Unipotent Representations of Exceptional Richardson Orbits
aa r X i v : . [ m a t h . R T ] S e p UNIPOTENT REPRESENTATIONS OF EXCEPTIONALRICHARDSON ORBITS
KAYUE DANIEL WONG
Abstract.
We study special unipotent representations attached to ex-ceptional Richardson orbits, and provide some evidences on a conjectureof Vogan for such orbits. Introduction
In [BV], Barbasch and Vogan studied special unipotent representa-tions for complex simple Lie groups G . These representations are of interestin various areas of representation theory. One application is on the OrbitMethod, which suggests that one can ‘attach’ some unitarizable representa-tions to each coadjoint orbit in the (dual of) Lie algebra Lie ( G ). Indeed, itis conjectured that special unipotent representations are the candidates ofrepresentations attached to special nilpotent orbits. A more precise formu-lation of such relations is given in Conjecture 1.2 below.For classical nilpotent orbits, Conjecture 1.2 was shown to be true in [B2]and [W2]. However, the case remains unknown for exceptional orbits. Oneof the goals of this manuscript is to establish such relations for exceptionalLie groups.Let G be a complex simple Lie group with Lie algebra g , and e ∈ O be anilpotent element in g with G e being the stabilizer subgroup of G . Lusztigin [L1] defined a quotient A ( O ) of the component group A ( O ) := G e / ( G e ) .Writing K as a maximal compact subgroup of G whose complexification isequal to K C ∼ = G , then the special unipotent representations are ( g C , K C )-modules given by { X O ,π | O special orbit, π irreducible representation of A ( O ) } . By Theorem III of [BV], one can write down the character formulas of allsuch representations, i.e. in terms of virtual K C ∼ = G -modules, we have:(1) X O ,π ∼ = X λ ∈ Λ + a O ,π ( λ ) Ind GT ( e λ ) , where Λ + is the collection of all dominant weights corresponding to G and a O ,π ( λ ) ∈ Z . However, the expression in the theorem hinders its simplicity for Richardson orbits. The main result of this manuscript is to simplifyEquation (1): Theorem 1.1.
Let G be a complex exceptional Lie group of adjoint type,and P = LU be a parabolic subgroup of G such that the moment map µ : T ∗ ( G/P ) → g is birational onto the Richardson orbit O ⊂ g , then (2) X O ,π ∼ = X w ∈ W ( L ) sgn( w ) Ind GT ( λ − wλ π ) for some λ, λ π ∈ Λ . In particular, X O ,π | K C ∼ = Ind GL ( V π ) for some irreducible,finite-dimensional representation V π of L . Moreover, if π is trivial, then V π is the trivial representation. It turns out that all but four exceptional orbits satisfy the hypothesis ofthe above theorem (Proposition 2.1).1.1.
A Conjecture of Vogan on Quantization.
One application of The-orem 1.1 is to give evidences on a conjecture of Vogan:
Conjecture 1.2 ([V2] Conjecture 12.1) . Let O be a complex special nilpotentorbit, and ψ be an irreducible representation of A ( O ) . Then the K -spectrumof the unipotent representation X O ,π satisfies the following: X O ,π ∼ = R ( O , ψ ) ∼ = Ind GG e ( ψ ) , where R ( O , ψ ) is the global section of the vector bundle G × G e ψ → G/G e ∼ = O , and ψ is an irreducible representation of G e that can be descended to G e / ( G e ) = A ( O ) . As mentioned above, the conjecture is known to be true for classical Liealgebras. For exceptional Lie algebras, one can compare the results in [S3]with the V π given in Theorem 1.1 that all V π are ‘lifts’ of some ψ ∈ A ( O ) ∧ inConjecture 1.2. Namely, we always have V π | G e ∼ = ψ . This gives an evidenceon the validity of Conjecture 1.2. Indeed, if π is the trivial representation,we can show that the conjecture holds: Theorem 1.3.
Let O be an exceptional Richardson orbit. Then X O , triv ∼ = R ( O ) , i.e. Conjecture 1.2 holds for all exceptional Richardson orbits. The Lusztig-Vogan map.
In [AS], Achar and Sommers gave a con-jecture on the image of the Lusztig-Vogan map Γ for classical nilpotent orbits(see Section 8 of [V3] for a definition of Γ). In particular, they conjecturedthe following:I. Fix a classical special nilpotent orbit O ; XCEPTIONAL RICHARDSON ORBITS 3
II. Look at the special piece S d ( O ) ([L2]) of the Lusztig-Spaltensteindual d ( O ) of O : S d ( O ) := {O ∨ ⊆ d ( O ) |O ∨ * O ∨ sp for other special O ∨ sp ( d ( O ) } ;III. For each O ∨ ∈ S d ( O ) , take the semisimple element h ∨ of a Jacobson-Morozov triple corresponding to O ∨ ; thenIV. There exists ψ ∈ A ( O ) ∧ such that Γ( O , ψ ) = h ∨ . The conjecture was proved in [W2] by • studying the ‘map’ (which, a priori, is not well-defined)Ψ : { ( O , π ) | O special orbit, π ∈ A ( O ) ∧ } → Λ + , with Ψ( O , π ) = λ max being the largest dominant element in Equation(1) for X O ,π such that a O ,π ( λ max ) = 0; and • using the validity of Conjecture 1.2 for classical nilpotent orbits torelate the two maps Ψ and Γ.By the formula of X O ,π in Theorem 1.1, one can see immediately thatΨ( O , π ) is equal to a conjugate of λ − w ( λ π ) in Equation (2), where w is the longest element of W ( L ). As a result, we obtain the following: Theorem 1.4.
Let G be a complex exceptional Lie group of adjoint type, and O be a Richardson orbit. For every semisimple element h ∨ in a Jacobson-Morozov triple corresponding to O ∨ ∈ S d ( O ) , there exists π ∈ A ( O ) ∧ suchthat Ψ( O , π ) = h ∨ . This is a generalization of the conjecture of Achar and Sommers for themap Ψ, which gives evidences on the relations between Γ and Ψ, a directconsequence for the validity of Conjecture 1.2.2.
Proof of Theorem 1.1
In this section, we write down the special unipotent representations cor-responding to almost all exceptional Richardson orbits O . Proposition 2.1.
Let O be an exceptional Richardson orbit other than A + A , D ( a ) in E , or E ( a )+ A , E ( a ) in E . Then there exists a parabolicsubgroup P such that the moment map µ : T ∗ ( G/P ) → g is birational onto O .Proof. It is known that µ is birational if [ G e : P e ] = 1. This holds when A ( O ) = 1. Also, by the results in [McG], the map is birational when O iseven (so that the Levi subgroup L is given by the nodes marked with 0’s).By checking the tables in [CM], we are only left with(3) D ( a ) + A in E ; D ( a ) , D ( a ) in E along with the four orbits mentioned in the proposition.Indeed, we can apply Lemma 5.5 in [FJLS] to check birationality of µ ,which says that for any e ∈ O , the number of components in µ − ( e ) is equalto X ψ ∈ d A ( e ) dim V ψ · [ σ O ,ψ : Ind WW ( L ) (sgn)] , where σ O ,ψ is the Springer representation H top ( B e ) ψ . One can use the tablein [Alv] to check that the above value is equal to 1 for the orbits in (3) (andis equal to 2 for the other four orbits). Since µ is birational if and only ifthis value is equal to 1, the result follows. (cid:3) We now write down the character formulas of the special unipotent rep-resentations X O ,π – in fact, the infinitesimal character of X O ,π for all π isequal to (a W × W -conjugate) of ( h ∨ , h ∨ ), where h ∨ is the semisimpleelement of a Jacobson-Morozov triple of d ( O ).Let λ = h ∨ . By Corollary 5.18 of [BV], the irreducible quotient X ( λ, wλ )of the principal series representation X ( λ, wλ ) has associated variety greaterthan or equal to O . The special unipotent representations are the X ( λ, wλ )’swith associated variety AV ( X ( λ, wλ )) = O . Note that by Theorem 1.5 of[BV], X ( λ, wµ ) ∼ = X ( xλ, xwµ ) for all x ∈ W . So we can begin with any W -conjugate of λ .The theorem below is a generalization of Proposition 9.11 of [BV]. As aconsequence, Theorem 1.1 follows. Theorem 2.2.
Let O be an exceptional Richardson orbit not equal to fourorbits specified in Proposition 2.1. Fix a Levi subalgebra l correspondingto the parabolic subgroup appearing in the Proposition, then the followingstatements hold: (a) The number of elements in P ( O ) := { wλ | wλ | l is dominant and regular } is equal to the number of conjugacy classes of A ( O ) . In particular,there exists a unique λ such that ( λ , α ∨ ) = 1 for all simple rootsin l , and λ − λ is a sum of positive roots for all λ ∈ P ( O ) . (b) The elements in P ( O ) = { λ , λ , . . . , λ r } can be arranged such that | λ i − λ | < | λ i +1 − λ | for all i . (c) The representations X ( λ , λ i ) are the special unipotent representa-tions attached to O . More precisely, suppose A ( O ) = S k ( k =2 , , , ), and C > · · · > C r is the ordering of all partitions of k , with π i ∈ A ( O ) ∨ parametrized by the partition C i , then X ( λ , λ i ) = X O ,π i ∼ = Ind GL ( V i ) XCEPTIONAL RICHARDSON ORBITS 5 for some finite-dimensional representations V i of L with highest weight ( λ − λ i ) | l . In particular, when i = 1 , then π is the trivial represen-tation of A ( O ) ∧ and V is the trivial representation of L .Proof. The case-by-case proofs of statements (a) and (b) will be postponedto Section 4. Assuming the results of (a) and (b) hold, then for each λ i ,let I ( λ , λ i ) be as defined in Section 9 of [BV] such that I ( λ , λ i ) has thecharacter formula given by Equation (1) and is isomorphic to Ind GL ( V λ − λ i )as K C -modules. Obviously, we have AV ( I ( λ , λ i )) = Ind gl (0) = O , andits composition factors must consist of special unipotent representationsattached to O . Using the arguments in 9.11 - 9.21 of [BV], one can see I ( λ , λ i ) = X ( λ , λ i ) and (c) follows. (cid:3) Proof of Theorem 1.3 and Theorem 1.4
General Case.
We begin by proving Theorem 1.3 and Theorem 1.4for all orbits satisfying the hypothesis of Theorem 1.1.Indeed, when O is not equal to the four specified orbits in Proposition2.1, the map µ : T ∗ ( G/P ) → O is the normalization of the orbit closure O . By standard arguments in algebraic geometry (see [J] for example), R ( O ) ∼ = C [ T ∗ ( G/P )]. By a result of [McG], the latter is isomorphic to
Ind GL (triv) as G -modules. This means R ( O ) ∼ = Ind GL (triv) ∼ = X O , triv andhence Theorem 1.3 holds for these orbits.Furthermore, as mentioned in the paragraph above Theorem 1.4, Ψ( O , π i )can be easily computed by conjugating λ − w ( λ i ) to the dominant Weylchamber. The values of Ψ( O , π i ) are given by the tables in Section 4, whichverifies Theorem 1.4 for these orbits.3.2. Special Case.
We now study the four orbits A + A , D ( a ) in E ,and E ( a ) + A , E ( a ) in E . Note that all these orbits have Lusztig’squotient A ( O ) = S , and the special piece S d ( O ) = { d ( O ) } only contains oneelement. Also, the orbits A + A in E and E ( a ) + A in E are called exceptional in Section 4 of [BV].Using Proposition 2.1, there is a generically 2 – 1 map µ : T ∗ ( G/P ) → O ,where semisimple part of the Levi subgroup L of P is of Type A for O = D ( a ) , E ( a ); and of Type A + A for O = A + A , E ( a ) + A .In the following subsections, we will write down the character formulasof X O , triv , X O , sgn for all these orbits. It turns out that the sum of the twospecial unipotent representations is isomorphic to Ind GL (triv). Consequently,Theorem 1.3 holds for these orbits: By [V2], we have X O , triv = R ( O ) − Y and X O , sgn = R ( O , sgn) − Y for some genuine K C -modules Y , Y . Therefore, Ind GL (triv) = X O , triv ⊕ X O , sgn = R ( O ) + R ( O , sgn) − Y − Y . KAYUE DANIEL WONG
By Proposition 4.6.1 of [B2], the left hand side is equal to R ( O ) + R ( O , sgn).So Y = Y = 0 and hence we have X O , triv ∼ = R ( O ) and X O , sgn ∼ = R ( O , sgn). Remark 3.1.
The method we used above can also be applied to verify Con-jecture 1.2 for other Richardson orbits with non-birational moment maps.For example, let O = F ( a ) in F . If we take the the parabolic subgroupwhose Levi is of type A + f A , then one can show that X O , h i ∼ = R ( O , h i ) .On the other hand, if we take the the parabolic subgroup whose Levi is oftype B , then one can show that X O , h i ∼ = R ( O , h i ) . O = A + A in E . By checking the tables of [AL] directly (thecalculations for other orbits can be found in [W3]), the two left cell rep-resentations attached to O are equal to j E D + A ( σ ), j E D + A ( σ ), where σ , σ are the two left cell representations attached to O ′ = [332211] + [11] in D + A . Using Proposition 6.6 of [BV], the character formulas for X O ,π can be derived from that of O ′ in D + A = (1 , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , , , − , − , − , − , , , , )(0 , , , , , − , , , , , , , , , − . Using Theorem 3.4 in [W2], Ψ( O ′ , h i ) = 0 2 0 20 0 2 ; Ψ( O ′ , h i ) =0 2 1 01 1 2 . Therefore, one can compute thatΨ( A + A , h i ) ∼ (1 , , − , − , − , − , , ∼ ( 12 , , − , − , − , − , − ,
112 ) , Ψ( A + A , h i ) ∼ (1 , , − , − , − , − , , ∼ ( 12 , − , − , − , − , − , − ,
112 ) . In terms of Dynkin diagram, we haveΨ( A + A , h i ) = 0 1 0 10 0 1 ; Ψ( A + A , h i ) = 1 0 0 01 1 1 . Note that we have Ψ( A + A , h i ) = h ∨ A + A = h ∨ d ( A + A ) and Theorem 1.4holds.3.2.2. O = D ( a ) in E . As in the above subsection, one can derive thecharacter formulas of X O ,π from that of O ′ = [332211] in D . More precisely,we have Ψ( O ′ , h i ) = 0 2 0 20 0 0 ; Ψ( O ′ , h i ) = 0 2 1 01 1 0 . Therefore, XCEPTIONAL RICHARDSON ORBITS 7 one can compute thatΨ( D ( a ) , h i ) ∼ (1 , , − , − , − , − , , ∼ (1 , , − , − , − , − , − , , Ψ( D ( a ) , h i ) ∼ (1 , , − , − , − , − , , ∼ (1 , , , − , − , − , − , . In terms of Dynkin diagram, we haveΨ( D ( a ) , h i ) = 0 2 0 00 0 2 ; Ψ( D ( a ) , h i ) = 1 0 1 00 0 2 . Note that Ψ( D ( a ) , h i ) = h ∨ A = h ∨ d ( D ( a )) , and Theorem 1.4 holds.3.2.3. O = E ( a ) + A in E . As before, the character formulas for X O ,π can be derived from that of O ′ = D ( a ) + [11] in E + A , which, by lastsubsection, can be derived from that of [332211] + [11] in D + A . Usingthe same argument as above, we haveΨ( E ( a ) + A , h i ) = 1 0 1 0 00 0 1 ; Ψ( E ( a ) + A , h i ) = 1 1 0 0 01 0 0 . We have Ψ( E ( a ) + A , h i ) = h ∨ A + A = h ∨ d ( E ( a )+ A ) and Theorem 1.4holds.3.2.4. O = E ( a ) in E . Finally, we can derive the character formula of X O ,π from that of O ′ = [332211] in D . As in the previous subsections, wehave Ψ( E ( a ) , h i ) = 2 0 0 0 00 0 2 ; Ψ( E ( a ) , h i ) = 2 0 0 0 00 1 0 . We have Ψ( E ( a ) , h i ) = h ∨ A = h ∨ E ( A ) , and Theorem 1.4 holds. KAYUE DANIEL WONG Tables
In this section, we provide details for the proofs in Theorem 2.2 andSection 3.1. For each exceptional Lie group, we give • all Richardson orbits O satisfying the hypothesis of Theorem 1.1; • the Levi subalgebra l where O is induced from, by specifying a sub-diagram of the Dynkin diagram of g ; • all irreducible representations π i of A ( O ); • the values of λ i appearing in Theorem 2.2(a)–(b).This verifies Theorem 2.2.The second last column of the tables gives the value of Ψ( O , π ) in termsof the weighted Dynkin diagram of L g . And the last column records theorbit O ∨ whose Dynkin element h ∨ is given by the previous column. Thisfinishes the proof of Theorem 1.4 in Section 3.1.4.1. G Orbits.
The exceptional group G has 3 Richardson orbits. Fixthe Dynkin diagram of g by: ( − , ,
1) (1 , − , O l π i λ i Ψ( O , π i ) O ∨ G φ h i (0 , ,
0) 0 03 0 G ( a ) (1,-1,0) h i (0 , − ,
1) 0 13 A h , i (1 , − ,
0) 1 03 e A h , , i (1 , , −
1) 0 23 G ( a )0 (-2,1,1) (1,-1,0)3 h i ( − , − ,
3) 2 23 G F Orbits.
The exceptional group F has 9 Richardson orbits. Fix theDynkin diagram of g by 1 223 4 with simple roots: (0 , − , , , , − (0 , , , , − − , − ) . O l π i λ i Ψ( O , π i ) O ∨ F φ h i (0 , , ,
0) 0 0 2 0 0 0 F ( a ) 3 h i ( , , , ) 0 0 2 0 1 A h , i ( − , , , ) 1 0 2 0 0 f A F ( a ) 1 3 h i (0 , , − , ) 0 0 2 1 0 A + e A C h i ( , − , − , ) 0 0 2 0 2 A B h i (0 , , , −
1) 2 0 2 0 0 e A F ( a ) 1 3 4 h i ( , , − , ) 0 1 2 0 0 A + f A h , i (1 , , − ,
1) 1 0 2 1 0 f A + A h , i ( , − , ) 1 0 2 0 2 B h , , i ( , − − , ) 0 1 2 0 1 C ( a ) h , , , i ( − , − − , ) 0 0 2 2 0 F ( a )˜ A h i ( , − , , ) 2 2 2 0 0 B A h i (0 , , , ) 2 1 2 0 1 C h i ( , , , ) 2 2 2 2 2 F E Orbits.
The adjoint exceptional group E has 15 Richardson orbits.Fix the Dynkin diagram 1 2 34 5 6 with simple roots: (1 , − , , , , , , − , , , , , , − , , , , , , , , , , , − , − , − , − , − , , −√ ) . O l π i λ i Ψ( O , π i ) O ∨ E φ h i (0 , , , , ,
0) 0 0 00 0 0 0 E ( a ) 2 h i (0 , , − , , ,
0) 0 0 01 0 0 A D h i (0 , , − , , ,
0) 1 0 00 0 1 2 A E ( a ) 2 4 5 h i (0 , , , , ,
0) 0 0 10 0 0 3 A h i (0 , , − , , ,
0) 0 0 02 0 0 A D ( a ) 1 2 4 h i (1 , , − , , ,
0) 1 0 01 0 1 A + A A + A h i (1 , , − , , ,
0) 1 0 00 1 0 A + 2 A D h i ( , − , − , , − , −√ ) 2 0 00 0 2 2 A A h i ( , , − , − , ,
0) 1 0 02 0 1 A D ( a ) 1 2 4 5 6 h i (1 , , − , , , −√
3) 0 1 00 1 0 2 A + A h i ( , − , − , , , −√ ) 1 0 02 0 1 A + A XCEPTIONAL RICHARDSON ORBITS 11 O l π i λ i Ψ( O , π i ) O ∨ h i (0 , − , − , , ,
0) 0 0 20 0 0 D ( a ) A h i (2 , , , − , ,
0) 2 0 02 0 2 A A + 2 A h i ( , , − , − , , −√ ) 1 1 01 1 1 A + A A h i (0 , , , , ,
0) 0 0 22 0 0 D A h i ( , , − , − , − , − √ ) 1 1 02 1 1 A h i (1 , , − , − , − , −√
3) 2 0 20 0 2 E ( a )2 A h i (4 , , , , ,
0) 2 0 22 0 2 D h i (4 , , , , , − √
3) 2 2 22 2 2 E E Orbits.
The adjoint exceptional group E has 27 Richardson orbitsexcluding A + A and D ( a ). Fix the Dynkin diagram 1 2 3 45 6 7 withsimple roots: (1 , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , , , − , − , − , − , , , , )(0 , , , , , − , , , , , , , , − , . O l π i λ i Ψ( O , π i ) O ∨ E φ h i (0 , , , , , , ,
0) 0 0 0 00 0 0 0 E ( a ) 4 h i (0 , , , , − , , ,
0) 0 0 0 00 0 1 A E ( a ) 2 4 h i (0 , , − , , − , , ,
0) 0 1 0 00 0 0 2 A E ( a ) 1 5 7 h i ( , − , − , − , , , − , ) 0 0 0 00 1 0 (3 A ) ′ h i (0 , − , , , , , ,
0) 0 0 0 00 0 2 A E h i ( , − , , − , , , , ) 2 0 0 00 0 0 (3 A ) ′′ E ( a ) 1 3 5 6 h i ( , − , , − , , − , , ) 1 0 0 01 0 0 4 A h i (0 , − , , − , , , ,
0) 0 1 0 00 0 1 A + A E ( a ) 1 2 5 7 h i ( , − , − , − , , , − , ) 0 0 0 10 0 0 A + 2 A D ( a ) 1 2 3 h i ( , , − , − , , , ,
0) 0 1 0 00 0 2 A D + A h i (1 , , − , , , , − ,
0) 0 2 0 00 0 0 2 A A h i ( , − , , − , , , − , ) 0 0 0 02 0 0 A + 3 A E ( a ) 1 2 5 6 7 h i (1 , , − , − , , , − ,
1) 0 1 0 00 1 0 2 A + A h i ( , − , − , − , , , − , ) 0 0 0 10 0 1 ( A + A ) ′ XCEPTIONAL RICHARDSON ORBITS 13 O l π i λ i Ψ( O , π i ) O ∨ h i ( , − , − , , , , − , − ) 0 0 0 00 2 0 D ( a ) D h i ( − , − , , − , − , , , ) 2 0 0 00 0 2 ( A + A ) ′′ E ( a ) 1 2 3 5 6 h i ( , , − , − , , − , − , ) 1 0 1 00 0 1 A + 2 A h i ( , , − , − , , − , , − ) 1 0 0 01 1 0 D ( a ) + A D ( a ) + A h i (1 , , − , , , − , − ,
0) 0 1 0 10 0 0 A + A A + A h i ( , , − , − , , , − , ) 0 0 2 00 0 0 A + A + A ( A ) ′′ h i ( − , − , , , − , − , , ) 0 0 0 00 2 2 D A + A + A h i (0 , − , − , , , , − ,
1) 0 0 0 20 0 0 A + A A h i ( − , − , , , − , − , ,
2) 1 0 0 01 1 2 D + A h i ( − , − , , , − , − , ,
1) 0 1 0 10 0 2 D ( a ) D ( a ) + A h i ( , , , − , − , − , − , ) 0 2 0 10 0 1 ( A ) ′ h i ( , , , − , − , − , , − ) 0 2 0 00 2 0 E ( a ) D h i ( , , − , − , − , , , ) 2 2 0 00 0 2 ( A ) ′′ D ( a ) 1 2 3 45 7 h i ( , , − , − , − , , , ) 2 1 0 10 0 1 A + A h i ( , , − , − , − , , , ) 2 0 1 01 1 0 D ( a ) h i ( , , − , − , − , , , ) 2 0 0 20 0 0 E ( a )( A + A ) ′′ h i ( − , − , , , , − , − , ) 0 2 0 00 2 2 D A + 3 A h i (3 , , , , − , − , − ,
0) 0 2 0 20 0 0 A A h i ( − , − , , , , − , − , ) 0 1 1 01 1 2 D + A O l π i λ i Ψ( O , π i ) O ∨ A h i ( , , − , − , − , − , , ) 2 2 1 01 1 2 D h i ( , , − , − , − , − , , ) 2 2 0 20 0 2 E ( a )(3 A ) ′′ h i ( − , , , , − , − , − , ) 0 2 0 20 2 2 E h i ( , , − , − , − , − , − , ) 2 2 2 22 2 2 E E Orbits.
The exceptional group E has 32 Richardson orbits exclud-ing E ( a ) + A and E ( a ). Fix the Dynkin diagram 1 2 3 4 56 7 8 withsimple roots: (1 , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , , , , , , , , − , − , − , − , − , − , − , − ) . O l π i λ i Ψ( O , π i ) O ∨ E φ h i (0 , , , , , , ,
0) 0 0 0 0 00 0 0 0 E ( a ) 1 h i ( , − , , , , , ,
0) 1 0 0 0 00 0 0 A E ( a ) 1 3 h i ( , − , , − , , , ,
0) 0 0 0 0 00 0 1 2 A E ( a ) 3 6 7 h i (0 , , , , , , ,
0) 0 1 0 0 00 0 0 3 A h i (0 , , , − , , , ,
0) 2 0 0 0 00 0 0 A E ( a ) 1 3 6 7 h i ( , − , , , , , ,
0) 0 0 0 0 01 0 0 4 A h i ( , − , , − , , , ,
0) 1 0 0 0 00 0 1 A + A E ( b ) 1 2 4 6 h i (1 , , − , , − , , − ,
0) 0 0 1 0 00 0 0 A + 2 A XCEPTIONAL RICHARDSON ORBITS 15 O l π i λ i Ψ( O , π i ) O ∨ E ( a ) 1 2 4 6 7 h i (1 , , − , , , , ,
0) 0 0 0 0 00 1 0 A + 3 A h i (1 , , − , , − , , ,
0) 0 0 0 0 00 0 2 2 A E ( a ) 1 2 3 h i ( , , − , − , , , ,
0) 2 0 0 0 00 0 1 A E ( b ) 1 2 4 5 8 h i (1 , , − , , , − , − , −
1) 0 1 0 0 00 0 1 2 A + A h i ( , − , − , , − , − , − , − ) 1 0 1 0 00 0 0 A + A h i ( , − , − , , − , − , , ) 0 2 0 0 00 0 0 D ( a ) E ( a ) 1 2 4 6 7 8 h i (1 , , − , − , − , , , −
1) 0 0 0 1 00 0 0 2 A + 2 A h i ( , − , − , , − , , − , − ) 1 0 0 0 00 1 0 A + 2 A h i (0 , − , − , , − , , ,
0) 0 1 0 0 01 0 0 D ( a ) + A D ( a ) 1 2 3 56 h i ( , , − , − , , , − ,
0) 0 0 1 0 00 0 1 A + A E ( b ) 1 2 3 6 7 8 h i (1 , , − , − , , , , −
1) 0 0 0 0 10 0 0 A + A + A h i (1 , , − , − , − , , ,
0) 0 0 0 0 02 0 0 D ( a ) + A D ( a ) 1 2 3 56 7 h i ( , , − , − , , , , −
1) 0 0 0 1 00 0 1 2 A h i ( , , − , − , , , ,
1) 1 0 0 0 10 0 0 A + 2 A E h i (0 , , , , , , ,
0) 2 2 0 0 00 0 0 D D + A h i (0 , − , − , , , − , , −
1) 0 0 2 0 00 0 0 A + A E ( a ) 2 4 56 7 h i ( − , , − , , , , , − ) 2 1 0 0 01 0 0 D + A h i ( , , − , , , , , ) 2 0 1 0 00 0 1 D ( a ) O l π i λ i Ψ( O , π i ) O ∨ A + A h i ( , , − , − , − , , , − ) 0 0 1 0 00 1 0 A + A + A D ( a ) 1 2 3 4 5 h i ( , , , − , − , − , , − ) 1 0 1 0 00 0 2 A h i ( , , , − , − , − , − , ) 0 2 0 0 00 0 2 E ( a ) A h i (1 , , − , , , , ,
0) 2 0 0 0 02 0 0 D + A E ( a ) 1 2 3 56 7 8 h i (1 , , − , − , , , , −
3) 0 1 0 0 10 0 0 A + A h i (1 , , − , − , , , , −
2) 1 0 1 0 00 1 0 D ( a ) + A h i ( , − , − , − , , , − , − ) 1 0 0 0 10 0 1 A + A h i (1 , , − , − , , , , −
1) 0 1 0 1 00 0 1 E ( a ) + A h i (0 , − , − , − , , , , −
1) 0 1 0 0 01 1 0 D ( a ) h i (1 , − , − , − , , , ,
0) 0 0 1 0 10 0 0 E ( a ) h i (0 , − , − , − , , , ,
1) 0 0 0 2 00 0 0 E ( a ) D h i (0 , , , , , , ,
0) 2 2 0 0 00 0 2 D E ( a ) 2 4 56 7 8 h i ( − , − , − , , , , , − ) 2 1 0 1 00 0 1 D + A h i ( − , − , − , , , , , − ) 2 1 0 0 01 1 0 D ( a ) D + A h i (3 , , , , − , − , − ,
0) 0 0 2 0 00 0 2 A A + A + A h i ( , , , − , − , − , − , − ) 0 0 1 0 10 0 1 A + A A + A h i ( − , − , − , , , , , −
2) 2 0 0 2 00 0 0 D + A XCEPTIONAL RICHARDSON ORBITS 17 O l π i λ i Ψ( O , π i ) O ∨ D ( a ) + A h i (2 , , , − , − , − , , −
3) 0 1 1 0 10 0 1 A h i ( , , − , − , − , − , , − ) 2 0 0 0 20 0 0 E ( b ) A h i ( , , , , , , , − ) 2 1 0 0 01 1 2 D h i ( − , , , , , , , ) 2 0 1 0 10 0 2 E ( a ) D h i ( − , − , , , , , , −
4) 2 2 2 0 00 0 2 E D ( a ) 1 3 4 56 7 8 h i ( − , − , , , , , , −
5) 2 2 1 0 10 0 1 E + A h i ( − , − , , , , , , −
4) 2 2 0 1 01 1 0 E ( a ) h i ( − , − , , , , , , −
3) 2 2 0 0 20 0 0 E ( b )2 A h i (6 , , , , , , , −
1) 1 0 1 1 01 1 2 D h i (6 , , , , , , ,
1) 0 2 0 0 20 0 2 E ( a ) A h i ( − , , , , , , , −
9) 2 2 2 1 01 1 2 E h i ( − , , , , , , , −
8) 2 2 2 0 20 0 2 E ( a )0 1 2 3 4 56 7 8 h i (6 , , , , , , , −
23) 2 2 2 2 22 2 2 E References [A] P. Achar,
Equivariant coherent sheaves on the nilpotent cone for complex reduc-tive Lie groups , Ph.D. Thesis, Massachusetts Institute of Technology 2001[AS] P. Achar and E. Sommers,
Local systems on nilpotent orbits and weighted Dynkindiagrams , Represent. Theory (2002), 190-201[Alv] D. Alvis, Induce/restrict matrices for exceptional Weyl groups , arXiv:0506377 [AL] D. Alvis, G. Lusztig, On Springer’s correspondence for simple groups of type E n ( n = 6 , , , 65-72[B1] D. Barbasch, Representations with maximal primitive ideal , Operator algebras,unitary representations, enveloping algebras, and invariant theory (Paris, 1989),Progr. Math. (1990), 317-331 [B2] D. Barbasch, Unipotent representations and the dual pair correspondence , arXiv:1609.08998 , to appear in volume in honor of Roger Howe[BV] D. Barbasch, D. Vogan, Unipotent representations of complex semisimple Liegroups , Ann. of Math. (1985), 41-110[Ca] R. W. Carter,
Finite groups of Lie type , Wiley & Sons (1993)[CO] T. Chmutova, V. Ostrik,
Calculating canonical distinguished involutions in theaffine Weyl groups , Experiment. Math. (2002), 99-117[CM] D. Collingwood, W. M. McGovern, Nilpotent orbits in semisimple Lie algebras ,Van Norstrand Reinhold Mathematics Series (1993)[dGE] W. A. de Graaf, A. Elashvili,
Induced nilpotent orbits of the simple Lie algebrasof exceptional type , Georgian Mathematical Journal (2009), 257-278[FJLS] B. Fu, D. Juteau, P. Levy, E. Sommers, Generic singularities of nilpotent orbitclosures , Advances in Mathematics (2017), 1-77.[J] J. Jantzen,
Nilpotent orbits in representation theory , Progress in Mathematics , Birkhauser Boston (2004)[L1] G. Lusztig,
Characters of reductive groups over a finite field , Ann. Math. Studies (1984)[L2] G. Lusztig,
Notes on unipotent classes , Asian J. Math. (1997), 194-207[L3] G. Lusztig, Unipotent classes and special Weyl group representations , J. Algebra. (2009), 3418-3449[LS] G. Lusztig, N. Spaltenstein,
Induced unipotent classes , J. London Math. Soc. (2) (1979), no. 1, 41-52[McG] W. M. McGovern, Rings of regular functions on nilpotent orbits and their covers ,Invent. Math. (1989), 209-217[S1] E. Sommers, A generalization of the Bala-Carter theorem , Int. Math. Res. Not. (1998), 539-562,[S2] E. Sommers, Lusztig’s canonical quotient and generalized duality , J. Algebra (2001), 790-812[S3] E. Sommers,
Irreducible local systems on nilpotent orbits , arXiv:1610.07645 , toappear in Bull. Inst. Math. Acad. Sinica[V1] D. Vogan, The orbit method and primitive ideals for semisimple Lie algebras , LieAlgebras and related topics (Windsor, Ont., 1984), CMS Conf. Proc. (1986),281-316[V2] D. Vogan, Associated varieties and unipotent representations , Harmonic Analysison Reductive Groups (W. Barker and P. Sally, eds.), Birkhauser, Boston-BaselBerlin (1991)[V3] D. Vogan,
The method of coadjoint orbits for real reductive groups , Park CityMathematics Series (1998)[W1] K. D. Wong, Quantization of Special Symplectic Nilpotent Orbits and Normalityof their Closures , J. Algebra (2016), 37-53[W2] K. D. Wong,
Some Calculations of the Lusztig-Vogan Bijection for ClassicalNilpotent Orbits , J. Algebra (2017), 317-339[W3] K. D. Wong,
Special Unipotent Representations with Half Integral InfinitesimalCharacters , https://arxiv.org/abs/1702.01503 The Chinese University of Hong Kong, Shenzhen, Longgang District, Shen-zhen, Guangdong 518172, China
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