Springer correspondence for the split symmetric pair in type A
Tsao-Hsien Chen, Kari Vilonen and Ting Xue
Compositio Math. 154 (2018), 2403–2425.
doi:10.1112/S0010437X18007443
4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Compositio Math. 154 (2018) 2403–2425 doi:10.1112/S0010437X18007443
Springer correspondence for the split symmetric pair in type A
Tsao-Hsien Chen, Kari Vilonen and Ting Xue
Abstract In this paper we establish Springer correspondence for the symmetric pair (SL(N), SO(N)) using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at q = 1. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at q = 1 arise in geometry.
1. Introduction In this paper we consider the Springer correspondence in the case of symmetric spaces. We will concentrate on the split case of type A, i.e., the case of SL(n, R). The case of SL(n, H) was considered by Henderson in [Hen01], Grinberg in [Gri98] and Lusztig in [Lus11a], and the case of U(p, q) was considered by Lusztig in [Lus11b] where he treats the general case of semisimple inner automorphisms. In both of these cases Springer theory closely resembles the classical situation. This turns out not to be so in the split case we consider here. In [CVX15a, CVX15b]wehave computed Fourier transforms of IC sheaves supported on certain nilpotent orbits using resolutions of singularities of nilpotent orbit closures. In this paper we study the problem in general in the split case of type A replacing the resolutions with a nearby cycle sheaf construction in [GVX18] based on earlier ideas of Grinberg [Gri01, Gri98]. Let us call an irreducible IC sheaf supported on a nilpotent orbit a nilpotent orbital complex. We show that the Fourier transform gives a bijection between nilpotent orbital complexes and certain representations of (extended) braid groups. We identify these representations of (extended) braid groups and construct them explicitly in terms of irreducible representations of Hecke algebras of symmetric groups at q = 1. This bijection can be viewed as Springer correspondence for the symmetric pair (SL(N), SO(N)). Let us note that the fact that representations of (a ne) Hecke algebras at q = 1 arise in this situation was already observed by Grojnowski in his thesis [Gro92]. The proof of our main result, Theorem 4.1, makes use of a nearby cycle sheaf construction in [GVX18] and smallness property of maps associated to certain ✓-stable parabolic subgroups. The nearby cycle sheaves produce nilpotent orbital complexes whose Fourier transforms have
Received 28 January 2017, accepted in final form 25 May 2018, published online 11 October 2018. 2010 Mathematics Subject Classification 20G99, 14L99 (primary). Keywords: Springer correspondence, nearby cycle sheaves, symmetric pairs, Hecke algebras, Hessenberg varieties, Fourier transform. The first author was supported in part by the AMS-Simons travel grant and the NSF grant DMS-1702337. The second author was supported in part by the ARC grants DP150103525 and DP180101445, the Academy of Finland, the Humboldt Foundation, the Simons Foundation, and the NSF grant DMS-1402928. The third author was supported in part by the ARC grants DE160100975, DP150103525 and the Academy of Finland. This journal is c Foundation Compositio Mathematica 2018. 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue
full support. Those IC sheaves behave like ‘cuspidal sheaves’ in the sense that they do not appear as direct summands of parabolic inductions. On the other hand, the smallness property mentioned above implies a simple description of the images of parabolic induction functors (Propositions 3.1, 3.2). Those results together with a counting lemma (Lemma 4.2)imply Theorem 4.1. As corollaries, we obtain criteria for nilpotent orbital complexes to have full support Fourier transforms (Corollaries 4.8, 4.9) and results on cohomology of Hessenberg varieties (Theorem 5.2). Our method appears to be applicable to general symmetric pairs and, more generally, polar representations studied in [Gri98] and we hope to return to this in future work. Let us mention that in [LY17], the authors show that one can obtain all nilpotent orbital complexes using spiral induction functors introduced in [LY17] (in fact, they consider more general setting of cyclically graded Lie algebras). Using their results and Theorem 4.1,weshow that all irreducible representations of Hecke algebras of symmetric groups at q = 1 appear in the intersection cohomology of Hessenberg varieties, with coe cient in certain local systems (see Theorem 6.3). This gives geometric constructions of irreducible representations of Hecke algebras of symmetric groups at q = 1 and provides them with a Hodge structure. The paper is organized as follows. In 2 we recall some facts about symmetric pairs and § introduce a class of representations of equivariant fundamental groups. Moreover, we recall the definition of the parabolic induction functor. In 3 we study parabolic induction functors for § certain ✓-stable parabolic subgroups. In 4, we prove Theorem 4.1: the Fourier transform defines § a bijection between the set of nilpotent orbital complexes and the class of representations of equivariant fundamental groups introduced in 2.In 5 and 6, we discuss applications of our § §§ results to cohomology of Hessenberg varieties and representations of Hecke algebras of symmetric groups at q = 1. Finally, in 7, we propose a conjecture that gives a more precise description § of the bijection in Theorem 4.1.
2. Preliminaries
For convenience we work over C. We adopt the usual convention of cohomological degrees for perverse sheaves by having them be symmetric around 0. We also use the convention that all functors are derived, so we write, for example, ⇡ instead of R⇡ .IfX is smooth we write CX [ ] ⇤ ⇤ for the constant sheaf placed in degree dim X so that CX [ ]isperverse.IfU X is a smooth ⇢ open dense subset of a variety X and is a local system on U,wewriteIC(X, ) for the L L IC-extension of [ ]toX; in particular, it is perverse. For simplicity of notation, when we have L a pair ( , ), where is an orbit and is a local system on , we also write IC( , ) instead O E O E O O E of IC( ¯, ). O E 2.1 Notation For e > 2, a partition of a positive integer k is called e-regular if the multiplicity of any part of is less than e. In particular, a partition is 2-regular if and only if it has distinct parts. Let us denote by (k) the set of all partitions of k and by (k) the set of all 2-regular partitions of k. P P2 We denote by k, 1 the Hecke algebra of the symmetric group Sk with parameter 1. More H precisely, k, 1 is the C-algebra generated by Ti,i=1,...,k 1, with the following relations H
TiTj = TjTi if i j > 2,i,j [1,k 1],TiTi+1Ti = Ti+1TiTi+1,i[1,k 2], | | 2 2 (2.1) T 2 = q +(q 1)T , where q = 1,i [1,k 1]. i i 2
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It is shown in [DJ86] that the set of isomorphism classes of irreducible representations of k, 1 H is parametrized by 2(k). For µ 2(k), we write Dµ for the irreducible representation of k, 1 P 2 P H corresponding to µ. For a real number a,wewrite[a] for its integer part.
2.2 The split symmetric pair (SL(N), SO(N)) Let G = SL(N) and ✓ : G G an involution such that K = G✓ =SO(N) and write g = Lie G. ! We have g = g g ,where✓ =( 1)i. The pair (G, K) is a split symmetric pair, i.e., there 0 1 |gi exists a maximal torus A of G that is ✓-split, where ✓-split means that for all x A, ✓(x)=x 1. 2 We also think of the pair (G, K) concretely as (SL(V ), SO(V )), where V is a vector space of dimension N equipped with a non-degenerate quadratic form Q such that SO(V )=SO(V,Q). We write the non-degenerate bilinear form associated to Q as , . rs h i rs rs Let g denote the set of regular semisimple elements in g and let g1 = g1 g . Similarly, let reg \ greg denote the set of regular elements in g and let g = g greg. 1 1 \ Let be the nilpotent cone of g and let = g.WhenN is odd, the set of K-orbits N N1 N \ in is parametrized by (N). When N is even, the set of O(N)-orbits in is parametrized N1 P N1 by (N), moreover, each O(N)-orbit remains one K-orbit if has at least one odd part, and P splits into two K-orbits otherwise. For (N), we write for the corresponding nilpotent 2 P O K-orbit in when has at least one odd part, and write I and II for the corresponding two N1 O O nilpotent K-orbits in when has only even parts. Suppose that =( > 0). N1 1 > 2 > ···> s For x (or !, ! =I, II), we have 2 O O s dim Z (x)= (i 1) . (2.2) K i Xi=1 Let a be a maximal abelian subspace of g1, which is also a Cartan subspace of g.Wehave the ‘little’ Weyl group W = NK (a)/ZK (a)=SN .
2.3 Equivariant fundamental group and its representations As was discussed in [CVX15a], the equivariant fundamental group K rs N 1 ⇡1 (g1 ) ⇠= ZK (a) o BN ⇠= (Z/2Z) o BN ,
where BN is the braid group of N strands and it acts on N N N 1 ZK (a) = (i1,...,iN ) (Z/2Z) ik =0 = (Z/2Z) ⇠ 2 ⇠ ⇢ k=1 X via the natural map B S . For simplicity we write N ! N N 1 N 1 BN =(Z/2Z) o BN and IN =(Z/2Z) . It is easy to see that the action of B on I has [N/2] + 1 orbits. We choose a set of e N N_ representatives I_ , 0 m [N/2], m 2 N 6 6 N of the BN -orbits as follows. Let ⌧ 0 (Z/2Z) be the element with all entries 0 except the ith i 2 position. Then ⌧ = ⌧ +⌧ ,i=1,...,N 1 , is a set of generators for I . For 0 m [N/2], { i i0 i0+1 } N 6 6 we define a character m as follows: (⌧ )= 1 and (⌧ ) = 1 for i = m. (2.3) m m m i 6
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For I ,weset 2 N_ B = StabBN . Let s , i =1,...,N 1, be the simple reflections in W = S . It is easy to check that i N StabS ( m)= si,i= m = Sm SN m if m = N/2, and N h 6 i ⇠ ⇥ 6 (2.4) Stab ( ) contains S S as an index-2 normal subgroup if m = N/2. SN m m ⇥ m Let us define
Bm,N m = the inverse image of Sm SN m = si,i= m under the map BN SN . ⇥ ⇠ h 6 i ! Then it follows from (2.4) that
B m = Bm,N m, when m = N/2, 6 (2.5) and B m contains Bm,N m as an index-2 normal subgroup when m = N/2. Let i,i=1,...,N 1, be the standard generators of BN which are lifts of the si under the 2 map BN SN .ThenBm,N m is generated by i, i = m, and m. We have a natural quotient ! 6 map 2 2 C[Bm,N m] ⇣ m, 1 N m, 1 = C[Bm,N m]/ ( i 1) ,i= m, m 1 . (2.6) H ⌦ H ⇠ h 6 i Note that in the above formula the corresponds to T of the Hecke algebra in (2.1). For i i us the i are more natural generators since they arise from geometry as unipotent monodromy operators. Let us write
m, 1 N m, 1 = m, 1. H ⌦ H H We consider a family of representations of BN as follows. For 0 6 m 6 [N/2], we define
C[BN ] L := Ind , 1 = [BN ] , 1, (2.7) m C[Bm,N m] m ⇠ C C[Bm,N m] m H e ⌦ H
where in the tensor product C[Bm,N m] acts on m, 1 via the quotient map (2.6) and on C[BN ] H by right multiplication. The module L m has a natural BN -action defined as follows. We let BN act on L by left multiplication and we let I act on L via a.(b v)=((b. )(a))(b v) for m N m ⌦ m ⌦ a IN , b BN and v m, 1.WeviewL m as a representatione of the equivariant fundamental 2 2 2 H group BN in this manner. We will next identify the composition factors of the modules L . Let µ1 (m) and m 2 P2 µ2 e(N m), m [0, [N/2]]. Proceeding just as in the definition of L , one obtains the 2 P2 2 m following representation of BN :
C[BN ] V 1 2 := Ind (D 1 D 2 ) = [BN ] (D 1 D 2 ). (2.8) µ ,µ C[Bm,N m] µ µ ⇠ C C[Bm,N m] µ µ e ⌦ ⌦ ⌦
Using (2.5), one readily checks that Vµ1,µ2 is an irreducible representation of BN when m = N/2, 1 2 1 2 6 or when m = N/2 and µ = µ .Whenm = N/2 and µ = µ , V 1 2 breaks into the direct sum 6 µ ,µ e I II of two non-isomorphic irreducible representations of BN , which we denote by Vµ1,µ2 and Vµ1,µ2 , i.e., we have V = V I VeII . (2.9) µ,µ ⇠ µ,µ µ,µ Moreover, 1 2 1 2 when m = N/2,V1 2 = V 1 2 if and only if (µ ,µ )=(⌫ , ⌫ ); 6 µ ,µ ⇠ ⌫ ,⌫ when m = N/2,Vµ1,µ2 ⇠= V⌫1,⌫2 if and only if either (µ1,µ2)=(⌫1, ⌫2) or (µ1,µ2)=(⌫2, ⌫1).
As the Dµ1 Dµ2 are the composition factors of m, 1 we conclude the following. ⌦ H
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1 2 1 Lemma 2.1. The composition factors of L consist of the V 1 2 , µ = µ , µ (m), m µ ,µ 6 2 P2 µ2 (N m), and when N =2m we have two additional composition factors V I and 2 P2 µ,µ V II for µ (m). µ,µ 2 P2 2.4 Parabolic induction functor and Fourier transform Let L be a ✓-stable Levi subgroup contained in a ✓-stable parabolic subgroup P G.Wewrite ⇢ l = Lie L, p = Lie P, L = L K, P = P K, l = l g , p = p g . K \ K \ 1 \ 1 1 \ 1 We will make use of the parabolic induction functor
Indg1 : D (l ) D (g ) l1 p1 LK 1 K 1 ⇢ ! defined in [Hen01, Lus11b]. In the following we recall its definition. Let
pr : p = l (n ) l 1 1 P 1 ! 1 be the natural projection map, where n is the nilpotent radical of p and (n ) = n g . P P 1 P \ 1 Consider the diagram
pr p1 p2 ⇡ˇ l o p o K p / K PK p / g , (2.10) 1 1 ⇥ 1 ⇥ 1 1 where p and p are natural projection maps and⇡ ˇ :(k, x) Ad(k)(x). 1 2 7! The maps in (2.10) are K P -equivariant, where K acts trivially on l and p ,byleft ⇥ K 1 1 multiplication on the K-factor on K p and on K PK p , and by adjoint action on g , and P ⇥ 1 ⇥ 1 1 K acts on l by a.l = pr(Ad a(l)), by adjoint action on p ,bya.(k, p)=(ka 1, Ad a(p)) on K p , 1 1 ⇥ 1 trivially on K PK p and g . ⇥ 1 1 Let A be a complex in D (l ). Then (pr p ) A p A for a well-defined complex A in LK 1 1 ⇤ ⇠= 2⇤ 0 0 P DK (K K p1). Define ⇥ g1 Ind A =ˇ⇡ A0[dim P dim L]. l1 p1 ! ⇢ Let F : D (g ) D (g ) be the Fourier transform functor (we also use the same notation K 1 ! K 1 F for the functor defined for (LK , l1)). Here we have identified g1 with g1⇤ via a K-invariant non-degenerate bilinear form on g1.Itisshownin[Hen01, Lus11b] that the induction functor commutes with Fourier transform, i.e.,
F(Indg1 A) Indg1 (F(A)). (2.11) l1 p1 = l1 p1 ⇢ ⇠ ⇢
3. Maximal ✓-stable parabolic subgroups and parabolic induction In this section, we study the parabolic induction functor with respect to a chosen family of Lm P m,1 m 0 0 N 0 V V ? C , ⇢ m ⇢ m ⇢ 2407 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue where V 0 = span e , 1 i m .WedefineLm to be the ✓-stable Levi subgroup of P m which m { i 6 6 } consists of diagonal block matrices of sizes m, N 2m, m.WhenN =2n, for ! =I, II, we define P n,! to be the parabolic subgroup of G that stabilizes the flag ! ! 2n 0 V V ? C , ⇢ n ⇢ n ⇢ where V I = span e , 1 i n and V II = span e , 1 i n 1,e . Let Ln,! be a ✓-stable n { i 6 6 } n { i 6 6 n+1} Levi subgroup of P n,!. According to [BH00], every maximal ✓-stable parabolic subgroup of G is K-conjugate to one of the above form. m m m m Let p = Lie P , p = p g , and (n m ) = n m g ,wheren m is the nilpotent radical 1 \ 1 P 1 P \ 1 P of pm,etc. Proposition 3.1. We have the following. (i) The map N P m ⇡ : K K (n m ) , (k, x) Ad k(x) m ⇥ P 1 ! N1 7! is a small map onto its image, generically one-to-one. (ii) The map N P m m ⇡ˇ : K K p g , (k, x) Ad k(x). m ⇥ 1 ! 1 7! is a small map onto its image, generically one-to-one. 2n,! 2n,! n,! The same holds for the two maps ⇡n and ⇡ˇn defined using P , ! =I, II. We define m N n,! 2n,! g1 =Imˇ⇡m, 1 6 m P Y r = xP greg x has eigenvalues a ,a ,...,a ,a ,a ,j [2m +1,N], m 2 1 | 1 1 m m j 2 where a = a for i = j . i 6 j 6 r m One checks readily that Ym = g1 . Consider the case m = N/2=n. For ! =I, II, let r,! reg Yn = x g1 x has eigenvalues a1,a1,...,an,an, where ai = aj for i = j, 2 | 6 ! 6 and the nilpotent part of x lies in the orbit n , O2 ! m where 2n is the nilpotent orbit given by the partition 2 and defined by the equation 2n,!O ! r,! n,! Im ⇡n = ¯ n .ThenYn is an open dense subset in g . O2 1 Let (pm)r = pm Y r and (lm)rs = lm (lm)rs. 1 1 \ m 1 1 \ Proposition 3.2. (1) Suppose that 1 6 m m K r LK m rs ⇡1 (Ym) ⇣ ⇡1 ((l1 ) ) = Bm BN 2m (3.2) ⇠ ⇥ Lm such that for an Lm-equivariant local system on (lme)rs associated to a ⇡ K ((lm)rs)- K T 1 1 1 representation E,wehave g1 m m Indlm pm IC(l1 , ) = IC(g1 , 0), 1 ⇢ 1 T ⇠ T 2408 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A where is the K-equivariant local system on Y r associated to the representation of ⇡K (Y r ) T 0 m 1 m which is obtained from E by pull-back under the map (3.2). (2) We have a natural surjective map n,! K r,! LK n,! rs ⇡1 (Yn ) ⇣ ⇡1 ((l1 ) ) ⇠= Bn, ! =I, II, (3.3) Ln,! such that for an Ln,!-equivariant local system on (ln,!)rs associated to a ⇡ K ((Ln,!)rs)- K T 1 1 1 representation E,wehave g1 n,! n,! Ind n,! n,! IC(l , ) IC(g , ), l p 1 ⇠= 1 0 1 ⇢ 1 T T r,! K r,! where is the K-equivariant local system on Yn associated to the representation of ⇡ (Yn ) T 0 1 which is obtained from E by pull-back under the map (3.3). 3.1 Proof of Proposition 3.1 We begin with the proof of (1). Consider the following projection N N ⌧ : (x, 0 Vm V ? V = C ) x g1,xVm =0,xV? Vm 1. m { ⇢ ⇢ m ⇢ | 2 m ⇢ } ! N N N When m = N/2, the map ⌧m can be identified with the map ⇡m.WhenN =2m, the image of 6 2m I II the map ⌧m has two irreducible components, i.e., closures of the two orbits 2m and 2m .The N,I N,II 2m O 2m 1 OI two maps ⇡m and ⇡m can be identified with the map ⌧m restricted to (⌧m ) ( 2m ) and 2m 1 II O (⌧ ) ( m )respectively.Thusitsu ces to show that m O2 N the map ⌧m is small over its image and generically one-to-one. (3.4) When m = N/2, one can check that the image of ⌧ N is as follows 6 m N ¯ N ¯ Im ⌧m = 3m1N 3m if N > 3m, Im ⌧m = 3N 2m23m N if N<3m. O O N 1 Assume that N 3m and x m N 3m .Then(⌧ ) (x) consists of the flag 0 V V V , > 3 1 m m m? 2 2 O N 1 ⇢ ⇢ ⇢ where V =Imx . Assume that N<3m and x N 2m 3m N .Then(⌧ ) (x) consists of the m 3 2 m 2 O N flag 0 Vm Vm? V ,whereVm =kerx. This proves that ⌧m is generically one-to-one. ⇢ ⇢ ⇢ N i j N 3i 2j m N 3m Let x 3i2j 1N 3i 2j Im ⌧m . We assume that 3 2 1 =3 1 if N > 3m, and 2 O ⇢ 6 3i2j1N 3i 2j =3N 2m23m N if N<3m.Itsu ces to show that 6 N 1 dim(⌧ ) (x) < codim N i j N 3i 2j /2. m Im ⌧m O3 2 1 N 3i N 3i Let x0 2j 1N 3i 2j Im ⌧ . (Note that ⌧ is defined since m i 6 (N 3i)/2.) One 2 O ⇢ m i m i checks readily that (using (2.2 ) for the second equality) N 1 N 3i 1 (⌧m ) (x) = (⌧ ) (x0) and codimIm ⌧ N 3i2j 1N 3i 2j =codim N 3i 2j 1N 3i 2j . ⇠ m i m Im ⌧m i O O Thus it su ces to show that N 3i 1 dim(⌧ ) (x0) < codim N 3i 2j 1N 3i 2j /2. m i Im ⌧m i O Let us write N N 1 N ⌦ =(⌧ ) (⇣j) for ⇣j j N 2j Im ⌧ m,j m 2 O2 1 ⇢ m and N a =codim N j N 2j = m(2N 3m) j(N j). m,j Im ⇡m O2 1 2409 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue N To prove that the map ⌧m is small, we are reduced to proving that aN dim ⌦N < m,j . (3.5) m,j 2 To prove this we recall the partitioning of ⌦N into ⌦N,k given in [CVX15b, 2] as follows: m,j m,j § N,k N ⌦ = (0 Vm V ? V = C ) dim(Vm ⇣jV )=k . m,j { ⇢ ⇢ m ⇢ | \ } We have ⌦N,k = max m + j N/2,j/2 k min j, m . m,j 6 ; , { } 6 6 { } N,k Recall that we have a surjective map ⌦m,j OGr(j k, j) OGr(m k, N 2j)withfibers (m k)(j k) ! ⇥ being a ne spaces A .Wehave j2 +3m2 + j + m dim ⌦N,k = 2k2 +( N +3j +2m + 1) k + mN mj . m,j 2 One checks that N if j N 2m, dim ⌦N,k is maximal when k = m + j , > m,j 2 j +1 if j N N ! ! (⇡m) C[ ] = IC( , C)(respectively((⇡N/2) ) C[ ] = IC( , C), ! =I, II), (3.6) ⇤ ⇠ O ⇤ ⇠ O where m N 3m N 2m 3m N =3 1 if N > 3m, =3 2 if N<3m. P m m P m Note that K K p1 is the orthogonal complement of K K (nP m )1 in the trivial bundle K g1 m ⇥ ⇥ ⇥ over K/PK . By the functoriality of Fourier transform, we have that N N F((⇡m) C[ ]) = (ˇ⇡m) C[ ]. (3.7) ⇤ ⇠ ⇤ Since Fourier transform sends simple perverse sheaves to simple perverse sheaves, we can conclude from (3.6) and (3.7) that N N (ˇ⇡m) C[ ] = IC(Im⇡ ˇm, C). ⇤ ⇠ 2n ! This proves the claim (2) of the proposition. The argument for (ˇ⇡n ) , ! =I, II, is the same. The proof of the proposition is complete. 2410 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A 3.2 Proof of Proposition 3.2 Note that we have that Lm = GL(m) SO(N 2m) and (lm) = gl(m) sl(N 2m) . (3.8) K ⇠ ⇥ 1 ⇠ 1 m m N To ease notation, let us now write that L = L , P = P , and⇡ ˇ =ˇ⇡m etc. We first show the following. ! 1 r 1 r,! The map⇡ ˇ (respectively⇡ ˇn ), when restricted to⇡ ˇ (Y )(respectivelyˇ⇡ (Yn )), is one-to-one. (3.9) Each element in Y r is K-conjugate to an element x p (see [KR71, Theorem 7]), where 0 2 1 x0ei = aiei,x0eN+1 i = ei + aieN+1 i for i [1,m], 2 x0ej = bjej + cjeN+1 j,x0eN+1 j = cjej + bjeN+1 j for j [m +1, [N/2]] (3.10) 2 x0e(N+1)/2 = b(N+1)/2e(N+1)/2 if N is odd and the numbers ai,i=1, . . . , m, bj + cj,bj cj,j = m +1,...,[N/2],b(N+1)/2 are distinct. 1 1 Thus it su ces to show that⇡ ˇ (x0) consists of one point. Note that⇡ ˇ (x0) consists of x -stable m-dimensional isotropic subspaces of V . Assume that U ⇡ˇ 1(x ). Then U is a 0 m 2 0 m direct sum of generalized eigenspaces of x0. The generalized eigenspace of x0 with eigenvalue ai is span ei,eN+1 i , i =1,...,m, and the eigenspace of x0 with eigenvalue bj + cj, bj cj, b(N+1)/2 { } is span ejeN+1 j , span ej + eN+1 j , span e(N+1)/2 ,respectively.SinceUm is isotropic, Um { } { } { } has to equal span e ,i =1,...,m .Thisproves(3.9) for⇡ ˇ , m The image of pr under the map pr : p l is lrs. (3.11) 1 1 ! 1 1 Let x pr. By the above proof of (3.9) we can assume that Ad(k)x = x for some k K,where 2 1 0 2 x is as in (3.10). Thus (k, x) ⇡ˇ 1(x ). It follows from (3.9) that (k, x)=(1,x ) K PK p . 0 2 0 0 2 ⇥ 1 Hence k PK . Assume that k = lu where l LK and u UK = U K (U is the unipotent 2 21 1 2 1 \ 1 radical of P ). Then we have pr(x) = pr(Ad(u l )x0) = pr(Ad(l )x0) = Ad(l )pr(x0). It is clear that pr(x ) lrs.Thus(3.11) follows. 0 2 By (3.9) and (3.11), we have the following diagram, when restricting (2.10)toY r, pr p1 p2 ⇡ˇ lrs o pr o K pr / K PK pr / Y r. 1 1 ⇥ 1 ⇥ 1 Using (3.9), we see that K r K P r K P P r K P r P r ⇡ (Y ) = ⇡ ⇥ K (Y ) = ⇡ ⇥ K (K K p ) = ⇡ ⇥ K (K p ) = ⇡ K (p ). 1 ⇠ 1 ⇠ 1 ⇥ 1 ⇠ 1 ⇥ 1 ⇠ 1 1 PK r PK rs LK rs Finally, the canonical map ⇡1 (p1) ⇡1 (l1 ) ⇠= ⇡1 (l1 ) is surjective. We see this as ! PK r follows. First, the canonical map above can be identified with the canonical map ⇡1 (p1) PK 1 rs r 1 rs ! ⇡1 (pr (l1 )). Now, because p1 is an open subset in pr (l1 ), which is smooth, the map ⇡ (pr) ⇡ (pr 1(lrs)) is a surjection. To conclude that this property persists when we pass 1 1 ! 1 1 to the equivariant fundamental group it su ces to remark that the equivariant fundamental group is always a quotient of the ordinary fundamental group as long as the group is connected. We now conclude the argument making use of Proposition 3.1. 2411 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue 4. Fourier transform of nilpotent orbital complexes for (SL(N), SO(N)) Consider the symmetric pair (G, K) = (SL(N), SO(N)). Let us write for the set of all simple AN K-equivariant perverse sheaves on (up to isomorphism), that is, the set of IC complexes N1 IC( , ), where is a K-orbit in and is an irreducible K-equivariant local system on O E O N1 E O (up to isomorphism). An IC complex in is called a nilpotent orbital complex. AN Let n =[N/2]. We set ⌃ = (⌫; µ1,µ2) 0 m n, ⌫ (m), N { | 6 6 2 P 0 k n m, µ1 (k),µ2 (N 2m k) . 6 6 2 P2 2 P2 In the case when N is even, we identify the triple (⌫; µ1,µ2)with(⌫; µ2,µ1)if µ 1 = µ2 and | | | | µ1 = µ2, and the triples (⌫; µ, µ) attain two labels I and II. 6 1 2 ! Given a triple (⌫; µ ,µ ) ⌃N (respectively (⌫; µ, µ) ⌃N , ! =I, II), where ⌫ = m Theorem 4.1. The Fourier transform F :Perv (g ) Perv (g ) induces a bijection K 1 ! K 1 F : ⇠ IC(gm, (⌫; µ1,µ2)) (⌫; µ1,µ2) ⌃ ,µ1 = µ2, ⌫ = m k d(k)= q(s)q(2k +1 s), (4.1) s=0 X k 1 q(k)2 +3q(k) e(k)= q(s) q(2k s)+ . (4.2) 2 s=0 X 2412 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A Lemma 4.2. We have n = p(n k)d(k)= ⌃ , (4.3) |A2n+1| | 2n+1| Xk=0 n = p(n k)e(k)= ⌃ . (4.4) |A2n| | 2n| Xk=0 Proof. Note that 1 p(k)xk = and q(k)xk = (1 + xs). (4.5) 1 xs s 1 s 1 Xk>0 Y> Xk>0 Y> Let p(l, k) denote the number of partitions of l into (not necessarily distinct) parts of exactly k di↵erent sizes. We have (see for example [Wil83]) yxs p(l, k)xlyk = 1+ . (4.6) 1 xs s 1 l,kX>0 Y> ✓ ◆ Assume first that N =2n+1. Note that if is a partition of N with parts of k di↵erent sizes, k 1 then the component group AK (x) of the centralizer ZK (x) for x is (Z/2Z) .Thusthere 2 O are 2k 1 irreducible K-equivariant local systems on (up to isomorphism). Hence using (4.6), O we see that s k 1 2n+1 1 1+x = p(2n +1,k)2 =Coe cient of x in . |A2n+1| 2 1 xs s 1 Xk>0 Y> ✓ ◆ Using (4.5), we see that 1+xs 2 = p(k)x2k q(k)xk . (4.7) 1 xs s 1 Y> ✓ ◆ ✓ Xk>0 ◆✓ Xk>0 ◆ It then follows that is the desired number. The fact that ⌃ equals the same number |A2n+1| | 2n+1| is clear from the definition. Thus (4.3) holds. Assume now that N =2n. Suppose that is a partition of N with parts of exactly k di↵erent k 1 sizes. If has at least one odd part, then there are 2 irreducible K-equivariant local systems k on (up to isomorphism). If has only even parts, then there are 2 irreducible K-equivariant O ! local systems on each (up to isomorphism), ! =I, II. Thus we have thatO k 1 k 1 = p(2n, k)2 + p(n, k)3 2 |A2n| · Xk>0 Xk>0 1 1+xs 3 1+xs =Coe cient of x2n in +Coe cient of xn in 2 1 xs 2 1 xs s 1 s 1 Y> ✓ ◆ Y> ✓ ◆ n k 1 n n 1 3 = p(n k) 2 q(s)q(2k s)+q(k)2 + p(n k)q(k)= p(n k)e(k). 2 2 s=0 ✓ Xk=0 ✓ X ◆◆ Xk=0 Xk=0 Here we have used (4.7) and the following equation 1+xs = p(k)xk q(k)xk . 1 xs s 1 Y> ✓ ◆ ✓ Xk>0 ◆✓ Xk>0 ◆ Again the fact that ⌃ equals the desired number is clear from the definition. 2 | 2n| 2413 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue Note that the IC sheaves appearing on the right-hand side of the Fourier transform map F in Theorem 4.1 are pairwise non-isomorphic. Thus, in view of Lemma 4.2, Theorem 4.1 follows from the following. 1 2 ! Proposition 4.3. Let (⌫; µ ,µ ) ⌃N (respectively (⌫; µ, µ) ⌃N , ! =I, II) and write 2 2 n,! m = ⌫ . The Fourier transform of IC(gm, (⌫; µ1,µ2)) (respectively IC(gm, (⌫; µ, µ)!), IC(g , | | 1 T 1 T 1 (⌫; , )!)) is supported on a K-orbit in . T ; ; N1 Proof. Let n =[N/2]. We begin the proof by showing the following for ( ; µ1,µ2) ⌃ ; 2 N (respectively ( ; µ, µ)! ⌃ , ! =I, II). ; 2 N 1 2 ! The Fourier transform of IC(g1, ( ; µ ,µ ))(respectively IC(g1, ( ; µ, µ) )) T ; T ; (4.8) is supported on a K-orbit in . N1 Recall that ( ; µ1,µ2)(respectively ( ; µ, µ)!)istheirreducibleK-equivariant local system rs T ; T ; ! on g1 corresponding to Vµ1,µ2 (respectively Vµ,µ). We will now appeal to the nearby cycle construction in [GVX18]. Let us recall the characters I ,0 m n of (2.3). In [GVX18] we apply a nearby cycle construction to local systems m 2 N_ 6 6 associated to the and obtain a K-equivariant perverse sheaf on the nilpotent cone . m P m N1 More precisely, for each character m,letuswriteW m = StabW ( m). Consider the following base-change diagram of the adjoint quotient map. g1, m / g1 f m f (4.9) ✏ ✏ a/W m / a/W rs rs Let us write g1, m for the base change of the regular semisimple locus g1 . Denote by m the K- rs K rs F equivariant local system on g1, m corresponding to the representation of ⇡1 (g1, m )=IN o B m , where IN acts via the character m and B m acts trivially (recall that B m = StabBN ( m)). We form the nearby cycle sheaf = , appropriately shifted, so that Perv ( ). P m f m F m P m 2 K N1 Applying [GVX18, Theorem 3.2], we obtain that F( )=IC(g , ), P m 1 M m where is the K-equivariant local system on grs corresponding to the B -representation M m 1 N M = [B ] ( 0 ). m C N [B0 ] C m W e ⌦C m ⌦ H m e 0 Let us explain the notation in the abovee formula in our setting. Let W m be the Coxeter subgroup of W generated by s with (ˇ↵( 1)) = 1, ↵ (g, a), where (g, a) is the root system of g ↵ m 2 with respect to a. Note that↵ ˇ( 1) IN .Then 0 is the Hecke algebra associated to the W m 0 2 0 H 0 Coxeter group W m with parameter 1. Let B m BN be the inverse image of W m W under 0 ⇢ 0 ⇢ the natural map BN W = SN .ThenB m = IN o B m . In our setting, one readily checks that 0 ! 0 W = si,i = m , W 0 = m, 1 and B = Bm,N m (here we use the notation in 2.3). m h 6 i H m H m § 0 e 0 The action of B on ( 0 ) is given by IN acting via the character m and B acting m C m W m m 0 ⌦ H via the quotient map [B ] 0 . Thus we conclude C m W e ! H m F( ) = IC(g , ), (4.10) P m ⇠ 1 L m 2414 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A rs where m is the K-equivariant local system on g1 corresponding to the representations L m of K rsL ⇡1 (g1 )=BN defined in (2.8). By Lemma 2.1 the IC sheaves IC(g , ( ; µ1,µ2)) and IC(g , ( ; µ, µ)!) are composition 1 T ; 1 T ; factors of thee IC(g , ). Hence (4.8) follows from (4.10). 1 L m Now let (⌫; µ1,µ2) ⌃ with ⌫ = m>0. Let 2 N | | (⇢ V 1 2 ) K ⌫ ⇥ µ ,µ rs denote the irreducible LK -equivariant local system on l1 associated to the irreducible LK rs LK rs representation of ⇡1 (l1 ) obtained as a pullback of ⇢⌫ ⇥ Vµ1,µ2 via the map ⇡1 (l1 ) ⇠= Bm BN 2m ⇣ Sm BN 2m. ⇥ ⇥ By Proposition 3.2, we have that e e m 1 2 g1 IC(g1 , (⌫; µ ,µ )) = Indlm pm IC(l1, (⇢⌫ ⇥ Vµ1,µ2 )). (4.11) T 1 ⇢ 1 K Since Fourier transform commutes with induction (see (2.11)), it su ces to show that the Fourier transform of IC(l , (⇢ V 1 2 )) is supported on an L -nilpotent orbit in l .This 1 K ⌫ ⇥ µ ,µ K 1 follows from the classical Springer correspondence for gl(m) and (4.8) applied to the symmetric pair (SL(N 2m), SO(N 2m)) (see (3.8)). n,! The proof for IC(gm, (⌫; µ, µ)!), IC(g , (⌫; , )!) proceeds in the same manner; in the 1 T 1 T ; ; latter case one uses the corresponding ✓-stable Levi and parabolic subgroups. We omit the details. 2 4.2 More on induction Let (⌫; µ1,µ2) ⌃ . Assume that ⌫ = m>0. Let Lm P m be as in 3. Recall that Lm = 2 N | | ⇢ § K ⇠ GL(m) SO(N 2m) and lm = gl(m) sl(N 2m) . ⇥ 1 ⇠ 1 A nilpotent Lm-orbit in lm is given by a nilpotent orbit in gl(m) and a nilpotent SO(N 2m)- K 1 orbit in sl(N 2m) . Thus the nilpotent Lm-orbits in lm are parametrized by (m) (N 2m), 1 K 1 P ⇥P with extra labels I and II for partitions in (N 2m) with all parts even. For ↵ (m) and P 2 P (N 2m), we denote by (or ! ) the nilpotent Lm-orbit in lm given by the nilpotent 2 P O↵, O↵, K 1 orbit in gl(m) and the nilpotent SO(N 2m)-orbit (or !)insl(N 2m) . O↵ O O 1 In the following we will omit the labels I and II with the understanding that everything ! g1 ! should have corresponding labels, for example, =Indlm pm ↵, etc. O 1 ⇢ 1 O g1 Proposition 4.4. Let ↵ (m) and (N 2m). Let =Indlm pm ↵, ,i.e., 2 P 2 P O 1 ⇢ 1 O = +2↵ . Assume that u and v (u +(n m ) ). We have a natural surjective i i i 2 O↵, 2 O \ P 1 map : A (v) A m (u). K ⇣ LK m ˜ Moreover, let C ⇥ be an LK -equivariant irreducible local system on ↵, and let be the E O E ˜ K-equivariant local system on obtained from C ⇥ via the map above. Then IC( , ) is g1 O E O E a direct summand of Indlm pm IC( ⌫,µ, C ⇥ ). 1 ⇢ 1 O E Corollary 4.5. If moreover ( µ, ) N 2m is a pair such that F(IC( µ, )) has full support, O E 2 A O E then we have g1 ˜ Indlm pm IC( ⌫,µ, C ⇥ ) = IC( , ). 1 ⇢ 1 O E ⇠ O E As before let us now write L = Lm and P = P m etc. We begin the proof of the above proposition with the following lemma. 2415 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue Lemma 4.6. The map : K PK ( ¯ +(n ) ) ¯ ⇥ O↵, P 1 ! O is generically one-to-one. Proof. Let x . We can and will assume that x +(n ) . We show that 1(x )isa 0 2 O 0 2 O↵, P 1 0 point. Assume that (k, x)=x , i.e., Ad k(x)=x .Thenx +(n ) . Let (respectively 0 0 2 O↵, P 1 O )bethe(unique)G-orbit (respectively L-orbit) in g (respectively l) that contains O↵, O (respectively ). We have that e O↵, e =Indg O l O↵, in the notation of Lusztig and Spaltenstein [LS79]. By [LS79, Theorem 1.3], we have Z0 (x ) P . e e G 0 ⇢ In fact, we have that Z (x ) P . This can be seen by enlarging the group G to GL(N) and G 0 ⇢ using the fact that Z (x ) is connected. Thus Z (x ) P . Furthermore, ( + n ) GL(N) 0 K 0 ⇢ K O \ O↵, P is a single orbit under P .Thusthereexistsp P such that Ad p(x)=x . It follows that 2 0 k 1p Z (x ) P .Thusk P K = P . Now we have that (k, x)=(1, Ad k(xe)) = (1e,x ). 2 2 G 0 ⇢ 2 \ K 0 Proof of Proposition 4.4. Note that the proof of the above lemma shows that ZG(v)=ZP (v). We have Z (v) Z (u)U .ThusZ (v)=Z (v) Z (u)(U K). It follows that we have P ⇢ L P K PK ⇢ LK P \ a natural projection map Z (v)/Z 0 (v)=Z (v)/Z 0 (v) Z (u)/Z 0 (u). K K PK PK ! LK LK We show that this gives us the desired map . Following [LS79], we have that Z (u)(U K) LK P \ has a dense orbit, i.e., the orbit of v, in the irreducible variety u +(nP )1.ThusZPK (v)=ZK (v) meets all the irreducible components of Z (u)(U K), which implies that is surjective. LK P \ It is easy to see that supp(Indg1 IC( , )) = ¯ . (4.12) l1 p1 ↵, C ⇥ ⇢ O E O The proposition follows from the definition of parabolic induction functor and Lemma 4.6. 2 Remark 4.7. The proof of Lemma 4.6 and the existence and surjectivity of the map in Proposition 4.4 works for any ✓-stable Levi contained in a ✓-stable parabolic subgroup. Proof of Corollary 4.5. Note that the assumption implies that F(IC( ↵, , C )) has full O ⇥ E support, i.e., IC( ↵, , C ⇥ )=IC(l1, ) for some irreducible LK -equivariant local system rs O E G G on l1 . We have that F(Indg1 IC( , )) = Indg1 F(IC( , )) = Indg1 IC(l , ). l1 p1 ↵, C ⇥ l1 p1 ↵, C ⇥ l1 p1 1 ⇢ O E ⇢ O E ⇢ G g1 It su ces to show that Ind IC(l1, ) is irreducible. This follows from the definition of the l1 p1 G induction functor and Proposition⇢ 3.1. 2 Corollary 4.8. The Fourier transform of a nilpotent orbital complex IC( , ) N has full g1O E 2 A support, i.e., supp F(IC( , )) = g1, if and only if it is not of the form Ind IC( 0, 0) where O E l1 p1 O E supp F(IC( , )) = l , and L P is a pair chosen as in 3. ⇢ O0 E0 1 ⇢ § Proof. The only if part follows from the facts that Fourier transform commutes with parabolic induction and that supp Indg1 A g . The if part follows from (4.11), Corollary 4.5 and l1 p1 ( 1 Theorem 4.1. ⇢ 2 2416 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A Corollary 4.9. Let =( ) (N). 1 > 2 > ··· 2 P (1) If 3 for some i,thensupp F(IC( , )) = g for any K-equivariant local system i i+1 > O E 6 1 on . The same holds for ! if has only even parts. E O O (2) Suppose that 2 for all i. Let f be the number of di↵erent sizes of parts of , i i+1 6 and g the number of i such that =2. i i+1 f 1 g (a) If at least one part of is odd, then there are 2 irreducible K-equivariant local systems on such that supp F(IC( , )) = g . E O O E 1 (b) If all parts of are even, then there is exactly one irreducible K-equivariant local system ! on each orbit !, ! =I, II, such that supp F(IC( !, !)) = g . E O O E 1 In particular, if 1 for all i,thensupp F(IC( , )) = g for any K-equivariant local i i+1 6 O E 1 system on . E O i0 Proof. (1) Assume that i0 i0+1 > 3. Let m = i0, ↵ =1 , =( 1 2,..., i0 2, i0+1,...). g1 Then =Indm m ↵, . Let u ↵, and v (u+(nP m )1). Note that AK (v) = ALm (u). O l1 p1 O 2 O 2 O \ ⇠ K It then follows from⇢ Proposition 4.4 that for each irreducible K-equivariant local system on g1 E ,IC( , ) is a direct summand of Indlm pm IC( ↵, , 0) for some irreducible LK -equivariant O O E 1 ⇢ 1 O E local system 0 on ↵, . As before, this shows that F(IC( , )) has smaller support. E O ! O g1E ! In the case when has only even parts, we let =Indm m ↵, ,ifm f g 1 m 6 2 if has at least one odd part, ! (4.13) respectively m 6 1 if all parts of are even. 2417 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue Theorem 4.1 implies that the number of pairs IC( , ) such that supp F(IC( , )) = g O E 2 AN O E 1 is d(n)(see(4.1)), when N =2n + 1, and e(n)(see(4.2)), when N =2n. In view of (4.13) and claim (1) of the corollary, it su ces to show that f g 1 f g 1 2 = d(n), 2 +2q(n)=e(n). (4.14) (2n+1) (2n), i i+162, 2PX 2 not2P all partsX of even i i+16 This can be seen as follows. Note that when N is even, the number of orbits of the form !, O where all parts of are even and 2, is 2q(n). We know that i i+1 6 1 d(n)=Coe cient of x2n+1 in (1 + xs)2, 2 s 1 Y> 3 1 e(n)= q(n)+Coe cient of x2n in (1 + xs)2. 2 2 s 1 Y> A partition satisfies that 2 if and only if each part of the transpose partition i i+1 6 0 has multiplicity at most 2. We have f = f 0 and g equals the number of parts in 0 with multiplicity 2. It is easy to see that each 0 whose parts have multiplicity at most 2 appears in (1 + xs)2 exactly 2f g times. Hence (4.14) follows. 2 s>1 QRemark 4.10. In [CVX15a, Conjecture 1.2], we conjectured that one can obtain all nilpotent orbital complexes by induction from those of smaller groups whose Fourier transforms have full support. This conjecture follows from Corollary 4.8. 5. Cohomology of Hessenberg varieties Hessenberg varieties, defined generally in [GKM06], arise naturally in our setting (for details, see [CVX15b]). In particular, they arise as fibers of maps ⇡ and⇡ ˇ in the following diagram K/PK g1 7 ⇥ g K PK E K PK E ⇥ ⇥ ? ⇡ ⇡ˇ ✏ g✏ N1 1 where P = P K for a ✓-stable parabolic subgroup P of G, E is a P -stable subspace of K \ K g1 consisting of nilpotent elements, and E? is the orthogonal complement of E in g1 via a K-invariant non-degenerate form on g1. The generic fibers of maps⇡ ˇ are Hessenberg varieties. In this section we discuss an application of our result to cohomology of Hessenberg varieties. Let us fix s grs and consider the corresponding Hessenberg variety 2 1 1 1 Hess :=⇡ ˇ (s)= gP K/P g sg E? . { K 2 K | 2 } The centralizer ZK (s) acts naturally on Hess and it induces an action of the component group ⇡0(ZK (s)) ⇠= IN on the cohomology groups H⇤(Hess, C). Let H⇤(Hess, C)= H⇤(Hess, C) I M2 N_ be the eigenspace decomposition with respect to the action of IN . 2418 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A Definition 5.1. The stable part H⇤(Hess, C)st of H⇤(Hess, C) is the direct summand H⇤(Hess, C) triv where triv I_ is the trivial character. 2 N rs For simplicity we now assume⇡ ˇ is onto. In this case⇡ ˇ is smooth over g1 (e.g. see [CVX15b, K rs Lemma 2.1]) and the equivariant fundamental group ⇡1 (g1 ,s) ⇠= IN o BN acts on H⇤(Hess, C) by the monodromy action. Recall that for I , B stands for the stabilizer of in B . Clearly, 2 N_ N each summand H⇤(Hess, C) is stable under the action of B . Let m IN_ , B m , and Bm,N m 2 be as in 2.3. Assume that is in the BN -orbit of m. Then for any b BN with b. = m § 1 2 we have an isomorphism ◆b : B = B m ,u bub . Note that triv = 0 and B m = Bm,N m ⇠ ! except when N is even and m = N/2. In that case, Bm,N m is an index-2 subgroup of B m . Recall the algebra m, 1 = m, 1 N m, 1 and their representations Dµ1 Dµ2 H H ⌦ H ⌦ introduced in 2.3. Each m, 1 is a quotient of the group algebra C[Bm,N m] and 0, 1 = § H H , 1 = N, 1 is the Hecke algebra of SN at q = 1. H triv H Theorem 5.2. (i) Let I be the representatives of B -orbits in 2.3.Toevery I m 2 N_ N § 2 N_ in the orbit of and an element b B satisfying b( )= , the monodromy action of b on m 2 N m H (Hess, ) induces an isomorphism H (Hess, ) H (Hess, ) compatible with the actions ⇤ C ⇤ C ⇠= ⇤ C m ◆b of B B on both sides. ⇠= m (ii) The action of C[Bm,N m] on H⇤(Hess, C) m factors through the algebra m, 1 and the 1 2 H resulting representation is a direct sum of D 1 D 2 , µ (m), µ (N m). In particular, µ ⌦ µ 2 P2 2 P2 the stable part H⇤(Hess, C)st is generated by irreducible representations of the Hecke algebra of S at q = 1. N Proof. Part (1) is clear. To prove part (2) we proceed as follows. By the decomposition theorem ⇡ C is a direct sum of shifts of nilpotent orbital complexes. Since F(⇡ C) = ⇡ˇ C (up to shift), ⇤ ⇤ ⇠ ⇤ Theorem 4.1 implies that a generic stalk of⇡ ˇ C, which is isomorphic to H⇤(Hess, C), is a direct ⇤ C[BN ] sum of the local systems V 1 2 =Ind Dµ Dµ introduced in (2.8). Since IN acts on µ ,µ C[Bm,N m] 1 2 ⌦ V 1 2 by the formula a.(b v)=((b. )(a))(b v) for a I , b B and v D D ,we µ ,µ ⌦ m ⌦ 2 N 2 N 2 µ1 ⌦ µ2 have (V 1 2 ) = D D . The theorem follows. 2 µ ,µ ⇠ µ1 ⌦ µ2 Example 5.3. Let C be the hyper-elliptic curve with a ne equation y2 = N (x a )(here j=1 j ai = aj for i = j). Assume N =2n + 2 is even. Then according to [CVX17, 2.3] the Jacobian 6 6 Q§ rs Jac(C) is an example of Hessenberg variety and the monodromy action of ⇡1(g1 ,s) factors through BN , that is, H⇤(Jac(C), C)=H⇤(Jac(C), C)st. Let µk =(N k, k) 2(N) and Dµk 2 P be the corresponding representation of N, 1. Using [A’Ca79], one can check that the induced Hi action of the group algebra C[BN ] on H (Jac(C), C) factors through N, 1 and for i 6 n the H resulting representation of N, 1 is isomorphic to H [i/2] i H (Jac(C), C) = Dµi 2j ⇠ Mj=0 with the primitive part Hi(Jac(C), ) D . C prim ⇠= µi Remark 5.4. It would be nice to have an explicit decomposition of H⇤(Hess, C) m into irreducible representations of m, 1. For this one needs finer information for the bijection in Theorem 4.1 H (see 7). In [CVX15a, CVX17], we establish an explicit bijection for certain nilpotent orbital § complexes and we work out an explicit decomposition for the cohomology of the Hessenberg varieties that are isomorphic to Fano varieties of k-planes in smooth complete intersections of two quadrics in projective space. 2419 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue 6. Representations of N, 1 H In this section we show that all irreducible representations of the Hecke algebra N, 1 come H from geometry. Indeed they all appear in intersection cohomology of a Hessenberg variety with coe cient in a local system. In particular, this shows that all irreducible representations of N, 1 H carry a Hodge structure. In particular, the irreducible representations of N, 1 can be viewed H as variations of Hodge structure. Let be a nilpotent K-orbit on g and an irreducible K-equivariant local system on . O 1 L O We call ( , ) a nilpotent pair. Following [LY17], we associate to each nilpotent pair ( , )two O L O L families of Hessenberg varieties Hess , 1 g1 together with local systems ˆ 1 on open subsets L ± ! L± Hess , 1 Hess , 1. L ± ⇢ L ± Let x g be a nilpotent element in . Choose a normal sl -triple x, h, y and let 2 1 O 2 { } g(i)= v g [h, v]=iv , g (i)=g(i) g , and g (i)=g(i) g . { 2 | } 0 \ 0 1 \ 1 For any N Z we write N 0, 1 for its image in Z/2Z.Define 2 2 { } x x x x pN = gN (k), lN = gN (2N), and l = lN . k 2N N Z M> M2 x x x One can check that l g is a graded Lie subalgebra of g and x l1 = g1(2). Let L0 K be ⇢ x 2 ⇢ the reductive subgroup with Lie algebra l0 = g0(0). By [LY17, 2.9(c)], the restriction 0 := x L1 L|l1 x x x x is an irreducible L0-equivariant local system on the unique open L0-orbit l1 on l1. x According to [Lus95], there exists a graded parabolic subalgebra q = N Z qN of l , a Levi 2 subalgebra m = m of q, and a cuspidal local system on the open M -orbit m of m N Z N 1 L 0 1 1 2 x L (here M0 is the reductive subgroup of L with Lie algebra m0) such that L 0 x x l1 some shift of the IC-complex IC(l1, 10 ) is a direct summand of Indm1 q1 IC(m1, 1). L ⇢ L In addition, we have F(IC(m1, 1)) = IC(m 1, 1), L ⇠ L where 1 is a cuspidal local system on the unique open orbit m 1 m 1. L ⇢ Define ˆq to be the pre-image of q under the projection map px lx . Let Q K be the N N N ! N K ⇢ parabolic subgroup with Lie algebra ˆq0. Denote by ˆq 1 the preimage of m 1 under the projection ± ± map ˆq 1 q 1 m 1. The group QK acts naturally on ˆq 1 and ˆq 1 and we define ± ! ± ! ± ± ± QK QK Hess , 1 := K ˆq 1, Hess , 1 := K ˆq 1. L ± ⇥ ± L ± ⇥ ± Let 1 ⇡ , 1 : Hess , 1 g1, (x, v) xvx L ± L ± ! ! and let ⇡ , 1 be its restriction to Hess , 1. For any s g1, we denote by Hess , 1,s and Hess , 1,s L ± L ± 2 L ± L ± the fiber of ⇡ , 1 and ⇡ , 1 over s,respectively. L ± L ± 2420 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A There are natural maps h , 1 : Hess , 1 [m 1/M0], h , 1 : Hess , 1 [m 1/M0] L ± L ± ! ± L ± L ± ! ± Q sending (k, v) Hess , 1 = K K ˆq 1 tov ¯, the image of v ˆq 1 under the map ˆq 1 m 1 2 L ± ⇥ ± 2 ± ± ! ± ! [m 1/M0]. We define the following local system ± ˆ 1 := (h , 1)⇤ 1 L± L ± L± on Hess , 1. Here we view the M0-local systems 1 as sheaves on [m 1/M0]. L ± L± ± Example 6.1. Consider the nilpotent pair ( , = )where is the trivial local system O L Ltriv Ltriv on . Using [Lus95, Proposition 7.3] one can check that in this case q = N Z qN is a Borel O x 2 subalgebra of l and m = N Z mN is a Cartan subalgebra. Moreover the grading on m is 2 L concentrated in degree zero, i.e., m = m0, and the cuspidal local system 1 is the skyscraper L L± sheaf supported on m 1 = 0 . It follows that in this case Hess , 1 = Hess , 1 and ˆ 1 is ± { } Ltriv ± Ltriv ± L± the constant local system. In [LY17, 7], the authors prove the following: § (⇡ , 1) IC(Hess , 1, ˆ 1)is the Fourier transform of (⇡ ,1) IC(Hess ,1, ˆ1). (6.1) L ⇤ L L L ⇤ L L Some shift of IC( , ) (respectively the Fourier transform of IC( , )) appears in O L O L (6.2) (⇡ ,1) IC(Hess ,1, ˆ1)(respectively (⇡ , 1) IC(Hess , 1, ˆ 1)) as a direct summand. L ⇤ L L L ⇤ L L Assume from now on that ⇡ , 1 : Hess , 1 g1 is surjective. Then the sheaf ˆ L rs L ! (⇡ , 1) IC(Hess , 1, 1) is smooth over g1 . One sees this as follows. According to the first L ⇤ L L statement of (6.1) the characteristic variety of (⇡ , 1) IC(Hess , 1, ˆ 1) coincides with that of L ⇤ L L (⇡ ,1) IC(Hess ,1, ˆ1) as they are Fourier transforms of each other. But (⇡ ,1) IC(Hess ,1, ˆ1) L ⇤ L L L ⇤ L L is K-equivariant and supported on the nilpotent cone. A straightforward calculation then shows ˆ rs the smoothness of (⇡ , 1) IC(Hess , 1, 1) on g1 . Thus, by the decomposition theorem, we L ⇤ L L conclude that (⇡ , 1) IC(Hess , 1, ˆ 1) grs is a direct sum of shifts of irreducible local systems. (6.3) L ⇤ L L | 1 In addition, the IC(Hess , 1, ˆ 1) and hence (⇡ , 1) IC(Hess , 1, ˆ 1) has a canonical L L L ⇤ L L structure as a Hodge module and thus the direct summands are IC-extensions of irreducible variations of pure Hodge structure, see, [Sai88]. We fix a generic s grs and then 2 1 * H⇤((⇡ , 1) IC(Hess , 1, ˆ 1))s =IH (Hess , 1,s, ˆ 1). (6.4) L ⇤ L L L L K rs * ˆ Thus we obtain an action of the fundamental group ⇡1 (g1 ,s) on IH (Hess , 1,s, 1) and by L L the discussion above this action breaks into a direct sum of irreducible representations which are also variations of Hodge structure. * The component group ⇡0(ZK (s)) = IN acts on IH (Hess , 1,s, ˆ 1) and we write ⇠ L L * * IH (Hess , 1,s, ˆ 1)= IH (Hess , 1,s, ˆ 1) L L L L I M2 N_ for the corresponding eigenspace decomposition. 2421 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue * * Definition 6.2. The stable part IH (Hess , 1,s, ˆ 1)st of IH (Hess , 1,s, ˆ 1)isthedirect * ˆ L L L L summand IH (Hess , 1,s, 1) triv where triv IN_ is the trivial character. L L 2 * ˆ K rs Observe that IH (Hess , 1,s, 1)st is stable under the monodromy action of ⇡1 (g1 ,s). L L K rs Moreover, the action factors through the braid group BN via the quotient map ⇡1 (g1 ,s) BN . rs ! For every irreducible representation Dµ of N, 1,letVµ be the local system on g1 associated H to D . By Theorem 4.1, there exists a unique nilpotent pair ( , ) such that F(IC( , )) = µ Oµ Lµ Oµ Lµ ⇠ IC(g1,Vµ). Theorem 6.3. Let Dµ be an irreducible representation of N, 1 and let ( µ, µ) be the H O L associated nilpotent pair as above. We have the following. * ˆ (i) The map ⇡ µ, 1 is onto, the action of the braid group BN on IH (Hess µ, 1,s, µ, 1)st L * ˆ L L factors through the Hecke algebra N, 1 and IH (Hess µ, 1,s, µ, 1)st is a direct sum of H L L irreducible representations of N, 1. H * ˆ (ii) Dµ appears in IH (Hess µ, 1,s, µ, 1)st with non-zero multiplicity. L L * ˆ Proof. Since for every irreducible subrepresentation W of IH (Hess µ, 1,s, µ, 1)st the L L corresponding Fourier transform F(IC(g1, )) is supported on the nilpotent cone (here rs W W is the local system on g1 associated to W ), the same argument as in the proof of Theorem 5.2 implies part (1). Part (2) follows from (6.1), (6.2), and (6.4). 2 7. Conjecture on more precise matching In Theorem 4.1 we show that the Fourier transform establishes a bijection between two sets of intersection cohomology sheaves. In this section we formulate a conjecture which refines the bijection in Theorem 4.1. We also relate the conjecture to our earlier conjectures in [CVX15b]. Our conjecture is not strong enough to produce an exact matching. The exact description of the bijection is crucial for applications, for example, computing cohomologies of Hessenberg varieties as explained in 5. § We begin with associating to each nilpotent orbit (respectively !, ! =I, II) a subset O O ⌃ ⌃ (respectively ⌃! ⌃ ), if (N) has at least one odd part (respectively has only ⇢ N ⇢ N 2 P even parts). Let be a partition of N and let 0 be the transpose partition of . Suppose that 2m1 2m 2m 1 2m 1 0 =( 0 ) ( 0) l ( 0 ) l+1 ( 0 ) k , (7.1) 1 ··· l l+1 ··· k where mi > 1, i =1,...,k. Here and in what follows we write the parts in a partition in the order which is most convenient for us. In particular, in (7.1) we place the parts with even multiplicity before the parts with odd multiplicity. Let 0, 1 for i [1,l] and let i 2 { } 2 m1 1 m m 1 m 1 ⌫( ,..., )=( 0 ) ( 0) l l ( 0 ) l+1 ( 0 ) k , 1 l 1 ··· l l+1 ··· k 2 1 2 µ( ,..., )=( 0 ) ( 0) l ( 0 ) ( 0 ). 1 l 1 ··· l l+1 ··· k Note that 2 ⌫( ,..., ) + µ( ,..., ) = N. Let | 1 l | | 1 l | J J := l +1,...,k such that 0 < 0 . ⇢ 0 { } j j j J j J0 J X2 2X 2422 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 Springer correspondence for the split symmetric pair in type A We define 1 1 µ ( ,..., ; J)=( 0 ) ( 0) l ( 0 ) ( 0 ),J= j ,...,j . 1 l 1 ··· l j1 ··· js { 1 s} 2 1 l µ ( 1,..., l; J)=( 10 ) ( l0) ( i01 ) ( i0k l s ),J0 J = i1,...,ik l s . ··· ··· { } Note that l0+1 = 0 if and only if all parts of are even. In this case, J0 = = J and 1 2 i ; µ ( 1,..., l; J)=µ ( 1,..., l; J) and we write µ( 1,..., l)=µ ( 1,..., l; J), i =1, 2. If has at least one odd part, then let ⌃ := (⌫( ,..., ); µ1( ,..., ; J),µ2( ,..., ; J)) 0, 1 ,i=1,...,l, 1 l 1 l 1 l | i 2 { } ⇢ J l +1,...,k , such that 0 < 0 . ⇢ { } j j j J j J0 J X2 2X If all parts of are even (in which case l0+1 = 0), then let ⌃! = (⌫( ,..., ); µ( ,..., ),µ( ,..., ))! 0, 1 ,i=1,...,l , ! =I, II. { 1 l 1 l 1 l | i 2 { } } We have ⌃ =2k 1 (respectively ⌃! =2l), which equals the number of non-isomorphic | | | | irreducible K-equvariant local systems on (respectively !). O O Conjecture 7.1. Let be a partition of N. (1) If has at least one odd part, then the Fourier transform F induces the following bijection: F : IC( , ) irreducible K-equivariant local system on (up to isomorphism) { O E |E O } ⌫ ⇠ IC(g| |, (⌫; µ1,µ2)) (⌫; µ1,µ2) ⌃ . ! { 1 T | 2 } Moreover, ⌫0 1 2 F(IC( , C)) = IC(g| |, (⌫0; µ ,µ )) O 1 T 0 0 1 2 1 2 where (⌫0; µ0,µ0) ⌃ is the unique triple such that ⌫0 = max ⌫ , (⌫,µ ,µ ) ⌃ and the 1 2 2 | | {| | 2 i} parts of µ0 and the parts of µ0 have the opposite parity (in particular, all parts of µ0 have the same parity). (2) If all parts of are even, then the Fourier transform induces the following bijection F : IC( !, ) ! =I, II, irreducible K-equivariant local system on !(up to isom) { O E | E O } ⌫ ⇠ IC(g| |, (⌫; µ, µ)!) ! =I, II, (⌫; µ, µ)! ⌃!,µ= ! { 1 T | 2 6 ;} IC(gn,!, (⌫; , )) ! =I, II, (⌫; , ) ⌃! . [ 1 T ; ; | ; ; 2 Moreover, ! n,! F(IC( , C)) = IC(g , (⌫0; , )), O 1 T ; ; where ⌫ = n and (⌫ ; , ) ⌃ . | 0| 0 ; ; 2 Note that F(IC( , )) has full support if and only if ⌫( ,..., )= . Thus we see that the O E 1 l ; conjecture is compatible with Corollary 4.9. We also remark that special cases of the conjecture are verified by [CVX15a, Theorems 4.1 and 4.3]. Let us relate the conjecture above to our previous conjectures in [CVX15b]. In [CVX15b]we 2n+1 2n+1 rs constructed local systems Ei,j and Ei,j on g1 . In terms of the parametrization introduced e 2423 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3 T.-H. Chen, K. Vilonen and T. Xue in this paper, we have E2n+1 = ( ;(2i j, j), (2n +1 2i)) i,j T ; E2n+1 = ( ;(2i 1 j, j), (2n +2 2i)). i,j T ; 2n+1 Thus we see that Conjecturee 7.1 applied to Ei,j agrees with [CVX15b, Conjectures 6.1 and 6.3]. 2n+1 2n+1 Applied to Ei,j , Conjecture 7.1 implies that the supports of F(IC(g1, Ei,j )) are as follows: e 3j 22i 2j 112n+3 4i+j if 4i j 6 2n +3, e O 3j 22n+2 2i j 14i j 2n 3 if 2i + j 6 2n + 2 and 4i j > 2n +3, O 32n 2i+222i+j 2n 212i 2j 1 if 2i + j > 2n +2. O Note that the above orbits are all of even dimension and each of the even-dimensional orbits appears twice there. Acknowledgements We thank Misha Grinberg for helpful conversations. T.C. thanks Cheng-Chiang Tsai for useful discussions and thanks the Institute of Mathematics Academia Sinica in Taipei for support, hospitality, and a nice research environment. T.X. thanks Syu Kato for helpful discussions. 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Yun, Z/m-graded Lie algebras and perverse sheaves, I, Represent. Theory 21 (2017), 277–321. Sai88 M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995. Wil83 H. S. Wilf, Three problems in combinatorial asymptotics, J. Combin. Theory Ser. A 35 (1983), 199–207. Tsao-Hsien Chen [email protected] Department of Mathematics, University of Chicago, Chicago 60637, USA Kari Vilonen [email protected] School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia and Department of Mathematics and Statistics, University of Helsinki, Helsinki 00014, Finland Ting Xue [email protected] School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia and Department of Mathematics and Statistics, University of Helsinki, Helsinki 00014, Finland 2425 4787 C D 64 5C 7:8 C: 6 C8 2 8CD / 8D 4 . 5C4C 8D 06 4 D 5 86 8 ,4 5C 7:8 , C8 8C D D8 4 4 45 8 4 D 64 5C 7:8 C: 6 C8 8C D D 7 C: 1 3