Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures
aa r X i v : . [ m a t h . R T ] D ec QUANTIZATION OF SPECIAL SYMPLECTIC NILPOTENT ORBITS ANDNORMALITY OF THEIR CLOSURES
KAYUE DANIEL WONGA
BSTRACT . We study the regular function ring R ( O ) for all symplectic nilpotent orbits O with even column sizes. We begin by recalling the quantization model for all such orbits byBarbasch using unipotent representations. With this model, we express the multiplicitiesof fundamental representations appearing in R ( O ) by a parabolically induced module.Finally, we will use this formula to give a criterion on the normality of the Zariski closure O of O .
1. I
NTRODUCTION
Let G = Sp (2 n, C ) be the complex symplectic group. The G -conjugates of a nilpotentelement X ∈ g form a nilpotent orbit O ⊂ g . The idea of the orbit method, first pro-posed by Kirillov, suggests that one can ‘attach’ a unitary representation to each nilpotentorbit O . In [30], the author used tools from unipotent representations and dual pair cor-respondence to achieve this goal for spherical, special nilpotent orbits and their covers(see Theorem A of [30]). In the following work, we would like to study a larger class ofnilpotent orbits.It is well-known that all such nilpotent orbits are parameterized by partitions, wherethe partition corresponds to the size of the Jordan blocks. For Sp (2 n, C ) , nilpotent orbitsare identified with the partitions of n in which odd parts occur with even multiplicity.In fact it is sometimes more convenient to look at the column sizes, or the dual parti-tion , of a given partition. More precisely, let ψ = [ r , r , . . . , r i ] be a partition of n , with r ≥ r ≥ · · · ≥ r i > , its dual partition is given by ψ ∗ = ( c k , c k − , . . . , c ) , where c k +1 − j = { i | r i ≥ j } . We will use square bracket [ r , r , . . . ] to denote the partition of anilpotent orbit, and round bracket ( c k , c k − , . . . ) to denote the dual partition of the sameorbit.Given two partitions ς = [ r , . . . , r p ] , ψ = [ r ′ , . . . , r ′ q ] , we define their union ς ∪ ψ =[ s , . . . , s p + q ] , where { s . . . , s p + q } = { r , . . . , r p , r ′ , . . . , r ′ q } as sets and s ≥ s ≥ · · · ≥ s p + q . Also, define the join to be ς ∨ ψ = ( ς ∗ ∪ ψ ∗ ) ∗ , so that if ς = ( c m , . . . , c ) , ψ =( c ′ n , . . . , c ′ ) , then ς ∨ ψ = ( d m + n +1 , . . . , d ) , where { d m + n +1 . . . , d } = { c m , . . . , c , c ′ n , . . . , c ′ } as sets and d m + n +1 ≥ d m + n ≥ · · · ≥ d .Here is a restatement of the characterization of nilpotent orbits for Sp (2 n, C ) , whichis implicit in the construction of nilpotent orbit closures in [18]: Any nilpotent orbit for Sp (2 n, C ) can be parameterized by a partition of n with column sizes ( c k , c k − , . . . , c ) ,where c k ≥ c k − ≥ · · · ≥ c ≥ (by insisting c k to be the longest column, we put c = 0 if necessary), such that c i + c i − is even for all i ≥ (where c − = 0 ).For most parts of the following work, we study the ring of regular functions R ( O ) for O = (2 a k , . . . , a , a ) , i.e. all columns of O are even. For example, the following isknown to be true: Theorem 1.1 (Barbasch - [9] and [10], p.29) . Let O = (2 a k , . . . , a , a ) be a nilpotent orbitsuch that a i − > a i − for all i . Then as G ∼ = K C -modules, the spherical unipotent representation X triv attached to O satisfies X triv ∼ = R ( O ) . In fact, Barbasch in [9] proved a much more general statement than Theorem 1.1 forother classical Lie groups and other unipotent representations. More specifically, the otherunipotent representations X π (see Equation (2)) corresponds to the global sections of some G -equivariant vector bundle G × G e V π of O . This essentially verifies a conjecture of Vogan(Conjecture 12.1 of [26]) for such orbits. More details are given in Remarks 3.3.With Theorem 1.1, we can essentially compute the multiplicity of any irreducible repre-sentations appearing in R ( O ) . In particular, we focus on the fundamental representationsof G = Sp (2 n, C ) , given by µ i := ∧ i C n / ∧ i − C n for i = 1 , , . . . , n (if i − < , take ∧ i − C n = triv ). We have the following formula forthe multiplicities of fundamental representations for a larger class of nilpotent orbits thanin Theorem 1.1: Theorem A.
Let O = (2 a k , . . . , a , a ) be a nilpotent orbit for G . Remove all column pairsof same size ( α i , α i ) in O , leaving the orbit ( d l , d l − , . . . , d ) , i.e. O = ( d l , d l − , . . . , d ) ∨ ( α , α , . . . , α x , α x ) with d i +1 = d i for all i . Then the multiplicities of the fundamental representations µ i are given by [ R ( O ) : µ i ] = [ Ind Sp (2 n, C ) GL ( D ) (triv) : µ i ] , YMPLECTIC NILPOTENT ORBITS 3 where GL ( D ) = Π li =0 GL ( d i + d i − ) × Π xi =1 GL ( α i ) . For example, let O = (8 , , , , , ,
2) = (8 , , ∨ (4 , , , . So [ R ( O ) : µ i ] = [ Ind Sp (26) GL (6) × GL (1) × GL (4) × GL (2) (triv) : µ i ] for all i = 0 , . . . , . More examples can be found in Example 5.2.The second main theorem concerns about the Zariski closure O of O . In particular, weare interested in the normality of O . It is known in [17] that in the case of SL ( n, C ) , allnilpotent orbit closures are normal. For Sp (2 n, C ) , Kraft and Procesi proved the following: Theorem 1.2 (Kraft-Procesi - [18]) . Let O = ( c k , c k − , . . . , c ) be a nilpotent orbit for G = Sp (2 n, C ) . If there is a chain of column lengths of the form c i = c i − = c i − = · · · = c j − = c j − = c j − , then O is not normal. For instance, for G = Sp (2 n, C ) , the orbit closures (8 , , , , (6 , , , are normal,while (8 , , , is not normal. On the other hand, there is an algebro-geometric criterionof normality. Namely, Proposition 8.2 of [6] says R ( O ) ∼ = R ( O ) if and only if O is normal.Our second Theorem gives a more refined criterion on normality of O : Theorem B.
Let O = (2 a k , . . . , a , a ) be a nilpotent orbit for Sp (2 n, C ) . Then O is notnormal iff there exists a fundamental representation µ i such that [ R ( O ) : µ i ] < [ R ( O ) : µ i ] .
2. F
UNDAMENTAL G ROUP AND L USZTIG ’ S QUOTIENT OF N ILPOTENT O RBITS
In this Section, we focus on the structure of nilpotent orbits for G = Sp (2 n, C ) . Moreprecisely, we compute the G - equivariant fundamental group A ( O ) and the Lusztig’s quotient A ( O ) of a nilpotent orbit for G . All the materials presented in this Section can be found in[14], [24], [25].Let O be a nilpotent orbit for G , and e ∈ O . Then the stabilizer group G e are theelements in G keeping e fixed, i.e. G e = { g ∈ G | g · e = e } . Following [14], define the G - equivariant fundamental group of O as A ( O ) := G e / ( G e ) , where H is the identity component of a group H . The calculation of A ( O ) for G = Sp (2 n, C ) can be tracked easily from [14, Chapter 5] or [24] as follows: KAYUE DANIEL WONG
Proposition 2.1.
Let G = Sp (2 n, C ) and O is a nilpotent orbit for G . Then A ( O ) = ( Z / Z ) b , where b is thenumber of distinct even elements in the partition of O . To cater for our forthcoming calculations, it is desirable to express A ( O ) using dual partitions: Proposition 2.2.
Let O = ( c k , . . . , c , c ) be a nilpotent orbit in Sp (2 n, C ) . Remove all column pairs c i − = c i − , and let ( d j , . . . , d , d , d ) be the remaining columns, i.e. O = ( c k , . . . , c , c ) = ( d j , . . . , d , d , d ) ∨ ( ν , ν , . . . , ν y , ν y ) , with d i − = d i − for all i . Then A ( O ) ∼ = ( Z / Z ) j is generated by { s j − , s j − , . . . , s , s } . For example, (8 , , , , , , has trivial fundamental group, while (8 , , , , , , , has fundamental group isomorphic to ( Z / Z ) . Proof.
Let O = O ∗ ∨ ( ν , ν , . . . , ν y , ν y ) , with O ∗ = ( d j , . . . , d , d , d ) as in the Proposition. Then A ( O ) and A ( O ∗ ) have the samesize - Indeed, by removing ( c i − , c i − ) = ( ν, ν ) from O , the partition description of O changes in the form: [ r ≥ · · · ≥ r ν − ≥ r ν > r ν +1 ≥ . . . ] → [ r − ≥ · · · ≥ r ν − − ≥ r ν − ≥ r ν +1 ≥ . . . ] with r ν > r ν +1 + 2 , or r ν = r ν +1 + 2 with r ν ≡ r ν +1 ≡ . In both cases, the numberof distinct even numbers before and after removal are equal. Hence Proposition 2.1 saysthe G -equivariant fundamental groups are the same.We now focus on finding A ( O ∗ ) . Suppose the partition description of O ∗ is [ r ∗ ≥ r ∗ ≥· · · ≥ r ∗ d j > . Then r ∗ i must satisfy the following: • If one of r ∗ i and r ∗ i +1 is odd, r ∗ i − r ∗ i +1 = 0 or ; • If both r ∗ i and r ∗ i +1 are even, r ∗ i − r ∗ i +1 = 0 or ; and • r ∗ d j ≤ .Therefore the number of distinct even elements in O ∗ is equal to the number of evenelements in { , , . . . , r ∗ } , which is ⌈ r ∗ ⌉ = j . Consequently A ( O ) ∼ = A ( O ∗ ) = ( Z / Z ) j . (cid:3) There is a quotient group A ( O ) of A ( O ) , first introduced by Lusztig, that plays a vitalrole in the theory of unipotent representations. We will follow [25] to determine A ( O ) under our dual partition notation. YMPLECTIC NILPOTENT ORBITS 5
Proposition 2.3.
Let O = ( c k , . . . , c , c ) be a nilpotent orbit for Sp (2 n, C ) . Remove all columnpairs ( c i − , c i − ) = ( ν, ν ) , along with all odd columns c j = 2 µ + 1 . Let ( d ′ p , . . . , d ′ , d ′ , d ′ ) bethe remaining columns, i.e. O = ( c n , . . . , c , c ) = ( d ′ p , . . . , d ′ , d ′ ) ∨ (2 µ + 1 , . . . , µ x + 1) ∨ ( ν , ν , . . . , ν y , ν y ) , then A ( O ) ∼ = ( Z / Z ) p with generators { s ′ p − , s ′ p − . . . , s ′ } . Remark 2.4. (1) Note that the above construction automatically makes A ( O ) into a quotient of A ( O ) .Indeed, { d ′ p , . . . , d ′ , d ′ } is obtained by removing the odd columns in { d j , . . . , d , d } , sothe Lusztig’s quotient map p : A ( O ) → A ( O ) has kernel ker p = { s l − | d l − is odd } .(2) We will focus on the case when O = (2 a k , . . . , a , a ) in the following sections. Inthis case, A ( O ) is always equal to A ( O ) . Proof.
There is a description of A ( O ) on [25, Section 5] which we will use here. Write O = [ r ≥ r ≥ . . . ] in its partition description. Let S odd be the collection of even r i appearing odd number of times in O and similarly for S even . Write S odd = { α m > α m − > · · · > α } ,putting α = 0 if necessary ; S even = { β l > β l − > · · · > β } , then A ( O ) is generated by x i , with i equal to α r − and all β s lying between α r +1 ≥ β s ≥ α r for all r .It is obvious that the above description of A ( O ) only depends on the ordering betweenthe even r i ’s appearing in O , but not on the value of r i . So we can reduce our study to O ∗ = [ r ∗ ≥ r ∗ ≥ . . . ] as in Proposition 2.2. Also, the odd r ∗ i does not show up in thecalculation of A ( O ∗ ) , so we can remove all odd r ∗ i ’s in O ∗ and get O ∗ = ( d k , d k − , . . . , d , d , d ) → O ∗∗ = ([ d k − − k − X j =1 ( d j − d j − )] , . . . , [ d − ( d − d )] , d , , where all d i − − P i − j =1 ( d j − d j − ) , ≤ i ≤ k are distinct integers with the property A ( O ) = A ( O ∗ ) = A ( O ∗∗ ) . Since ( d j − d j − ) is even for all j , the i -th columns of O ∗ and O ∗∗ are of the same parity. So it suffices to prove the Proposition for O ∗∗ .Consider all orbits with partition and dual partition of the form Q = [(2 k ) m k > (2 k − m k − > · · · > m ] = ( v k − , . . . , v i − , . . . , v , KAYUE DANIEL WONG with all v i − distinct (so Q = O ∗∗ is an example). Note that(1) m k = v and m k − j = v j +1 − v j − for all j > . Claim: A ( Q ) = ( Z / Z ) p , where p is the number of even v i − ’s in A ( Q ) .The proof of the Claim is combinatorial. We give a sketch proof here. Suppose v , v , . . . , v x − are all even and v x +1 is odd. Then Equation (1) says m k , . . . , m k − x +1 are all even and m k − x is odd, i.e. k, . . . , (2 k − x + 2) ∈ S even and (2 k − x ) = 2 α m ∈ S odd . According tothe formulation of A ( O ) in the beginning of the proof, (2 k ) , (2 k − , . . . , (2 k − x + 2) > α m and they contribute to A ( Q ) , while (2 k − x ) = 2 α m does not contribute to A ( Q ) . So theeven columns contribute while the odd column does not contribute to A ( Q ) . This matcheswith our Claim.Now suppose v x +1 , . . . , v y − are all odd and v y +1 is even. Then Equation (1) says that (2 k − x − , . . . , (2 k − y + 2) ∈ S even and (2 k − y ) = 2 α m − ∈ S odd . According to theformulation in the beginning of the proof, α m > (2 k − x − , . . . , (2 k − y + 2) > α m − and they do not contribute to A ( Q ) , while (2 k − y ) = 2 α m − contributes to A ( Q ) . Sothe odd columns do not contribute while the even column contributes to A ( Q ) . This alsomatches with our Claim.One can continue the argument using induction to prove the Claim holds for all such Q . In other words, A ( O ∗∗ ) is generated by its even column pairs, so the Proposition holdsfor O ∗∗ and hence for O as well. (cid:3) SPECIAL UNIPOTENT REPRESENTATIONS
Recall the construction of special unipotent representations attached to a special classi-cal nilpotent orbit O with L O is an even orbit in [7] (or [30]). Algorithm 3.1.
Let O = (2 a k , . . . , a , a ) with a i − > a i − for all i :(I) The Spaltenstein dual is given by L O = S ki =1 [2 a i + 1 , a i − − ∪ [2 a + 1] . Hence λ O = 12 L h = ( λ ; . . . ; λ k ; a , . . . , , , where λ i = ( a i , . . . , , a i − − , . . . , , for each i .(II) Let γ ( O ) := {O ′ ⊆ O|O ′ * O spec for any other special orbit O spec ( O} . Lusztig in [21] defined an injection γ ( O ) ֒ → A ( O ) , such that the composition of maps σ : γ ( O ) ֒ → A ( O ) left cell −→ ˆ W YMPLECTIC NILPOTENT ORBITS 7 maps O ′ to its Springer representation. In our case, γ ( O ) is given by γ ( O ) = [ I ⊂{ s ,s ,...,s k − } {O I } , with O I = _ s j − / ∈ I (2 a j , a j − ) ∨ _ s i − ∈ I (2 a i + 1 , a i − − ∨ ( d ) . By Proposition 2.3, A ( O ) has k elements, which has the same cardinality as γ ( O ) . So the injec-tion γ ( O ) ֒ → A ( O ) is indeed a bijection.From now on, we denote elements in A ( O ) by the subset I ⊂ { s , s , . . . , s k − } . According tothe algorithm of computing Springer representations given in [24, Section 7] , σ ( O I ) = j WW I ( sgn ) where W I = Y s j − / ∈ I ( C a j × D a j − ) × Y s i − ∈ I ( D a i +1 × C a i − − ) × C a . (III) Let λ j = ( a j , . . . , , a j − − , . . . , , as before, and λ i = ( a i , . . . , , a i − − , . . . , , . Define R I = X w ∈ W I ( − l ( w ) X ∪ s j − / ∈ I λ j ; ∪ s i − ∈ I λ i ; a . . . , w ( ∪ s j − / ∈ I λ j ; ∪ s i − ∈ I λ i ; a . . . , ! , where X (cid:18) λ λ (cid:19) = K − finite part of Ind GB ( e ( λ ,λ ) ⊗ is the principal series representation with character ( λ , λ ) ∈ h C , the complexification of the maximal torus h in g (here we treat G as a real Lie group). In particular, the G ∼ = K C -types of X (cid:18) λ λ (cid:19) is equal to Ind GT ( e λ − λ ) ,which we will denote as Ind GT ( λ − λ ) subsequently (see Theorem 1.8 of [7] for more details onthe principal series representations).(IV) The special unipotent representations are parameterized by π ∈ A ( O ) ∧ . In fact, for any I ⊂ { s , . . . , s k − } , there exists an irreducible A ( O ) -representation π I with π I ( s j − ) = ( − , if s j − ∈ I , otherwise . All π ∈ A ( O ) ∧ can be obtained in this way, i.e. π = π I for some I . Then the special unipotentrepresentations are of the form: (2) X π I = 12 k X J ⊂{ s ,s ,...s k − } tr π I ( J ) R J , KAYUE DANIEL WONG where π I ( J ) = Π s j − ∈ J π I ( s j − ) . In particular, (3) X triv = X π φ = 12 k X J ⊂{ s ,s ,...s k − } R J and the sum of all special unipotent representations is given by (4) R φ = M π ∈ A ( O ) ∧ X π Example 3.2.
Let O ′ = [4 , , , , , , ,
1] = (8 , , , . Then A ( O ) = A ( O ) = { φ, { s } , { s } , { s , s }} , and above calculation gives λ O = (4321; 210; 21; 0) , with R φ = X w ∈ C × D × C × D ( − l ( w ) X (cid:18) w ( 4321; 210; 21; 0) (cid:19) ; R s = X w ∈ C × D × D ( − l ( w ) X (cid:18) w ( 4321; 210; 210) (cid:19) ; R s = X w ∈ D × C × C × D ( − l ( w ) X (cid:18) w ( 43210; 21; 21; 0) (cid:19) ; R s ,s = X w ∈ D × C × D ( − l ( w ) X (cid:18) w ( 43210; 21; 210) (cid:19) . and X triv = ( R φ + R s + R s + R s ,s ) . Remark 3.3.
According to Theorem 1.1, we have X triv ∼ = R ( O ) ∼ = Ind GG e (triv) . As a generalization of Theorem 1.1, by the last paragraph of [9], or more explicitly, p.29 of[10], it can be seen that as K C ∼ = G -modules, X π ∼ = Ind GG e ( π ) for all local systems π ∈ A ( O ) ∧ = ( G e / ( G e ) ) ∧ . In other words, we have attached a uni-tary representation X π (by [8]) to each orbit data ( O , π ) for all O satisfying the hypothesisof Theorem 1.1 (for more details on orbit data, see [26] or Section 2 of [5]).More generally, we can further extend our scheme to a larger class of nilpotent orbits- Consider nilpotent orbits O = (2 a k , . . . , a , a ) with no restrictions on the size ofcolumns. Separate the columns a i − = 2 a i − from O , i.e. O = O ′ ∨ ( ν , ν , . . . , ν y , ν y ) , YMPLECTIC NILPOTENT ORBITS 9 where O ′ satisfies the hypothesis of Theorem 1.1. Then O = Ind gg ′ × gl ( ν ) ×···× gl ( ν y ) ( O ′ ⊕ triv ⊕ · · · ⊕ triv) . By Proposition 2.2, A ( O ) = A ( O ′ ) and hence there is a natural one-to-onecorrespondence between Π ∈ A ( O ) ∧ and π ∈ A ( O ′ ) ∧ . By Corollary 1.3 of [19], Ind GG ′ × GL ( ν ) ×···× GL ( ν y ) ( Ind G ′ G ′ e ( π ) ⊠ triv ⊠ · · · ⊠ triv) = Ind GG e (Π) . Therefore, the unitarily induced module
Ind GG ′ × GL ( ν ) ×···× GL ( ν y ) ( X π ⊠ triv ⊠ · · · ⊠ triv) isthe corresponding unitary representation attached to the orbit data ( O , Π) .Using the formula of X π and the techniques in Proposition 4.2-4.3 of [30], we can com-pute the Lusztig-Vogan bijection γ ( O , π ) (Section 1 of [30]) for all local systems π of all O ’s discussed in Remark 3.3. For example, let O = (8 , , , , then γ ( O , π φ ) = (8 , , , , , , , , , ∼ (8 , , , , , ,
0; 4 , , γ ( O , π s ) = (8 , , , , , , , , , ∼ (8 , , , , , ,
0; 4 , , γ ( O , π s ) = (8 , , , , , , , , , ∼ (8 , , , , , ,
1; 4 , , γ ( O , π s ,s ) = (8 , , , , , , , , , ∼ (8 , , , , , ,
1; 4 , , Moreover, Theorem 5.1 of [30] can be readily verified as well.4. F
UNDAMENTAL M ULTIPLICITIES
By Theorem 1.1 and the character formula of X triv given in Algorithm 3.1, one canpractically compute the multiplicities of any irreducible G -representations appearing in R ( O ) . In this Section, we focus on the fundamental multiplicities µ i = V i n − i definedin the Introduction (the subscript β of V β denotes the highest weight of an irreduciblerepresentation of G ). Lemma 4.1.
For G = Sp (2 n, C ) and a ≥ b > both even, define the virtual G -modules U a,b = 12 [ X w ∈ C a × D b ( − l ( w ) Ind GT (( a, . . . , , b − , . . . , , − w ( a, . . . , , b − , . . . , , X w ∈ D a +1 × C b − ( − l ( w ) Ind GT (( a, . . . , , b − , . . . , , − w ( a, . . . , , b − , . . . , , U a = X w ∈ C a ( − l ( w ) Ind GT ( a, . . . , , − w ( a, . . . , , , then [ U a,b : µ i ] = [ Ind
GGL ( a + b ) (triv) : µ i ] = δ i , [ U a : µ i ] = [ Ind
GGL ( a ) (triv) : µ i ] = δ i for all i (where δ ij is the Kronecker delta function).Proof. Note that U a,b and U a are character formulas of the special unipotent representa-tion X triv , for nilpotent orbits O = (2 b a − b ) and triv = (1 n ) respectively. It is provedin Section 2 of [30] that for all such O , all G -representations of R ( O ) are of the form V p , p ,..., p b , ,..., with multiplicity one. In particular, no µ i = V i n − i appears in R ( O ) for i > . Hence [ U a,b : µ i ] = [ U a : µ i ] = 0 for all i > .On the other hand, one can use Frobenius reciprocity to conclude that [ Ind
GGL ( a + b ) (triv) : µ i ] = [ Ind
GGL ( a ) (triv) : µ i ] = 0 for all i > . Hence the result follows. (cid:3) Lemma 4.2.
Let G = Sp (2 p + 2 q, C ) , with G = Sp (2 p, C ) and G = Sp (2 q, C ) be subgroupsof G such that G × G embeds into G diagonally. Write T , T as Cartan subgroups of G and G respectively (so that T := T × T is a Cartan subgroup of G , and let C = X i a i Ind G T ( γ i ) , C = X j b j Ind G T ( δ j ) be virtual representations of G and G respectively. Then as virtual G -modules, X i,j a i b j Ind GT ( γ i ; δ j ) ∼ = Ind GG × G ( C ⊠ C ) Proof.
Suppose µ is a finite dimensional, irreducible representation of G , writing µ | G × G = ⊕ k ( π k ⊠ π k ) as the restricted representation to G × G , then [ Ind GG × G ( C ⊠ C ) : µ ] = [ C ⊠ C : µ | G × G ]= X k [ X i a i Ind G T ( γ i ) : π k ][ X j b j Ind G T ( δ j ) : π k ]= X i,j,k a i b j [ γ i : π k | T ][ δ j : π k | T ]= X i,j,k a i b j [( γ i ; δ j ) : ( π k ⊠ π k ) | T × T ]= X i,j a i b j [( γ i ; δ j ) : µ | T × T ]= X i,j a i b j [ Ind GT ( γ i ; δ j ) : µ ] (cid:3) Proposition 4.3.
Let O ′ = (2 a ′ p , . . . , a ′ , a ′ ) be a nilpotent orbit with a i − > a i − for all i .Then the multiplicities of the fundamental representations are given by [ R ( O ′ ) : µ i ] = [ Ind
GGL ( D ′ ) (triv) : µ i ] , with GL ( D ′ ) = Π pi =0 GL ( a ′ i + a ′ i − ) (Recall we take a ′− = 0 in the Introduction). Remark 4.4.
The above Proposition essentially shows Theorem A holds for all O ′ ’s with a ′ i − > a ′ i − for all i - According to Theorem A, one needs to remove all column pairs YMPLECTIC NILPOTENT ORBITS 11 of the same size. By the hypothesis of the above Proposition, a column pair can onlyexist when ( α, α ) = (2 a ′ i , a ′ i − ) . Therefore, GL ( a ′ i + a ′ i − ) = GL (2 a ′ i ) = GL ( α ) as inTheorem A. Proof.
We will prove the Theorem by induction on p . Note that O ′ satisfies the hypothesisin Theorem 1.1, therefore R ( O ′ ) ∼ = X triv as G -modules. For the rest of the proof, we willuse the notation X π, O to denote the unipotent representation X π attached to the orbit O . p = : O = ( d ′ ) . According to Algorithm 3.1, R ( O ) ∼ = X triv , O ∼ = U a ′ . So the resultfollows from Lemma 4.1. Induction Step:
Suppose the hypothesis holds for O r = (2 a ′ r , . . . , a ′ , a ′ ) and G = G r = Sp ( P ri =0 a ′ i , C ) , i.e. [ R ( O r ) : µ i ] = [ X triv , O r : µ i ] = [ Ind
GGL ( D ′ r ) (triv) : µ i ] , where GL ( D ′ r ) = Π ri =0 GL ( a ′ i + a ′ i − ) . Now study the orbit O r +1 = (2 a, b, a ′ r , . . . , d ′ , d ) and G = G r +1 . Algorithm 3.1 gives X triv , O r +1 = 12 r +1 X I ⊂{ s ′ ,s ′ ,...s ′ r +1 } R I = 12 r +1 ( X J ⊂{ s ′ ,s ′ ,...s ′ r − } R J + X J ⊂{ s ′ ,s ′ ,...s ′ r − } R J ∪{ s ′ r +1 } ) ∼ = 12 r +1 [ X J ⊂{ s ′ ,s ′ ,...s ′ r − } X C a × D b × W J ( − l ( w ) Ind GT (( λ r +1 ; λ O r ) − w ( λ r +1 ; λ O r ))+ X J ⊂{ s ′ ,s ′ ,...s ′ r − } X D a +1 × C b − × W J ( − l ( w ) Ind GT (( λ ′ r +1 ; λ O r ) − w ( λ ′ r +1 ; λ O r ))] , with λ r +1 = ( a, . . . , , b − , . . . , , , λ ′ r +1 = ( a, . . . , , b − , . . . , , and λ O r is as inStep (1) of Algorithm 3.1. Now apply Lemma 4.2 with G = Sp (2 a + 2 b ) and G = G r ,and C = U a,b = 12 [ X C a × D b ( − l ( w ) Ind G T ( λ r +1 − wλ r +1 ) + X D a +1 × C b − ( − l ( w ) Ind G T ( λ ′ r +1 − wλ ′ r +1 )]; C = 12 r X J ⊂{ s ′ ,s ′ ,...s ′ r − } X W J ( − l ( w ) Ind G T ( λ O r − wλ O r ) ∼ = X triv , O r , we will get X triv , O r +1 ∼ = Ind
GSp (2 a +2 b ) × G r ( U a,b ⊠ X triv , O r ) . Now all fundamental representations µ i in G decomposes as µ i | Sp (2 a +2 b ) × G r = ⊕ ( γ,δ ) µ γ ⊠ µ δ , with all µ γ and µ δ being fundamental representations of G and G respectively. [ R ( O r +1 ) : µ i ] = [ X triv , O r +1 : µ i ] = [ U a,b ⊠ X triv , O r : µ i | Sp (2 a +2 b ) × G r ]= M ( γ,δ ) [ U a,b : µ γ ][ X triv , O r : µ δ ]= M ( γ,δ ) [ Ind Sp (2 a +2 b ) GL ( a + b ) (triv) : µ γ ][ Ind G r GL ( D ′ r ) (triv) : µ δ ]= [ Ind Sp (2 a +2 b ) GL ( a + b ) (triv) ⊠ Ind G r GL ( D ′ r ) (triv) : µ i | Sp (2 a +2 b ) × G r ]= [ Ind
GGL ( D ′ r +1 ) (triv) : µ i ] , where the third line comes from Lemma 4.1 the induction hypothesis. So the proof iscomplete. (cid:3) The proof of Theorem A follows immediately from the above Proposition:
Corollary 4.5.
Let O = (2 a k , . . . , a , a ) be a nilpotent orbit for G . Remove all column pairsof same size ( α i , α i ) in O , leaving the orbit ( d l , d l − , . . . , d ) , i.e. O = ( d l , d l − , . . . , d ) ∨ ( α , α , . . . , α x , α x ) with d i +1 = d i for all i . Then the multiplicities of the fundamental representations are given by [ R ( O ) : µ i ] = [ Ind Sp (2 n, C ) GL ( D ) (triv) : µ i ] , where GL ( D ) = Π li =0 GL ( d i + d i − ) × Π xi =1 GL ( α i ) .Proof. We separate the columns a i − = 2 a i − in O as in Remark 3.3, i.e. O = O ′ ∨ ( ν , ν , . . . , ν y , ν y ) , where O ′ satisfies the hypothesis of 4.3. Taking π = triv in Remark 3.3,we get R ( O ) = Ind GG ′ × GL ( ν ) ×···× GL ( ν y ) ( R ( O ′ ) ⊠ triv ⊠ · · · ⊠ triv) . By Proposition 4.3, R ( O ′ ) = Ind G ′ GL ( D ′ ) (triv) , hence the result follows from induction instages. (cid:3) As mentioned in the beginning of this Section, we present an example to compute themultiplicities of irreducible G -representations V β appearing in R ( O ) other than the fun-damental representations V i n − i here. Example 4.6.
Let O = (8 , . Then the character formula X triv ∼ = R ( O ) can be expandedas: X triv = 12 [ X C × D ( − l ( w ) Ind GT (4321 , − w (4321 , X D × C ( − l ( w ′ ) Ind GT (43210 , − w ′ (43210 , . YMPLECTIC NILPOTENT ORBITS 13
To find the coefficient of
Ind GT (0 ) in the above expression, one needs to find out howmany w ∈ W ( C × D ) so that (4321 , − w (4321 , can be W -conjugated to haveweight (0 ) (and respectively for w ′ ∈ W ( D × C ) ). Obviously this forces w = w ′ = Id ,and hence R ( O ) ∼ = X triv | K C = 12 ( Ind GT (0 )+ Ind GT (0 ))+ X λ ∈ t ∗ c λ Ind GT ( λ ) = Ind GT (0 )+ X λ ∈ t ∗ c λ Ind GT ( λ ) , || λ || > . To find out the coefficients of
Ind GT (1 ) , one needs to find out which w ∈ W ( C × D ) sothat (4321 , − w (4321 , can be W -conjugated to have weight (1 ) (and respectivelyfor w ′ ∈ W ( D × C ) ). The list of all such w (4321 , and w ′ (43210 , are given below: w (4321 ,
10) (4321 , − w (4321 , w ′ (43210 ,
1) (43210 , − w ′ (43210 , Since each of the w and w ′ above is a simple reflection, i.e. l ( w ) = l ( w ′ ) = 1 , therefore ( − l ( w ) = ( − l ( w ′ ) = − and R ( O ) ∼ = Ind GT (0 ) + 12 [( −
5) + ( − Ind GT (1 ) + X λ ∈ t c λ Ind GT ( λ ) , || λ || > as virtual G -modules. Continuing the calculations, we get R ( O ) ∼ = Ind GT (0 ) − Ind GT (1 ) + 6 Ind GT (1 ) + 0 Ind GT (2 ) − Ind GT (1 ) + . . . For any irreducible G -representation µ , Frobenius reciprocity gives [ R ( O ) : µ ] = [(0 ) : µ | T ] − ) : µ | T ] + 6[(1 ) : µ | T ] + 0[(2 ) : µ | T ] − [(1 ) : µ | T ] + . . . so in practice this gives [ R ( O ) : µ ] for any µ - for example, if µ = V , then Weyl characterformula gives µ | T = 6 × (0 ) + 1 × (1 ) + 1 × (2 ) + . . . , with the remaining terms lyingoutside the dominant Weyl chamber C = { ( a , a , . . . , a ) ∈ t ∗ | a ≥ a ≥ · · · ≥ } . So [ R ( O ) : V ] = 1 × − × × × − × . Indeed, since O is a spherical orbit, the multiplicity of V in R ( O ) is known (e.g. Section2 of [30]) to be one.
5. N
ORMALITY OF O RBIT C LOSURES
One of the reasons we are interested in computing the multiplicity of fundamental rep-resentations appearing in R ( O ) is to detect non-normality of the orbit closure O . To doso, we will find an upper bound on [ R ( O ) : µ ] for all fundamental representations µ , andshow that this upper bound is strictly smaller than [ R ( O ) : µ ] if O is not normal.The upper bound we need is given in the Lemma below: Lemma 5.1.
Let G = Sp (2 n, C ) and O = ( c k , c k − , . . . , c ) be any nilpotent orbit. For any finite dimensional irreducible representations µ , [ R ( O ) : µ ] ≤ [ R ( O ♯ ) : µ ] where O ♯ = ( c k + c k − , c k + c k − , c k − + c k − , c k − + c k − , . . . , c + c , c + c , c , c ) .Proof. We only work in the case when G = Sp (2 n, C ) . Note that by the Kraft-Procesicriterion (Theorem 1.2), O ♯ is normal. Therefore R ( O ♯ ) = R ( O ♯ ) . On the other hand, notethat O ♯ ⊃ O . Consequently, we have a G -module surjection R ( O ♯ ) = R ( O ♯ ) ։ R ( O ) and hence [ R ( O ) : µ ] ≤ [ R ( O ♯ ) : µ ] for any finite dimensional G -representations µ . How-ever, the latter term is equal to [ R ( O ♯ ) : µ ] . Hence the result follows. (cid:3) Proof of Theorem B.
One direction is easy - if O is normal, then R ( O ) = R ( O ) as G -modules,hence [ R ( O ) : µ i ] = [ R ( O ) : µ i ] for all i .Now suppose O = (2 a k , a k − , . . . , a , a ) be a nilpotent orbit such that O is notnormal. Then Theorem A says [ R ( O ) : µ i ] = [ Ind
GGL ( D ) (triv) : µ i ] . According to Lemma 5.1, O ♯ = ( a k + a k − , a k + a k − , . . . , a + a , a + a , a , a ) . Sincethere may be some odd columns appearing in O ♯ , we cannot use Theorem A directly tocompute [ R ( O ♯ ) : µ i ] . However, O ♯ = Ind ggl ( a k + a k − ) ⊕···⊕ gl ( a + a ) ⊕ gl ( a ) (triv) is stronglyRichardson, therefore [ R ( O ♯ ) : µ i ] = [ Ind
GGL ( D ♯ ) (triv) : µ i ] with GL ( D ♯ ) = GL ( a k + a k − ) × · · · × GL ( a + a ) × GL ( a ) . By the Kraft-Procesi de-scription of non-normal orbit closures (Theorem 1.2), the two parabolic subgroups GL ( D ) and GL ( D ♯ ) of G are different. More precisely, define F := { i ∈ N | GL ( i ) is a factor of GL ( D ) } ; F ♯ := { i ∈ N | GL ( i ) is a factor of GL ( D ♯ ) } YMPLECTIC NILPOTENT ORBITS 15 with multiplicities. Rearrange the elements in F and F ♯ so that F = { f ≤ f ≤ f . . . } ; F ♯ = { f ♯ ≤ f ♯ ≤ f ♯ . . . } . Since O has non-normal closure, it has columns of the form a i > a i − = 2 a i − = · · · = 2 a j − = 2 a j − > a j − pick the smallest value of such c j − = c j − . Then { a , a + a , . . . , a j − + a j − } = { f , f , . . . , f j − } = { f ♯ , f ♯ , . . . , f ♯ j − } while f ♯ j − := a j − + a j − ∈ F ♯ but not in F . More precisely, it is easy to see f ♯ j − Let O = (8 , , , , , , for Sp (32 , C ) . Following Theorem 1.2, its closureis not normal.Using the notations in the above proof, F = { , , , } . Now O ♯ = (7 , , , , , , , ,and F ♯ = { , , , } . The first discrepancy between F and F ♯ occurs at = 2 . Hence [ R ( O ) : µ i ] < [ R ( O ) : µ i ] must occur at i = 2(1) + 2 = 4 . Using Frobenius reciprocity, wecomputed the multiplicities as follows: i [ R ( O ♯ ) : µ i ] [ R ( O ) : µ i ] The discrepancies of the two rows of numbers reflects the non-normality of O .We would like to end with a conjecture: Conjecture 5.3. Let O be a classical nilpotent orbit for G , and µ is any irreducible, finite dimen-sional representation of G . Then the multiplicities [ R ( O ) : µ ] can be computed. In particular, if µ = µ i is a fundamental representation, then [ R ( O ) : µ ] = [ R ( O ♯ ) : µ ] The proof of the above Conjecture for O = (2 a k , . . . , a , a ) is the content of an on-going work of Barbasch and the author in [11]. R EFERENCES [1] P. 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