Annihilators and associated varieties of Harish-Chandra modules for Sp(p,q)
aa r X i v : . [ m a t h . R T ] J a n Annihilators and associated varieties of Harish-Chandra mod-ules for
S p ( p , q ) WILLIAM MCGOVERN
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195
Introduction
The purpose of this paper is to extend the recipes of [5] from the group SU ( p , q )to the group Sp( p , q ); we will compute annihilators and associated varieties ofsimple Harish-Chandra modules for the latter group. We will appeal to the clas-sification of primitive ideals in enveloping algebras of types B and C in [4] viatheir generalized τ -invariants; thus (inspired by [5]) we will define a map H tak-ing (parameters of) simple Harish-Chandra modules X of trivial infinitesimalcharacter to pairs ( T , T ) where T is a domino tableau and T an equivalenceclass of signed tableaux of the same shape as T . Then T will parametrize theannihilator of X (via the classification of primitive ideals in [4]) while any rep-resentative of T , suitably normalized, parametrizes its associated variety (viathe classification of nilpotent orbits in Lie Sp( p , q ) in [1, 9.3.5]). The proof ofthese properties will rest primarily on the commutativity of the maps H withboth τ -invariants and wall-crossing operators T αβ , defining the latter operatorsas in [10]. We will parametrize our modules X via signed involutions of signature( p , q ) and construct the tableaux from the involutions. Section 1 Cartan subgroups and Weyl groups
For G = Sp( p , q ), n = p + q , p ≤ q we set g = Lie G and we let g be its complex-ification. Let θ be the usual Cartan involution of G or g and let k + p be thecorresponding Cartan decomposition. Denote by K the subgroup of the complexi-fication of G corresponding to k . Let H be a compact Cartan subgroup of G withcomplexified Lie algebra h . As a choice of simple roots in g relative to h we take2 e , e − e ,... , e n − − e n − , e n − e n − , following [4].There are p + G . If we take H to be a compact Cartan subgroup and define H i inductively for i > H i − through e p − i + − e p + i for 1 ≤ i ≤ p , then the H i furnisha complete set of representatives for the conjugacy classes of Cartan subgroupsof G . The real Weyl group W ( H i ) of the i th Cartan subgroup H i is isomorphicto W p − i ⋉ S p − i × W i × W q − p + i , where W r , S j respectively denote the hyperoctahe-dral group of rank r and the symmetric group on j letters; here W ( H i ) embedsinto the complex Weyl group W of g (relative to h ) by permuting, interchanging,and changing the signs of the first p − i pairs of coordinates and then permut-ing and changing the signs of the next i and then the next q − p + i coordinates[6, 7]. The subgroups H i are all connected and there is a single block of simpleHarish-Chandra modules for G with trivial infinitesimal character.1 ection 2 The D set and Cartan involutions Using Vogan’s classification of simple Harish-Chandra modules with trivial in-finitesimal character by Z /2 Z -data [11], we parametrize such modules for thesegroups combinatorially, as follows. Define S n , the set of signed involutions on n letters, to be the set of all sets { s ,... , s m } , where each s i takes one of the forms( i , + ),( i , − ),( i , j ) + ,( i , j ) − , where i , j lie between 1 and n , the pairs ( i , j ) are orderedwith i < j , and each index i between 1 and n appears in a unique pair of exactlyone of the above types. We say that σ ∈ S n has signature ( p , q ) or lies in S n , p , ifthe total number of singletons ( i , + ) and pairs ( i , j ) + or ( i , j ) − in σ is p . Let I n denote the set of all involutions in the complex Weyl group W n , Identify an ele-ment ι ∈ I n with an element σ ∈ S n by decreeing that ( j , + ) ∈ σ if ι ( j ) = j ,( j , − ) ∈ σ if ι ( j ) = − j ,( i , j ) + ∈ σ if the (positive) indices i , j are flipped by ι ,( i , j ) − ∈ σ if in-stead the indices i , − j are flipped by ι . Then W n acts on I n by conjugation and wemay transfer this action to S n via the above identification. The set of involutionsflipping p − r pairs of indices in { ± ± ( p + q ) } is stable under conjugation andhas cardinaility equal to the index of W ( H p − r ), whence we may take S n , p as aparametrizing set D for the set of simple Harish-Chandra modules for Sp( p , q )with trivial infinitesimal character. More generally, as needed for inductive ar-guments below, we define for any subset M of { n } the set S M , p in the sameway as S n , p , replacing the numbers 1,... , n by the numbers in M .The Cartan involution θ corresponding to any σ ∈ S n , p fixes a unit coordinatevector e i whenever ( i , + ) ∈ σ or ( i , − ) ∈ σ . If ( i , j ) + ∈ σ , then θ flips the vectors e i and e j , while if ( i , j ) − ∈ σ , then θ flips e i and − e j , unless j − i =
1, in which case θ sends both e i and e j to their negatives. Thus a simple root e i + − e i is compactimaginary for σ if and only if ( i , ǫ ) and ( i + ǫ ) both lie in σ for some sign ǫ , while2 e is compact imaginary if and only if (1, ǫ ) lies in σ for some sign ǫ . Section 3 The cross action and the Cayley transform
For an element σ of S n , p and a pair of indices i , j between 1 and n , we defineIn( i , j , σ ) as in [5, Definition 1.9.1], interchanging the unique occurrences of i and j in σ and leaving all others unchanged (so that σ is unchanged if ( i , j ) + or( i , j ) − occurs in it). We define SC(1, σ ) to be σ with the pair (1, i ) ǫ in it replaced by(1, i ) − ǫ , − ǫ the opposite sign to ǫ , if there is such a pair in σ ; otherwise SC( σ ) = σ .We define In(1,2, σ ) ′ to be σ with the pairs (1, i ) ǫ ,(2, j ) ǫ ′ replaced by (1, j ) − ǫ ′ ,(2, i ) − ǫ if such pairs occur in σ ; otherwise we set In(1,2, σ ) ′ = In(1,2, σ ).3.1. PROPOSITION. Let s be the simple root e i − e i − , σ ∈ S n , p . Then s × σ , thecross action of s on the parameter σ , is given by In( i − i , σ ). For t = e and σ ∈ S n , p we have t × σ = SC(1, σ ). Proof.
This may be computed directly, along the lines of [5, §§1.8,9].As in [5], we will also need to compute Cayley transforms of parameters σ through simple roots. It suffices to consider the simple root s = e i + − e i . Define c i ( σ ) to be σ with the pairs ( i , ǫ ),( i + − ǫ ) replaced by ( i , i + ǫ for σ ∈ S n , p , if such2airs occur in σ ; otherwise c i ( σ ) is undefined. Thus c i ( σ ) is defined if and only if e i + − e i is an imaginary noncompact root for σ . In a similar way, define c i ( σ ) if e i + − e i is real for σ to be the i through nvolution obtained from σ by replacing( i , i + ǫ by ( i , ǫ ),( i + − ǫ ).3.2. PROPOSITION. If e i + − e i is imaginary noncompact for a parameter σ in S n , p , then the Cayley transform of σ through this root is given by c i ( σ ). If thisroot is real for σ , then the Cayley transform through it is given by c i ( σ ). Proof.
Again this follows from a direct computation, along the lines of [5, 1.12].
Section 4 τ -invariants and wall-crossing operators We define the τ -invariant τ ( σ ) of a parameter σ in S n , p in the same way as in[5, 1.13]: it consists of the simple roots that are either real, compact imaginary,or complex and sent to negative roots by the Cartan involution θ . We extendthis definition to S M , p as in [5, 1.15]. Similarly, if α , β are nonorthogonal simpleroots of the same length, then we define the wall-crossing operator T αβ on S n , p as in [5, 1.14] and [10]. It is single-valued. If α , β are nonorthogonal but havedifferent lengths we define T αβ on S n , p in the same way; in this setting it takeseither one or two values. In both cases T αβ sends parameters with α not in the τ -invariant but β in it to parameters with the opposite property. In more detail,if a parameter σ includes (1,2) + , then the effect of T αβ on it is to replace (1,2) + by either 1 + ,2 − or 1 − ,2 + , and vice versa; otherwise we interchange 1 and 2 in σ if at least one of them is paired with another index but they are not paired witheach other, or we change the sign attached to the pair (1, i ) ǫ in σ , whichever (orboth) of these operations has the desired effect on the τ -invariant of σ . Section 5 The algorithm
We now describe the algorithm we will use to compute for a given σ ∈ S n , p theannihilator and associated variety of the corresponding Harish-Chandra modulefor Sp( p , q ). We will attach an ordered pair ( T , T ) to σ , where T is a dominotableau in the sense of [2] and T is an equivalence class of signed tableaux ofsignature (2 p ,2 q ) and the same shape as T . The shape of T will be a doubledpartition of 2( p + q ); that is, a partition of 2( p + q ) in which the parts occur inequal pairs. Any tableau in T will thus also have rows occurring in pairs, calleddouble rows, of equal length; moreover the two rows in a double row will beginwith the same sign. The equivalence relation defining T will be that we canchange all signs in any pair of double rows of the same even length, or in anypair D , D of double rows of different even lengths whenever there is an opencycle of T in the sense of [4, 3.1.1] (so not including its smallest domino) with itshole in one of D , D and its corner in the other. Here we allow the second doublerow D to have length 0, so that, for example, if3 = then the two signed tableaux of the same shape (one with its rows beginning with + , the other with − ) both lie in T , since if T is moved through the open cycle ofits 2-domino, then that domino is given a clockwise quarter-turn, so that it nowoccupies one square of a double row that was empty in T . On the other hand,if T is replaced by its transpose, then the two signed tableaux of this shape liein separate classes T , since in that case the open cycle of the 2-domino doesnot intersect the empty double row. In addition to this equivalence relation, wedecree as usual for signed tableaux that any two of them are identified wheneverone can be obtained from the other by interchanging pairs of double rows of thesame length. The signature of T (i.e. the number of + signs in it) will be 2 p ; notethat this is an invariant of T . To construct T and T we follow a similar recipeto [5, Chap. 3], replacing the tableaux occurring there with domino tableaux andusing insertion and bumping for domino tableaux as in [2].5.1. DEFINITION. Let σ ∈ S n , p . Order the elements of σ by increasing sizeof their largest numbers. We construct the pair H ( σ ) = ( T , T ) attached to σ inductively, starting from a pair of empty tableaux. At each step we insert thenext element ( i , ǫ ) or ( i , j ) ǫ into the current pair of tableaux. Assume first thatthe next element of σ is ( i , ǫ ) (with ǫ a sign) and choose any representative T of T .(1) If the first double row of T ends in − ǫ , then add ǫ to the end of both of itsrows and add a vertical domino labelled i to the end of the first double rowof T .(2) If not and if the first double row of T has (rows of) even length, then welook first for a lower double row of T with the same length ending in − ǫ ; ifthere is a such a double row, we interchange it with the first double row in T and then proceed as above. Otherwise we start over, trying to insert ( i , ǫ )into the highest double row of T strictly shorter than its first double row.(In the end, we may have to insert a domino labelled i into a new double rowof T , using ǫ for both signs in the new double row of T .)(3) If not and the first (or first available) double row of T has odd length butthere is more than one double row of this length, but none ending in − ǫ ,then we change all signs in the first two double rows of T of this length andthen proceed as in the previous case.(4) Otherwise the highest available double row R in T has even length, endsin ǫ , and is the only double row of this length. In this case we look at thedomino in T occupying the last square in the lower row of R . If we move T through the open cycle of this domino, we find that its shape changes byremoving this square and adding a square either at the end of the higherrow of some double row R ′ of T or else in a new row, not in T . If it lies in anew row, then change all signs in R and proceed as above. If it does not lie4n a new row and R ′ R , then change the signs of T in both R and R ′ andproceed as above (again not actually moving T through the cycle). Finally, if R = R ′ , then move T through the open cycle, place a new horizontal dominolabelled i at the end of the lower row of R in T , and choose the signs in T so that both rows of R now end in ǫ while all other rows of T have the samesigns as before.5.2. DEFINITION. Retain the notation of the previous definition but assumenow that the next element of σ is ( i , j ) ǫ . We begin by inserting a horizontaldomino labelled i at the end of the first row of T if ǫ = + , or a vertical dominolabelled i at the end of the first column of T if ǫ = − , following the procedure of[2] (and thus bumping dominos with higher labels as needed). We obtain a newtableau T ′ , whose shape is obtained that of T by adding a single domino D , lyingeither in some double row R of T or else in a new row (in which case D must behorizontal). Let ℓ be the length of R (before D was added).(1) If D is horizontal and ℓ is even, then add a domino labelled j to T ′ immedi-ately below the position of D , in the lower row of R . Choose signs in T sothat both rows of R now end in a different sign than they did before; leaveall other signs the same. If D lies in a new row, then we have a new doublerow in T , which can begin with either sign; to make a particular choice, wedecree that both rows in the new double row begin with − .(2) If D is horizontal and ℓ is odd, then T ′ does not have special shape in thesense of [2], but its shape becomes special if one moves through just oneopen cycle. Move through this cycle and choose the signs in T so that R is now a genuine double row and its rows end in a different sign than theydid before. Then T has either two more + signs than before or two more − signs. Insert a vertical domino labelled j to the first available double row in T strictly below R , following the procedure of the previous definition. Thesign attached to j is − if T gained two + signs and is + otherwise.(3) If D is vertical and ℓ is even, then R is still a double row; choose signs sothat its rows end in the same sign as they did before, leaving all other signsunchanged. If a new double row was created by inserting the new domino,then its rows can begin with either sign; to make a particular choice, wedecree that this sign is + . Now add a new vertical domino j to the firstavailable double row strictly below R , as in the previous case, giving it thesame sign as in that case.(4) If D is vertical and ℓ is odd, then proceed as in the previous case.One can check that either choice of sign made in Definition 5.2(1) or (3) gives riseto equivalent tableaux T , so that in the end we get a well-defined equivalenceclass T . To compute the associated variety of the Harish-Chandra module cor-responding to σ , we choose any representative of T and normalize it so that alleven rows begin with + ; this is because this variety is the closure of one nilpo-tent K -orbit in p ∗ [9, 5.2] and such orbits (via the Kostant-Sekiguchi bijectionbetween them and nilpotent orbits in g ) are parametrized by signed tableaux of5ignature (2 p ,2 q ) in which even rows begin with + [1, 9.3.5]. Later we will showthat the map H defines a bijection between S n , p and pairs ( T , T ) with T ofsignature (2 p ,2 q )We give two examples. First let σ = { (1,2) + } ∈ S . Then we get T = while T consists of both signed tableaux of this shape, as noted above. In theother example, we let σ = { ((1, + ),(2, − ),(3,4) + ,(5, + ) } ∈ S . Then T = and T = + − + − + where we denote signed tableaux by tableaux tiled by vertical dominos, eachlabelled with the (common) sign of each of its squares; note that here T consistsof a single tableau. Section 6 τ -invariants and T αβ on tableaux We define the τ -invariant τ ( T , T ) of a pair ( T , T ), or just of its domino tableau T , in the same way as [3, 2.1.9]; thus for example 2 e lies in the τ -invariant of T in type C if and only if the 1-domino in it is vertical. If α , β are simple rootsof the form e i − e i − , e i + − e i for some i , then we define T αβ on a domino tableau T lying in the domain D αβ of this operator as in [3, 2.1.10] and extend this toa pair ( T , T ) by making the operator act trivially on T . We now define T αβ in the other cases, following the notation of [3, 2.3.4], and as in [3] defining thisoperator on pairs rather than single tableaux.6.1. DEFINITION. Suppose that α = e , β = e − e . If T ∈ D αβ , then either F ⊆ T or ˜ F ⊆ T .(1) In the first case, let T ′ be obtained from T by replacing F by F . If the2-domino of T ′ lies in an open cycle not including the 1-domino and if theequivalence class T breaks up into two classes T ′ , T ′′ with respect to T ′ ,then we set T αβ ( T , T ) = (( T ′ , T ′ ),( T ′ , T ′′ )). If the 2-domino lies in a closedcycle c , then let ˜ T ′ be the tableau obtained from T ′ by moving through c andwe set T αβ ( T , T ) = (( ˜ T ′ , T ),( T ′ , T )). Otherwise set T αβ ( T , T ) = ( T ′ , T ).(2) If instead ˜ F ⊆ T , then the 2-domino of T lies in a closed cycle c , since T has the (special) shape of a doubled partition; if this cycle were open, itwould have to be simultaneously an up and down cycle in the sense of [4,§3], a contradiction. Let ˜ T be obtained from T by moving through c andlet ˜ T ′ be obtained from ˜ T by replacing F by F . Then set T αβ ( T , T ) =( ˜ T ′ , T ). 63) If instead α = e − e , β = e , then define T αβ ( T ) for T ∈ D αβ as above,interchanging F , F throughout by ˜ F , ˜ F .For example, if T is as in the first example in the last section, so that T con-sists of both signed tableaux of this shape, then T e , e − e sends ( T , T ) to thepair (( T ′ , T ′ ),( T ′ , T ′′ ), where T ′ the transpose of T and T ′ , T ′′ are the two signedtableaux in T . Note also that, unlike [3, 2.3.4], we must not move through anyopen cycles, as all of our tableaux must have doubled partition shape. There areno right domino tableaux and so there is no notion of extended cycle. Section 7 H commutes with τ -invariants As in [5], we prove that our algorithm H computes the annihilators of simpleHarish-Chandra modules by showing that it commutes with taking τ -invariantsand applying wall-crossing operators. In this section we deal with τ -invariants.7.1. PROPOSITION. Let σ ∈ S n , p and α a simple root for Sp( p , q ). Then α ∈ τ ( σ )if and only if α ∈ τ ( H ( σ ). Proof.
We enumerate all possible ways in which α can lie in τ ( σ ), or fail to lie inthis set, and then check directly that the conclusion holds in each case.Suppose first that α ∈ τ ( σ ).(1) If α = e i + − e i is compact imaginary, then we must have ( i , ǫ ),( i + ǫ ) ∈ σ forsome sign ǫ . The i -domino starts out vertical and in the first double row of T ; eventually either the i - and ( i + i + i -domino winds up above the ( i + α = e is compact imaginary, then (1, ǫ ) ∈ σ for some sign ǫ , sothat the 1-domino is vertical in T .(2) If α = e i + − e i is real, then we must have ( i , i + + ∈ σ . It is clear from thealgorithm that the i -domino winds up below the ( i + α is complex. If α = e , then we must have (1, j ) − ∈ σ for some j ,and then it is clear that the 1-domino winds up vertical in the first doublerow of T , while the j -domino lies below this double row.(4) We are now reduced to the case where α = e i + − e i , α complex. If( i , ǫ ),( j , i + ǫ ′ ∈ σ for some j < i for signs ǫ , ǫ ′ and if the i -domino is ver-tical when adjoined to T , then it is added to the end of some double row R such that the double rows above it end in the same sign as R in T (sincethe i -domino was not put into a higher row). When the j -domino is inserted,adding a domino D to the shape of T , the additional signs added to T , if D is vertical, are both − ǫ , whence the ( i + ǫ and winds up in a row below R (since all higher rows end withthe same sign as they did before the j -domino was inserted). If D is horizon-tal, then it lies in the bottom row of T , and once again, the ( i + i -domino ends up hori-zontal (lying directly below another horizontal domino) when it is adjoinedto T .(5) If ( i + ǫ ),( i , k ) + ∈ σ for some k > i +
1, then the ( i + T ; the i -domino is adjoined horizontallyeither to this double row or a higher one, and if to this double row bumpsthe ( i + i -domino, as desired.(6) If ( j , i + ǫ ,( i , k ) + ∈ σ and j < i < k , then the i -domino is added to the firstrow of T , while the ( i + j , i ) ǫ ,( i + k ) − ∈ σ .(7) If ( j , i ) + ,( j , i + ǫ ∈ σ and j < j < i , then adding the j -domino bumps thedominos that were previously bumped by adding the j -domino, togetherwith at least one domino in the double row of the i -domino, so that the( i + i -domino.(8) If ( j , i ) ǫ ,( j , i + − ∈ σ and j < j < i , then since the j -domino is insertedvertically into the first column of T , the ( i + i -domino, as in the previous case.(9) if ( j , i ) + ,( j , i + − ∈ σ and j < j , then again the ( i + i -domino, similarly to the previous case.(10) if ( i + k ) ǫ ,( i , k ) + ∈ σ and i + < k < k , then either the i -domino bumpsthe ( i + i -domino winds up horizontal in the first dou-ble row, while the ( i + i -domino.(11) if ( i , k ) − ,( i + k ) − ∈ σ and i + < k < k , then the ( i + i -domino.(12) if ( i , k ) + ,( i + k ) − ∈ σ and i + < k < k , then as above the i -dominowinds up in the first double row while the ( i + α ∈ τ ( σ ). Now suppose the contrary. The caseswhere α = e are easily dealt with, so assume that α = e i + − e i .(1) If α is noncompact imaginary, so that ( i , ǫ ),( i + − ǫ ) ∈ σ , then the ( i + i -domino or in a higher row.(2) If ( j , i ) − ,( i + ǫ ) ∈ σ and j < i , then either the ( i + − ǫ in T , whence the ( i + i -domino.(3) If ( i , ǫ ),( i + k ) ′ + ∈ σ and + i < k , then the ( i + i -domino.(4) If ( i + ǫ ),( i , k ) − ∈ σ and i < k then the ( i + i -domino.85) If ( j , i ) ǫ ,( i + m ) + ∈ σ and j < i , then the ( i + i -domino.(6) If ( j , i ) ǫ ,( j , i + + ∈ σ and j < j < i , then either the ( i + i -domino appears, or the i -domino is in the first double row and the( i + i + i -domino.(7) If ( j , i ) − ,( j , i + + ∈ σ and j > j < i , then the i -domino is in the lowestdouble row of T and the ( i + j , i ) − ,( j , i + − ∈ σ and j < j < i , then the i -domino is in the lowestdouble row of T and ( i + i , k ) ǫ ,( i + k ) + ∈ σ and i + < k < k , then the ( i + T and does not lie below the i -domino.(10) If ( i + k ) − ,( i , k ) − ∈ σ and i + < k < k , then the i -domino bumps the( i + i + k ) + ,( i , k ) − ∈ σ , i + < k < k , then the ( i + i -domino.This exhausts all cases and concludes the proof. Section 8 H commutes with T αβ .Again following [5], we complete our program of showing that the map H com-putes annihilators by showing that it commutes with wall-crossing operators.8.1. PROPOSITION. Let α , β be nonorthogonal simple roots and let σ ∈ S n , p liein the domain of the operator T αβ . Then H ( T αβ ( σ )) = T αβ ( H ( σ ). Proof.
As in [5] we enumerate all ways in which σ can lie in the domain of T αβ and check that the conclusion holds in all cases. Let T , T respectively denotethe domino tableau and a representative of the class of signed tableaux attachedto σ by the algorithm.Suppose first that { α , β } = { e , e − e } . If (1,2) + ∈ σ , then F ⊆ T and T αβ ( σ )consists of the two involutions σ , σ obtained from σ by replacing (1,2) + by(1, + ),(2, − ) and (1, − ),(2, + ) in turn. Clearly the domino tableaux attached to σ , σ are both equal to the tableau T ′ defined in Definition 6.1 (1). The signedtableaux attached to these involutions at the second step of the algorithm arenot equivalent at that step, whence by the algorithm they remain inequivalentat its end. Hence H ( σ ) H ( σ ), as desired. The cases where (1, + ),(2, − ) ∈ σ or(1, − ),(2, + ) ∈ σ are similar. Finally, in the case where at least one of the indices1 and 2 is paired with another index but 1 and 2 are not paired with each other,one clearly moves the 1- and 2-dominos in T in the desired fashion, whence onecan check that if any other dominos move, they are the ones in the closed cycle9ontaining the 2-domino of T and in fact the two domino tableaux produced arethose specified by Definition 6.1 (cf. [3, 2.2.9,2.3.7]). If the 2-domino of T doesnot lie in a closed cycle, then only one domino tableau is produced, which againagrees with that given by this definition.Henceforth we assume that α = e i − e i − , β = e i + − e i for some i ≥
2. Set σ ′ = T αβ ( σ ) and let T ′ , T ′ respectively denote the domino tableau and a repre-sentative of the class of signed tableaux attached to σ ′ by the algorithm. Thecases in our discussion below are parallel to the corresponding cases in the proofof [5, Proposition 4.2.1]. That proof shows that the desired result holds whenevernone of the indices i − i , i + a , b ) − in σ and T αβ doesnot act on T by an F -type interchange in the sense of [3], using [3, 2.1.20,2.1.21]in place of the results in Section 2.5 of [5]: in all cases either the ( i − i - or i - and ( i + T , whichever of these has the desiredeffect on τ -invariants. Apart from this one changes signs and moves throughopen cycles in the same way in the constructions of T , T ′ and T , T ′ , so that T ′ is equivalent to T , as desired. If an ordered pair ( a , b ) − involving one of theindices i − i , i + σ , then one checks directly that T ′ is obtainedfrom T by either interchanging the ( i − i - or i - and ( i + T ′ = T , as desired; note that ordered pairs ( a , b ) − have no analogue in[5].. It only remains to show that the desired result holds whenever T αβ acts on T by an F -type interchange (again, there is no analogue of such an interchangein [5]). In each case below, we indicate how many subcases involve an F -typeinterchange; then the result follows by a direct calculation in each such subcase.Throughout we denote by j , j , j , j indices less than i − j < j < j , andsimilarly by k , k , k , k indices greater than i + k < k < k .(1) Suppose first that ( i − ǫ ),( i , − ǫ ),( i + − ǫ ) all lie in σ for some sign ǫ , whichfor definiteness we take to be + . Then σ ′ is obtained from σ by replacing theterms ( i − + ),( i , − ) by the single term ( i − i ) + . Let ˜ σ consist of the termsof σ involving only indices less than i − T , ˜ T be the domino andrepresentative of the class of signed tableaux attached to ˜ σ by the algorithm.There are four subcases, according as the top double row of ˜ T has even orodd length and ends with + or − , but only one of these has T αβ acting on T by an F -type interchange. One checks directly that the conclusion holds inthis case.(2) If instead ( i − ǫ ),( i , i + + ∈ σ , so that ( i − ǫ ),( i , ǫ ),( i + − ǫ )) ∈ σ ′ , then againonly one subcase out of four has T αβ acting on T by an F -type interchange,and the conclusion holds in that case.(3) If ( j , i − ǫ ,( i , i + + ∈ σ , so that ( j , i ) ǫ ,( i − i + + ∈ σ ′ , then no F -type in-terchange ever takes place.(4) If ( i − i + ǫ ,( i , k ) + ∈ σ , so that ( i − i ) ǫ ,( i + k ) + ∈ σ ′ , then no F -typeinterchange ever takes place.(5) If ( j , i − ǫ ,( i , ǫ ′ ),( i + ǫ ′ ) ∈ σ , so that ( i − ǫ ′ ),( j , i ) ǫ ),(( i + ǫ ′ ) ∈ σ ′ , thenthere are eight subcases, depending as in case 1 on the length parity andsign at the end of the top double row, and this time also on whether the10 −
1- and i + F -type interchange and the desired result holds in both of them.(6) If ( i − ǫ ),( i , − ǫ ),( j , i + ǫ ′ ∈ σ , so that ( i − ǫ ),( j , i ) ǫ ′ ,( i + − ǫ ) ∈ σ ′ , then no F -type interchange takes place.(7) If ( i − ǫ ),( i + ǫ ),( i , k ) + ∈ σ , so that ( i − ǫ ),( i , ǫ ),( i + k ) + ∈ σ ′ , then thereis one case where an F -type interchange occurs and the result holds in thatcase.(8) If ( i − ǫ ),( i + − ǫ ),( i , k ) + ∈ σ , so that ( i , ǫ ),( i + − ǫ ),( i − k ) + ∈ σ ′ , then an F -type interchange always occurs and the result holds in all cases.(9) If ( j , i − + ,( i , ǫ ′ ),( j , i + + ∈ σ , so that ( i − ǫ ),( j , i ) + ,( j , i + + ∈ σ ′ , thenthere is one case where an F -type interchange occurs and the result holdsin it.(10) If ( j , i − + ,( i , ǫ ),( j , i + + ∈ σ , so that ( j , i − + ,( j , i ) + ,( i + ǫ ) + ∈ σ ′ , thenan F -type interchange never arises.(11) If ( j , i − + ,( i + ǫ ),( i , k ) + ∈ σ , so that ( j , i ) + ,( i + ǫ ),( i − k ) + ∈ σ ′ , then an F -type interchange never arises.(12) If ( i − ǫ ),( j , i + + ,( i , k ) + ∈ σ , so that ( i − ǫ ),( j , i ) + ,( i + k ) + ∈ σ ′ , then an F -type interchange never arises.(13) If (( i + ǫ ),( i − k ) + ,( i , k ) + ∈ σ , so that ( i , ǫ ),( i − k ),( i + k ) ∈ σ ′ , thenthere are two subcases where an F -type interchange arises and the resultholds in both of them.(14) If ( i − ǫ ),( i + k ) + ,( i , k ) + ∈ σ , so that (( i , ǫ ),( i + k + ),( i − k ) + ∈ σ ′ , thenthere are two subcases where an F -type interchange arises and the resultholds in both of them.(15) If (( j , i − + ,( j , i ) + ,( j , i + + ∈ σ , so that ( j , i − + ,( j , i ) + ,( j , i + + ∈ σ ′ ,then no F -type interchange occurs.(16) If ( j , i − + ,( j , i ) + ,( j , i + + ∈ σ , so that ( j , i − + ,( j , i ) + ,( j , i + ∈ σ ′ ,then no F -type interchange toccurse.(17) If ( j , i − + ,( j i , i + + ,( i , k ) + ∈ σ , so that ( j , i − + ,( j , i ) + ,( i + k ) + ∈ σ ′ ,then no F -type interchange occurs.(18) If ( j , i − + ,( j , i ) + ,( i , k ) + ∈ σ , so that ( j , i ) + ,( j , i + + ,( i − k ) + ∈ σ ′ , thenno F -type interchange occurs.(19) If ( j , i + + ,( i − k ),( i , k ) + ∈ σ , so that ( j , i ) + ,( i − k ) + ,( i + k ) + ∈ σ ′ ,then no F -type interchange occurs.(20) If ( j , i − + ,( i + k ) + ,( i , k ) + ∈ σ , so that ( j + i ,( i + k ) + ,( i − k ) + ∈ σ ′ , thenno F -type interchange occurs.(21) If ( i + k ) + ,( i − k ) + (, i , k ) + ∈ σ , so that ( i , k ) + ,( i − k ) + ,( i + k ) + ∈ σ ′ ,then no F -type interchange occurs.(22) If ( i − k ) + ,( i + k ) + ,( i , k ) + ∈ σ , so that (( i , k ) + ,( i + k ) + ,( i − k ) + ∈ σ ′ ,then no F -type interchange occurs.11his exhausts all cases and concludes the proof.8.2. THEOREM. Let σ ∈ S n , p . Then the first coordinate T of H ( σ ) parametrizesthe annihilator of the simple Harish-Chandra module corresponding to σ via theclassification of [4, Theorem 3.5.11]. Proof.
Thanks to Propositions 7.1 and 8.1, we know that the primitive ideal I corresponding to T has the same generalized τ -invariant as the Harish-Chandramodule X corresponding to σ , whence by [4, Theorem 3.5.9] I is indeed the anni-hilator of X , since primitive ideals of trivial infinitesimal character in type C areuniquely determined by their generalized τ -invariants.We also see that, since the wall-crossing operators T αβ generate the Harish-Chandra cells for Sp( p , q ) [7, Theorem 1], modules in the same Harish-Chandracell for this group (and trivial infinitesimal character) have the same signedtableaux T attached to them, up to changing the signs in double rows whoserows have even length. Section 9 H is a bijection H defines a bijection between S n , p and ordered pairs( T , T ), where T is a domino tableau with shape a doubled partition of 2( p + q )and T is an equivalence class of signed tableaux of signature (2 p ,2 q ) and thesame shape as T . Proof.
We first show that any ordered pair ( T , T ) as in the hypothesis lies in therange of H , by induction on p + q . Assuming that this holds for all pairs ( T , T )if T has fewer than n = p + q dominos, let T , T be a pair with n dominos in T .Let T ′ be T with the n -domino removed.If the n -domino in T is horizontal and lies in a row of even length, then thenext to last row R of T ′ has two more squares than its last row. By [2, 1.2.13],there is a domino tableau T whose shape is that of T ′ with the last two squaresremoved from R such that inserting a horizontal i -domino for a suitable index i into the first row of T produces the tableau T ′ , or else there is such a tableau T and an index i such that inserting a vertical i -domino into the first column of T produces T ′ . In either case there is a pair ( T ′′ , T ′′ ) = H ( σ ′ ) in the range of H , andif we add ( i , n ) + or ( i , n ) − to σ ′ to get σ (the first pair if the i -domino is horizontal,the second if it is vertical), then H ( σ ) = ( T , T ), as desired. If instead the n -domino in T is horizontal but lies in a row of odd length then we can move T ′ through a suitable open cycle to produce a new tableau T ′′ with shape a doubledpartition such that the shape of T differs from that of T by a single verticaldomino. We then reduce to the case covered in the following paragraph.Now suppose that the n -domino in T is vertical, so that the shape of T ′ isthat of a doubled partition. Let T be a representative of T ; assume for def-initeness that the squares in T corresponding to those of the n -domino in T are labelled + . Look at all the double rows in T above the one corresponding12o the double row with the n -domino in. T . If every such double row consists ofrows of odd length ending in + , then one checks immediately that there is a class T ′ such that the pair ( T , T ′ ) = H ( σ ′ ) lies in the range of H and if we add ( n , + )to σ ′ the resulting involution σ satisfies H ( σ ) = ( T , T ) as desired. Otherwise,if the lowest such double row D has rows of odd length and ends in − , let ˜ T ′ be obtained from T ′ by removing the last squares of the rows of D . There is adomino tableau T with the same shape as ˜ T ′ and an index i such that inserting asuitably oriented i -domino into T gives T ′ . As above there is a class T ′′ of signedtableaux such that ( T , T ′′ ) = H ( σ ′ ) and then there is σ with H ( σ ) = ( T , T ), asdesired. If the lowest such double row has rows of even length ending in + , thenlook at the open cycle through the largest domino in the corresponding doublerow. The argument of the last paragraph produces the desired σ . If the lowestsuch double row D has rows of even length ending in − , then look at the opencycle of T ′ through the largest domino in the corresponding double row. If thisopen cycle has its hole and corner in different double rows D , D , then changeall signs in these double rows of T and argue as in the previous case. Finally, ifthis open cycle has its hole and corner both in D , then move through this opencycle in T ′ and argue as in the case where the n -domino in T is horizontal. Inthis case adjoining the i -domino initially produces a domino tableau where thefirst row of D has length two more than its second row; moving through the opencycle, as specified by Definition 5.2 (2), gives D the shape it has in T and thenbumping the n -domino into the next lower double row yields T , as desired.Now we know that H is surjective. To show that it is injective, it is enoughto show that its domain and range have the same cardinality. To this end, weappeal to [7]. The cells of Harish-Chandra modules for Sp( p , q ) span complexvector spaces which carry the structure of representations of the Weyl group W of type C p + q . Let p be a doubled partition of 2( p + q ), with Lusztig symbol s ,and let π be the corresponding irreducible representation of W . Enumerate thedistinct even parts of p as r ,... , r k and denote by p ,... , p k the 2 k partitionsobtained from p by either replacing the block r i ... , r i of parts of p equal to r i by r i + r i ,... , r i , r i − i between 1 and k . Then the p i correspond (via their Lusztig symbols) to the representations inthe complex double cell of π of Springer type in the sense of [7]. From this and[7, Corollary 3] it follows that the number of equivalence classes T of shape p relative to a fixed domino tableau T of this shape equals the number of modulesin any Harish-Chandra cell C with annihilator the primitive ideal correspondingto T , provided that C has at least one such module. Hence the domain and rangeof H have the same cardinality and H is a bijection.Fix a signed tableau T ′ whose rows of even length all begin with + . It follows thatsigned involutions σ such that the normalization (in the sense of the paragraphafter Definition 5.2) of the second coordinate of H ( σ ) is T ′ correspond bijectivelyto modules in a Harish-Chandra cell for Sp( p , q ) and that all such cells (of mod-ules with trivial infinitesimal character) arise in this way; in particular, and inaccordance with [7, Theorem 6], there are as many such cells as there are nilpo-13ent orbits in g . It remains to show that all modules in the cell corresponding to T ′ have associated variety equal to the closure of the corresponding K -orbit in p via [1, 9.3.5]. This we will do in the next and final section. Section 10 Associated varieties
Our final result is10.1. THEOREM. Let σ ∈ S n , p correspond to the Harish-Chandra module Z .Then the associated variety of Z is the closure of the K -orbit corresponding to T ′ ,where H ( σ ) = ( T , T ), T is a representative of T , and T ′ is its normalization asdefined after Definition 5.2 (obtained from T by changing signs as necessary inall rows of even length so that they begin with + ). Proof.
Let q be a θ -stable parabolic subalgebra of g whose corresponding Levisubgroup of G is Sp( p ′ , q ′ ) × U ( p , q ) × ··· × U ( p r , q r ), where the p i , q i are suchthat p ′ + P i p i = p , q " + P i q i = q . There is a simple derived functor module A q of trivial infinitesimal character whose associated variety is the closure of theRichardson orbit O attached to q in the sense of [8]. The corresponding clan σ ′ is obtained as follows. Its first block of terms corresponds to the factor Sp( p ′ , q ′ ),taking the form (1, + )... ,( p ′ − q ′ , + ),( p ′ − q ′ + p ′ − q ′ + − ,... ,( p ′ + q ′ − p ′ + q ′ ) − if p ′ ≥ q ′ or (1, − ),... ,( q ′ − p ′ , − ),( q ′ − p ′ + q ′ − p ′ + − ,... ,( q ′ + p ′ − q ′ + p ′ ) − if q ′ > p ′ . Its next block of terms takes the form( m + + ),... ,( m + p − q , + ),( m , m + p − q + + ,... ,( p ′ + q ′ + p ′ + q ′ + p + q ) + , if p ≥ q , or ( m + − ),... ,( m + q − p ) − ,( m , m + q − p + + ,...( p ′ + q ′ + p ′ + q ′ + p + q ) + if q > p (where m + = ⌊ ( p ′ + q ′ + p + q + ⌋ );the remaining blocks of σ correspond similarly to the remaining factors U ( p i , q i ).Letting H ( σ ′ ) = ( T , T ) and defining T ′ as above, one checks immediately thatthe orbit corresponding to T ′ is indeed O . More generally, let X ′ be any sim-ple Harish-Chandra module for Sp( p ′ , q ′ ) with trivial infinitesimal character andassociated variety ¯ O . Then there is a simple Harish-Chandra module X for G obtained from X ′ by cohomological parabolic induction from q , whose associatedvariety is the closure of the orbit induced from O in the sense of [8]. Its signedinvolution σ ( X ) is obtained from that of X ′ by adding the blocks of terms corre-sponding to the U ( p i , q i ) factors in the above construction of σ , and if the theoremholds for X ′ and its associated variety, then the same is true for X .Given Z as in the theorem, let ¯ O be its associated variety. If O is the closureof a Richardson orbit, say the one attached to the θ -stable parabolic subalgebra q , then the module A q above lies in the same Harish-Chandra cell as Z and thetheorem holds for A q , whence it holds for Y . In general, using [8, Proposition2.3 (3)] and induction by stages, we can induce O to an orbit O ′ for a higherrank group G ′ such that all even parts in the partition corresponding to O ′ havemultiplicity at most 4, whence O ′ is Richardson by [8, Corollary 5.2]. Then theresult holds for the module Z ′ correspondingly induced from Z . But now theorbit O of Sp( p , q ) is the only one inducing to O ′ relative to a suitable θ -stableparabolic subalgebra q of Lie G ′ with Levi subgroup having Sp( p , q ) as its onlyfactor of type C . It follows that the theorem holds for Z , as desired.14 eferences [1] D. Collingwood and W. McGovern. Nilpotent orbits in semisimple Lie alge-bras, Chapman and Hall, 1993.[2] D. Garfinkle. On the classification of primitive ideals for complex classicalLie algebras, I, Compositio Math. , 75:135–169, 1990.[3] D. Garfinkle. On the classification of primitive ideals for complex classicalLie algebras, II,
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