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5 Coadjoint orbits for A+ ,B+, and D+ 0 n−1 n n 0 2 Shantala Mukherjee ∗ n a Dept. of Mathematics J DePaul University 0 Chicago, IL 60614 2 [email protected] ] T R . h t Abstract a m A complete description of the coadjoint orbits for A+ , the nilpo- n−1 [ tentLiealgebraofn×nstrictlyuppertriangularmatrices,hasnotyet been obtained, though there has been steady progress on it ever since 1 theorbitmethodwasdevised. We applymethods developedbyAndré v 2 to find defining equations for the elementary coadjoint orbits for the 3 maximalnilpotent Lie subalgebrasof the orthogonalLie algebras,and 3 wealsodetermineallthepossibledimensionsofcoadjointorbitsinthe 1 case of A+ . 0 n−1 5 0 MSC2000: 17B30,17B35 / h t 1 Introduction a m : The orbit method was created by Kirillov in an attempt to describe the v i unitary dual Nˆn for the nilpotent Lie group Nn of n×n upper triangular X matrices with 1’s on the diagonal (the unitriangular group). It turned out r a that the orbit method had much wider applications. In Kirillov’s words: ‘...all main questions of representation theory of Lie groups: construction ofirreduciblerepresentations,restriction-inductionfunctors,generalizedand infinitesimal characters, Plancherel measure, etc., admit a transparent description in terms of coadjoint orbits’ ([K]). ∗This work is part of the author’s doctoral dissertation, written at the University of Wisconsin-Madison under the supervision of Prof. Georgia Benkart, and financially supported in part byNSFgrant #DMS-0245082 1 The Lie algebra of N is n , which consists of all n×n strictly upper n n triangular matrices. The group N acts on the dual space n∗ by the coad- n n joint action, which will be explained later. A complete description of the set of coadjoint orbits n∗/N ≃ Nˆ in general is still not available, though n n n progress had been made in the case of the unitriangular group over a finite field. André in [A2] defined “basic sums” of elementary coadjoint orbits and their defining equations for the unitriangular group over an arbitrary field and showed that the dual space n∗ is a disjoint union of these basic n sums of orbits. Similar results have been obtained by N.Yan, in his work on double orbits and cluster modules ([Y]) of the unitriangular group over a finite field. Using André’s results, we determine all possible dimensions of the coadjoint orbits of the unitriangular group over C. Elementary coadjoint orbits (defined later) are in some sense the “smallest” coadjoint orbits. Adapting the methods of André, we derive the defining equations for the elementary coadjoint orbits in the case that the Lie algebra is a maximal nilpotentsubalgebraofanorthogonalLiealgebraover thefieldCofcomplex numbers. The case of the symplectic Lie algebra has so far failed to yield a consistent pattern, therefore it is not discussed here. 2 The Lie algebras A+ , B+, D+ n−1 n n LetL = sl (C), theLiealgebra ofn×ncomplex matrices of trace0. LetE n ij denotethestandardmatrixunitwith1inthe(i,j) positionand0elsewhere. Thus L has a basis {E | 1≤ i 6= j ≤ n}∪{E −E | i= 1,...,n−1}. ij ii i+1,i+1 Relative to the Cartan subalgebra h spanned by the diagonal matrices E − E , i = 1,...,n −1; L decomposes into root spaces (common ii i+1,i+1 eigenspaces). Thus L = h⊕ L , ǫi−ǫj 1≤i6=j≤n M where ǫ : h → C denotes the projection onto the (i,i) entry, and i L = {x ∈ L |[h,x] = (ǫ −ǫ )(h)x ∀ h ∈ h} = CE . ǫi−ǫj i j ij The roots ǫ −ǫ are linear combinations of the simple roots ǫ − ǫ ,ǫ − i j 1 2 2 ǫ ,...,ǫ −ǫ , and the coefficients are either all nonpositive or all non- 3 n−1 n negative integers. The positive roots are given by Φ+(A )= {ǫ −ǫ | 1 ≤ i< j ≤ n}. n−1 i j 2 Thus, there are 1(n−1)n positive roots. 2 The sum L of the root spaces corresponding to the posi- α∈Φ+(An−1) α tive roots is the nilpotent Lie algebra n of strictly upper triangular matri- n L ces. Here we denote this Lie algebra by A+ . It has a basis of root vectors n−1 {e |α ∈ Φ+(A )} where e = E for α = ǫ −ǫ , 1 ≤ i < j ≤n. α n−1 α ij i j Now we establish our conventions for the root systems B and D . The n n following definitions can be found in [FH, Sec. 18.1]. Let J be the m×m m matrix with 1’s along the antidiagonal and 0’s elsewhere. The orthogonal Lie algebra so (C) is the Lie algebra of m×m matrices X satisfying the m relation XtJ +J X = 0. Thus the matrices in so (C) are antisymmetric m m m about the antidiagonal. Relative to the Cartan subalgebra h spanned by the diagonal matrices E −E , the odd orthogonal Lie algebra L = so (C) decom- ii 2n+2−i,2n+2−i 2n+1 poses into root spaces L = h⊕ L ⊕ L , ǫi±ǫj ǫi 1≤i6=j≤n 1≤i≤n M M where ǫ : h → C denotes the projection onto the (i,i) entry. The roots are i linear combinations of the simple roots ǫ −ǫ ,ǫ −ǫ ,...,ǫ −ǫ ,ǫ , with 1 2 2 3 n−1 n n coefficients that are either all nonnegative or all nonpositive integers. The positive roots are given by Φ+(B ) = {ǫ ±ǫ | 1≤ i < j ≤ n}∪{ǫ | 1 ≤ i≤ n} n i j i and |Φ+(B )| = n2. n We partition the set of positive roots into two subsets Definition 2.1. Φ+(B )= {ǫ −ǫ | 1 ≤ i< j ≤ n}∪{ǫ |i = 1,...,n} 1 n i j i Definition 2.2. Φ+(B )= {ǫ +ǫ | 1 ≤ i< j ≤ n} 2 n i j The sum of spaces L is a finite-dimensional nilpotent Lie α∈Φ+(Bn) α algebra consisting of all strictly upper triangular matrices of size 2n + 1 L which are anti-symmetric about the antidiagonal (and have zeroes on the antidiagonal). We denote this Lie algebra by B+. It has a basis of root n vectors {e |α ∈ Φ+(B )} where α n E −E , if α = ǫ −ǫ , 1 ≤ i< j ≤ n; ij 2n+2−j,2n+2−i i j e = E −E , if α = ǫ , 1≤ i ≤ n; (1) α  i,n+1 n+1,2n+2−i i E −E , if α = ǫ +ǫ , 1 ≤ i< j ≤ n. i,2n+2−j j,2n+2−i i j   3 Relative to the Cartan subalgebra h spanned by the matrices H = E − i ii E , the even orthogonal Lie algebra L = so (C) decomposes 2n+1−i,2n+1−i 2n into root spaces L = h⊕ L ǫi±ǫj 1≤i6=j≤n M where ǫ : h → C denotes the projection onto the (i,i) entry. The roots are i linearcombinationsofthesimplerootsǫ −ǫ ,ǫ −ǫ ,...,ǫ −ǫ ,ǫ +ǫ , 1 2 2 3 n−1 n n−1 n andthecoefficientsareeitherallnonnegativeorallnonpositiveintegers. The positive roots are given by Φ+(D )= {ǫ ±ǫ |1 ≤ i< j ≤ n} n i j and |Φ+(D )|= n2−n. n The elements of Φ+(D ) can be partitioned into two subsets: n Definition 2.3. Φ+(D ) ={ǫ −ǫ |1 ≤ i < j ≤ n}, 1 n i j Definition 2.4. Φ+(D ) ={ǫ +ǫ |1 ≤ i < j ≤ n} 2 n i j The sum of root spaces L is a finite-dimensional nilpotent α∈Φ+(Dn) α Lie algebra consisting of all strictly upper triangular matrices of size 2n L which are anti-symmetric about the antidiagonal (and have zeroes on the antidiagonal). We denote this Lie algebra by D+. It has a basis of root n vectors {e |α ∈ Φ+(D )} where α n E −E , if α = ǫ −ǫ , 1 ≤ i< j ≤ n; ij 2n+1−j,2n+1−i i j e = (2) α (Ei,2n+1−j −Ej,2n+1−i, if α = ǫi+ǫj, 1 ≤ i< j ≤ n. 3 Singular and Regular Roots For a positive root α in any one of the sets Φ+(A ),Φ+(B ), or Φ+(D ), n−1 n n we will define two sets: S(α), the set of α-singular roots; and R(α), the set of α-regular roots. The set S(α) is the union of all pairs of positive roots which sum up to α. For a positive root of the form ǫ − ǫ , 1 ≤ i < j ≤ n, in the sets i j Φ+(A ), Φ+(B ) , and Φ+(D ), we have n−1 1 n 1 n j−1 {ǫ −ǫ ,ǫ −ǫ }, if j −i> 1 S(ǫ −ǫ )= k=i+1 i k k j i j (∅S, otherwise We see that |S(ǫ −ǫ )| = 2(j −i−1). i j 4 For a positive root of the form ǫ , 1 ≤ i≤ n, in Φ+(B ) we have i 1 n n {ǫ −ǫ ,ǫ }, if 1 ≤ i≤ n−1 S(ǫ )= k=i+1 i k k i (∅S, if i = n. Here |S(ǫ )| = 2(n−i). i For the roots ǫ +ǫ , 1 ≤ i< j ≤ n, in Φ+(B ) we have: i j 2 n j−1 n S(ǫ +ǫ ) = {ǫ −ǫ ,ǫ +ǫ }∪ {ǫ −ǫ ,ǫ +ǫ } i j i k k j i k j k k=i+1 k=j+1 [ [ (3) n ∪{ǫ ,ǫ }∪ {ǫ +ǫ ,ǫ −ǫ } i j i k j k k=j+1 [ Notice that |S(ǫ +ǫ )| = 2(2n−(i+j)). i j For the positive roots ǫ +ǫ , 1 ≤ i < j ≤ n, in Φ+(D ) we have: i j 2 n j−1 n S(ǫ +ǫ ) = {ǫ −ǫ ,ǫ +ǫ }∪ {ǫ −ǫ ,ǫ +ǫ } i j i k k j i k j k k=i+1 k=j+1 [ [ (4) n ∪ {ǫ +ǫ ,ǫ −ǫ } i k j k k=j+1 [ In this case, |S(ǫ +ǫ )| = 2(2n−i−j −1). i j For α ∈ Φ+(A ),Φ+(B ), or Φ+(D ), we define R(α) to be the com- n−1 n n plement of the set S(α) in the respective set of positive roots. Clearly, α ∈ R(α). 4 Elementary Coadjoint Orbits Thedefinitionsandresultsinthissubsectionaretakenfrom[A2,Sec.1]. Let Φ+ denote one of the sets of positive roots Φ+(A ),Φ+(B ), or Φ+(D ). n−1 n n Let g denote the correspondingnilpotent Lie algebra A+ ,B+, or D+. The n−1 n n group G = exp(g) acts on the dual space g∗ by the coadjoint action (g.f)(x) = f(g−1xg), ∀g ∈ G, f ∈ g∗, x∈ g. Then, by [D, Thm. 6.2.4], there is a one-one correspondence between the G-orbits in g∗ and the primitive ideals of U(g), which are the annihilators 5 of the simple U(g)-modules, constructed as follows: For any f in the G- orbit, a simple g-module is obtained by inducing a one-dimensional module of a maximal subalgebra of g that is subordinate to f, up to a g-module ([D, Thm. 6.1.1]). The annihilator in U(g) of this module is the primitive ideal I(f) that corresponds to the G-orbit Ω of f. By [D, Thm. 4.7.9(iii)], f factoring U(g) by a primitive ideal gives a Weyl algebra A , which is the m non-commutative algebra of algebraic differential operators on a polynomial ring C[x ,...,x ]. 1 m Definition 4.1. gf = {x ∈ g | f([x,y]) = 0 ∀y ∈ g} is the radical of the form f. Then dimg/gf = dimΩ = 2m, where U(g)/I(f) = A . f m For any α ∈ Φ+, let e∗ denote the element of the dual vector space g∗ α defined as follows: for any β ∈ Φ+, 1, if α = β; e∗(e ) = α β (0, otherwise. Then{e∗ |α ∈ Φ+}is a basis of g∗. Let c∈ C benon-zero. Then,underthe α coadjoint action of the group G (= exp(g)) on g∗, the coadjoint orbit O (c) α that contains the element ce∗ is called the α-th elementary orbit associated α with c. Note that if f = ce∗, then gf = {x ∈ g | f([x,y]) = 0 ∀ y ∈ g} is α spanned by the e , β ∈ R(α), so β dim(O (c)) = dim(g/gf) = dimg−dimgf = |Φ+|−|R(α)| = |S(α)|. (5) α Let t be an arbitrary scalar in C and let β ∈ Φ+ . Then, the matrix exp(te )∈ G. So, by the definition of the coadjoint representation, we have β for any γ ∈Φ+: exp(te ).e∗(e )= e∗(exp(−te )e exp(te )) β α γ α β γ β = e∗(exp(ad(−te )(e )) α β γ (6) 1 = e∗(e −t[e ,e ]+ t2[e ,[e ,e ]]+...). α γ β γ 2 β β γ It is clear that for any simple root α ∈ Φ+ and any c∈ C, the coadjoint i orbit containing ce∗ is equal to {ce∗ }, because ce∗ ([g,g]) = 0. αi αi αi By Prop. 8.2 in [H], any coadjoint orbit is an irreducible variety in g∗, so in particular, the elementary coadjoint orbit O (c) is an irreducible variety α of dimension |S(α)|. 6 André in [A1, Lem. 2] describes the elementary orbit O (c), for any α α ∈ Φ+(A ) and any non-zero scalar c. n−1 If g ∈ G, then g.(ce∗)= c(g.e∗), so α α 1 f ∈ O (c) if and only if f ∈ O (1). α α c Thus, it is enough to determine the defining equations for O (1). Adapting α André’s proof,weobtain thedefiningequations forelementary orbits O (1), α where α ∈ Φ+(A ),Φ+(B ), or Φ+(D ): n−1 1 n 1 n Theorem 4.2. (a) Letα= ǫ −ǫ ∈Φ+(A ),Φ+(B ), orΦ+(D ), where i j n−1 1 n 1 n 1 ≤ i < j ≤ n. Let g denote the corresponding nilpotent Lie algebra A+ ,B+, or D+. Then O (1) consists of all elements f ∈ g∗ which n−1 n n α satisfy the equations 1, if β = α; f(e ) = f(e )f(e ), if β = ǫ −ǫ , i < r < s < j; (7) β  ǫi−ǫs ǫr−ǫj r s 0 otherwise, for β ∈ R(α), and f takes arbitrary values on e for β ∈ S(α). β (b) Let g denote the nilpotent Lie algebra B+. For ǫ ∈ Φ+(B ), 1 ≤ i ≤ n, n i 1 n the elementary orbit O (1) consists of all elements f ∈ g∗ that satisfy ǫi the equations 1, if β = ǫ ; i f(e )= f(e )f(e ), if β = ǫ −ǫ , i < r < s≤ n; (8) β  ǫi−ǫs ǫr r s  0 otherwise, for β ∈ R(ǫi), and f takes arbitrary values on eβ for β ∈ S(ǫi). Proof. (a) Let V be the variety in g∗ consisting of all f ∈ g∗ that satisfy the equations (7) and let f ∈ V. Then j−1 j−1 f = exp(f(e )e ) exp(−f(e )e ) .e∗ ∈ O (1), ǫi−ǫk ǫk−ǫj ǫk−ǫj ǫi−ǫk α α ! k=i+1 k=i+1 Y Y so V ⊆ O (1). To show that equality holds, let T : V → C2(j−i−1) be the α map defined by applying f to pairs in S(α) as follows: T(f)= f(e ),f(e ),...,f(e ),f(e ) ǫi−ǫi+1 ǫi+1−ǫj ǫi−ǫj−1 ǫj−1−ǫj (cid:0) (cid:1) 7 for all f ∈ V. (For example, if g = A+ and α= ǫ −ǫ , then 3 1 4 T(f)= (f(e ),f(e ),f(e ),f(e )).) ǫ1−ǫ2 ǫ2−ǫ4 ǫ1−ǫ3 ǫ3−ǫ4 Then T is an isomorphism of algebraic varieties, and since C2(j−i−1) is an irreducible variety, it follows that V is irreducible and dimV = 2(j−i−1). The coadjoint orbit O (1) is also an irreducible algebraic variety of di- α mension 2(j −i−1). We have two irreducible varieties V and O (1) of the α same dimension and V ⊆ O (1), so it follows that V = O (1). α α (b) Let V be the variety in g∗ consisting of all f ∈ g∗ that satisfy the equations (8). Let f ∈V. Then n n f = exp(f(e )e ) exp(−f(e )e ) .e∗ ∈ O (1), ǫi−ǫk ǫk ǫk ǫi−ǫk α α ! k=i+1 k=i+1 Y Y so V ⊆ O (1). To show that equality holds, let T : V → C2(n−i) be the map α defined by: T(f)= f(e ),f(e ),...,f(e ),f(e ) ǫi−ǫi+1 ǫi+1 ǫi−ǫn ǫn (cid:0) (cid:1) for all f ∈ V. (For e.g., if g = B+ and α= ǫ , then 3 1 T(f)= (f(e ),f(e ),f(e ),f(e )).) ǫ1−ǫ2 ǫ2 ǫ1−ǫ3 ǫ3 Then T is an isomorphism of algebraic varieties, and because C2(n−i) is an irreducible variety, V is irreducible and dimV =2(n−i). The coadjoint orbit O (1) is also an irreducible algebraic variety of di- α mension 2(n−i). We have two irreduciblevarieties V andO (1) of the same α dimension and V ⊆ O (1), so it follows that V = O (1). α α Next, we can describe the defining equations of the elementary orbit O (1) for the positive roots α = ǫ + ǫ , 1 ≤ i < j ≤ n in Φ+(B ) or α i j 2 n Φ+(D ). 2 n Theorem 4.3. (i) Let g = B+. For a positive root ǫ +ǫ , 1 ≤ i < j ≤ n n i j 8 in Φ+(B ), f ∈ O (1) if and only if f satisfies: 2 n ǫi+ǫj 1 if β = ǫ +ǫ ; i j f(e )f(e ) if β = ǫ −ǫ , i≤ r < s ≤ j;  ǫi−ǫs ǫr+ǫj r s f(eǫr+ǫj) −12f(eǫi)2+ nk=j+1(−1)kf(eǫi−ǫk)f(eǫi+ǫk)   (cid:16) if β =Pǫr −ǫj, i ≤ r < j; (cid:17)   f(e )f(e ) if β = ǫ ±ǫ , i< r < j < s ≤ n; f(e )=  ǫi±ǫs ǫr+ǫj r s β   f(e )f(e )−f(e )f(e )  ǫj±ǫs ǫi+ǫr ǫi±ǫs ǫj+ǫr if β = ǫ ±ǫ , j < r < s ≤ n; r s  f(e )f(e ) if β = ǫ , i< r < j;  ǫi ǫr+ǫj r  f(e )f(e )−f(e )f(e ) if β = ǫ , j < r ≤ n;  ǫj ǫi+ǫr ǫi ǫj+ǫr r  0 otherwise,     (9)   for all β ∈ R(ǫ +ǫ ), and f takes arbitrary values on e for all β ∈ i j β S(ǫ +ǫ ). i j (ii) Let g = D+. For a positive root ǫ + ǫ , 1 ≤ i < j ≤ n in Φ+(D ), n i j 2 n f ∈ O (1) if and only if f satisfies: ǫi+ǫj 1 if β = ǫ +ǫ ; i j f(e )f(e ) if β = ǫ −ǫ , i ≤r < s ≤ j;  ǫi−ǫs ǫr+ǫj r s f(eǫr+ǫj) nk=j+1(−1)kf(eǫi−ǫk)f(eǫi+ǫk)  f(e )=  (cid:16)P if β = ǫr −ǫj, i ≤ r <(cid:17)j; β   f(e )f(e ) if β = ǫ ±ǫ , i <r < j < s ≤ n;  ǫi±ǫs ǫr+ǫj r s f(e )f(e )−f(e )f(e ) ǫj±ǫs ǫi+ǫr ǫi±ǫs ǫj+ǫr   if β = ǫ ±ǫ , j < r < s ≤ n;  r s   0 otherwise,     (10)   for all β ∈ R(ǫ +ǫ ), and f takes arbitrary values on e for all β ∈ i j β S(ǫ +ǫ ). i j 9 Proof. (i) Recall that j−1 n S(ǫ +ǫ ) = {ǫ −ǫ ,ǫ +ǫ }∪ {ǫ −ǫ ,ǫ +ǫ } i j i k k j i k j k k=i+1 k=j+1 [ [ n ∪{ǫ ,ǫ }∪ {ǫ +ǫ ,ǫ −ǫ }, i j i k j k k=j+1 [ and |S(ǫ +ǫ )| = 2(2n−(i+j)) = dimO (1). The set S(ǫ +ǫ ) can be i j ǫi+ǫj i j written as the disjoint union of two subsets S and S defined as follows: (i) (j) j−1 n n S = {ǫ −ǫ }∪ {ǫ −ǫ }∪{ǫ }∪ {ǫ +ǫ } (i) i k i k i i k k=i+1 k=j+1 k=j+1 [ [ [ and j−1 n n S = {ǫ +ǫ }∪ {ǫ +ǫ }∪{ǫ }∪ {ǫ −ǫ }. (j) k j j k j j k k=i+1 k=j+1 k=j+1 [ [ [ For γ ∈ S , let γ′ be the unique element of S such that γ+γ′ = ǫ +ǫ . (i) (j) i j Let n(γ,γ′) be the integer such that [eγ,eγ′]= n(γ,γ′)eǫi+ǫj It follows from equations (1) that n(γ,γ′)= ±1. LetV bethevarietying∗ consistingofallf ∈ g∗ satisfyingequations(9). Let f ∈ V. Then f =  exp(n(γ,γ′)f(eγ)eγ′) exp(−n(γ,γ′)f(eγ′)eγ).e∗ǫi+ǫj, γ,γ′ γ,γ′ Y Y   where γ ∈S and γ′ ∈ S . (i) (j) Thus V ⊆ O (1). To show equality, let us define a map T : V → ǫi+ǫj C2(2n−(i+j)) by T(f)= f(eγ1),f(eγ1′),f(eγ2),f(eγ2′),...,f(eγ2n−(i+j)),f(eγ2′n−(i+j)) (cid:16) (cid:17) where γ ∈ S , for all k. k (i) (For example, if we have ǫ +ǫ ∈ Φ+(B ), then 1 3 2 3 T(f)= (f(e ),f(e ),f(e ),f(e )).) ǫ1−ǫ2 ǫ2+ǫ3 ǫ1 ǫ3 Then T is an isomorphism of algebraic varieties, hence V is irreducible and dimV = 2(2n−(i+j)). 10