the polynomialp ( λ , λ , λ , λ ) = −
12 ( λ + λ + λ + − λ λ + λ + λ λ + λ λ + λ − λ + λ − . (4) λ > , λ < , λ + λ ≤ , where it is given by the polynomialp ( λ , λ , λ , λ ) =
12 ( − λ + λ − − λ λ + λ λ + λ λ + λ − λ − λ − λ − . The automorphism ( λ , λ , λ , λ ) −→ ( − λ , − λ , − λ , − λ ) , which corre-sponds to taking a representation to its dual, interchanges the regions (1)and (2), and their polynomials, and similarly regions (3) and (4) and theirpolynomials. Further, we havep − p = ( λ + λ ) − ( λ + λ ) . Proof.
The method we follow to prove this theorem is also based on therestriction of a GL ( C ) representation to GL ( C ), as we did in the previoussection for the calculation of the zero weight space for GL ( C )-representations.We start with an irreducible representation of GL ( C ) with highest weight( λ ≥ λ ≥ λ ≥ λ ) , and with trivial central character, i.e., λ + λ + λ + λ = . We look at irreducible representations of GL ( C ) with highest weight( µ ≥ µ ≥ µ ) appearing in this representation of GL ( C ). Thus, we have λ ≥ µ ≥ λ ≥ µ ≥ λ ≥ µ ≥ λ . For analyzing the zero weight space, it suffices to consider only those repre-sentations of GL ( C ) with highest weight ( µ , µ , µ ) with µ + µ + µ = ( C ))that complicates our analysis. Denote the dimension of the zero weight space in the irreducible repre-sentation of GL ( C ) with highest weight ( λ ≥ λ ≥ λ ≥ λ ) by d ( λ , λ , λ , λ ).Similarly, denote the dimension of the zero weight space in the irreduciblerepresentation of GL ( C ) with highest weight ( µ , µ , µ ) by d ( µ , µ , µ );we will always assume that the central character of this representation ofGL ( C ) is trivial, and so d ( µ , µ , µ ) is a positive integer, explicitly givenby Lemma 5.1. We remind the reader from loc. cit. that the value of d ( µ , µ , µ ) is a polynomial in ( µ , µ , µ ) (of degree 1) which depends onwhether µ is non-negative or non-positive.Denote the interval [ λ , λ ] by I (we abuse the notation [ λ , λ ] which iscustomarily denoted by [ λ , λ ]), the interval [ λ , λ ] by I , and the interval[ λ , λ ] by I . Our problem consists in choosing integers µ i ∈ I i such that µ + µ + µ = I j + I k ⊂ − I ℓ , for a triple { i , j , k } = { , , } , in which case, one can choose µ j ∈ I j , µ k ∈ I k arbitrarily, and then µ ℓ = − ( µ j + µ k ) automatically belongs to I ℓ . This is what happens in casesI and II below; but the other cases that we deal with in III, . . . , VI, theanalysis is considerably more complicated. Case I: λ ≤
0, and therefore λ ≥ ≥ λ ≥ λ ≥ λ .This implies that µ ≥ ≥ µ ≥ µ ; in particular, in this case µ is always ≤
0. Further, I + I = [ λ + λ , λ + λ ] is contained in − I = [ − λ , − λ ] . Therefore, by Lemma 5.1, d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X µ i ∈ I i ( µ − µ + = X λ ≥ µ ≥ λ λ ≥ µ ≥ λ ( µ − µ + = X λ ≥ µ ≥ λ " ( λ + λ )( λ − λ + − ( µ − λ − λ + = ( λ + λ )( λ − λ + λ − λ + − ( λ − λ + λ + λ − λ − λ + =
12 ( λ − λ + λ − λ + λ − λ + . Case II: λ ≥
0, and therefore λ ≥ λ ≥ λ ≥ ≥ λ .This implies that µ ≥ µ ≥
0; in particular, in this case µ is always ≥ I + I = [ λ + λ , λ + λ ] is contained in − I = [ − λ , − λ ] . IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 15
Therefore, by Lemma 5.1, d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X µ i ∈ I i ( µ − µ + = X λ ≥ µ ≥ λ λ ≥ µ ≥ λ ( µ − µ + =
12 ( λ − λ + λ − λ + λ − λ + . Rest of the cases: λ > > λ , and therefore λ ≥ λ > > λ ≥ λ .Given that µ i ∈ I i with µ + µ + µ =
0, we find that λ ≥ − ( µ + µ ) ≥ λ , and therefore, − λ − µ ≥ µ ≥ − λ − µ . Since we already have λ ≥ µ ≥ λ ,µ is in the intersection of the two intervals − λ − µ ≥ µ ≥ − λ − µ and λ ≥ µ ≥ λ . Therefore, µ must belong to the interval I ( µ ) = [ min ( λ , − λ − µ ) , max ( − λ − µ , λ )] . Conversely, it is clear that if µ ∈ I , µ ∈ I ( µ ) , and µ = − ( µ + µ ), theneach of the µ i belongs to I i , and µ + µ + µ = ( C ) with highest weight ( λ ≥ λ ≥ λ ≥ λ ) , and with trivial central character as d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ > µ ∈ I ( µ ) ( µ − µ + + X ≥ µ ≥ λ µ ∈ I ( µ ) ( µ − µ + . At this point, we assume that λ + λ ≥
0. In this case, if µ ≥
0, then λ ≥ − λ − µ . On the other hand, under the same condition (i.e., λ + λ ≥ µ ≤
0, then − λ − µ ≥ λ . This means that for µ ≥ I ( µ ) = [ min ( λ , − λ − µ ) , max ( − λ − µ , λ )] = [ − λ − µ , max ( − λ − µ , λ )], andfor µ ≤ I ( µ ) = [ min ( λ , − λ − µ ) , − λ − µ ]. Therefore, we get d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ > − λ − µ ≥ µ ≥ max ( − λ − µ , λ ) ( µ − µ + + X ≥ µ ≥ λ min ( λ , − λ − µ ) ≥ µ ≥ − λ − µ ( µ + µ + . At this point, we assume that besides λ + λ ≥
0, we also have 2 λ + λ ≥
0; this latter condition has the effect that the region [ λ ,
0] where µ issupposed to belong, splits into two regions where max ( λ − µ , λ ) takes thetwo possible options. Similarly, the region [0 , λ ] where µ belongs in thesecond sum gets divided into two regions. Case III: λ + λ ≥ λ + λ ≥ d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ ≥ − ( λ + λ ) − λ − µ ≥ µ ≥ λ ( µ − µ + + X − ( λ + λ ) > µ > − λ − µ ≥ µ ≥ − λ − µ ( µ − µ + + X ≥ µ ≥ − ( λ + λ ) − λ − µ ≥ µ ≥ − λ − µ ( µ + µ + + X − ( λ + λ ) > µ ≥ λ λ ≥ µ ≥ − λ − µ ( µ + µ + =
14 [ − λ − λ λ − λ − λ λ + λ λ λ − λ λ − λ λ − λ − λ λ − λ λ λ − λ λ λ − λ λ λ − λ λ + λ λ + λ λ + λ − λ − λ λ + λ − λ λ + λ λ + λ − λ λ − λ λ − λ λ − λ + λ + λ + λ − λ + =
14 [2 λ λ − λ λ − λ − λ λ − λ λ − λ λ − λ λ − λ + λ + λ λ − λ − λ λ − λ + λ + λ + λ + , = −
12 ( λ + λ + λ + − λ λ + λ + λ λ + λ λ + λ − λ + λ − , where in the second last equality, we have used the equation λ = − ( λ + λ + λ ) to write the polynomial in only λ , λ , λ . IMENSION OF ZERO WEIGHT SPACE: AN ALGEBRO-GEOMETRIC APPROACH 17
Case IV: λ + λ ≥ λ + λ < d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ > − λ − µ ≥ µ ≥ − λ − µ ( µ − µ + + X ≥ µ ≥ − ( λ + λ ) − λ − µ ≥ µ ≥ − λ − µ ( µ + µ + + X − ( λ + λ ) > µ ≥ λ λ ≥ µ ≥ − λ − µ ( µ + µ + =
14 [ − λ − λ λ − λ λ + λ λ λ − λ λ − λ λ − λ λ − λ − λ λ − λ λ λ + λ λ − λ λ λ − λ λ λ − λ λ + λ λ − λ λ + λ − λ − λ λ − λ + λ λ + λ − λ λ − λ λ − λ + λ − λ + λ − λ + =
14 [2 λ λ − λ λ − λ − λ λ − λ λ − λ λ − λ λ − λ + λ + λ λ − λ − λ λ − λ + λ + λ + λ + = −
12 ( λ + λ + λ + − λ λ + λ + λ λ + λ λ + λ − λ + λ − , where again in the second last equality, we have used the equation λ = − ( λ + λ + λ ) to write the polynomial in only λ , λ , λ . Case V: λ + λ < λ + λ ≤ d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ > ( λ + λ ) − λ − µ ≥ µ ≥ λ ( µ − µ + + X ( λ + λ ) ≥ µ ≥ λ ≥ µ ≥ λ ( µ − µ + + X > µ > ( λ + λ ) λ ≥ µ ≥ λ ( µ + µ + + X λ + λ ≥ µ ≥ λ λ ≥ µ ≥ − λ − µ ( µ + µ + =
14 [ − λ + λ λ + λ λ + λ − λ λ + λ λ λ + λ λ − λ λ + λ λ + λ λ λ + λ λ + λ λ λ − λ λ λ + λ λ + λ λ + λ λ + λ − λ + λ − λ λ + λ λ − λ − λ λ − λ λ − λ + λ − λ + λ − λ + =
14 [2 λ λ − λ λ − λ λ + λ λ − λ λ + λ λ + λ + λ λ − λ − λ λ − λ + λ + λ + =
12 ( − λ + λ − − λ λ + λ λ + λ λ + λ − λ − λ − λ − . Case VI: λ + λ < λ + λ ≥ d ( λ , λ , λ , λ ) = X µ i ∈ I i d ( µ , µ , µ ) = X λ ≥ µ > ( λ + λ ) − λ − µ ≥ µ ≥ λ ( µ − µ + + X ( λ + λ ) ≥ µ ≥ λ ≥ µ ≥ λ ( µ − µ + + X > µ ≥ λ λ ≥ µ ≥ λ ( µ + µ + =
14 [ − λ + λ λ + λ λ + λ − λ λ + λ λ λ + λ λ − λ λ + λ λ + λ λ λ + λ λ + λ λ λ − λ λ λ + λ λ + λ λ + λ λ + λ − λ + λ − λ λ + λ λ − λ − λ λ − λ λ − λ + λ − λ + λ − λ + =
14 [2 λ λ − λ λ − λ λ + λ λ − λ λ + λ λ + λ + λ λ − λ − λ λ − λ + λ + λ + =
12 ( − λ + λ − − λ λ + λ λ + λ λ + λ − λ − λ − λ − . (cid:3) Remark (6.2). (a) The fact that the answers in the cases III and IV aboveare the same (similarly in the cases V and VI) seems not obvious aprioribefore the final answer is calculated via a software.(b) It is curious to note that the polynomials p and p are equal for λ =
0, and the polynomials p and p are equal for λ =
0, but the polynomials p and p are not equal for λ =
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