Conormal variety on exotic Grassmannian

Transcription

1 Conormal variety on exotic Grassmannian joint work in progress with Lucas Fresse Kyo Nishiyama Aoyama Gakuin University New Developments in Representation Theory IMS, National University of Singapore (8 28, March 2016) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 1 / 27

2 Plan Plan of talk 1 Conormal variety Review the properties of conormal varieties 2 Double flag variety for symmetric pair Introduce double flag variety Moment maps, nilpotent varieties, Steinberg theory 3 Exotic Grassmannian and nilpotent variety K-orbits on Exotic Grassmannian Exotic nilpotent variety 4 Some combinatorics Marked Young tableaux & Striped signed Young diagram Correspondence via Young diagrams 5 Orbit correspondence Bijecton of orbits: Hermitian symmetric case General correspondence Nishiyama (AGU) Exotic Grassmannian 2016/03/18 2 / 27

3 Conormal variety Moment map Conormal variety: a review G : algebraic group / C g = Lie (G) X : smooth variety G T X : cotangent bdle (symplectic) G by Hamiltonian action = moment map µ : T X g (x, ξ) (z ξ(z x )) z x : vector field at x X generated by z g Definition 2.1 Y X := µ 1 (0) T X : conormal variety X /G O : G-orbit TO X = x O (T xo) : conormal bdle Lagrangian subvar of dim = dim X Nishiyama (AGU) Exotic Grassmannian 2016/03/18 3 / 27

4 Conormal variety Conormal bundle X /G O : G-orbit TO X : conormal bdle Lemma 2.2 Y X = O X /G T O X (hence the name of conormal variety) Proof. (x, ξ) µ 1 (0) µ(x, ξ)(z) = ξ(z x ) = 0 ( z g) ξ (T x O) (O := G x) Corollary 2.3 Assume #X /G <. 1 Y X is equi-dimensional of dim X and 2 Y X = O X /G T O X gives irred decomposition as an alg variety ( T O X : irreducible and dim T O X = dim X ) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 4 / 27

5 Double flag variety for symmetric pair Double flag variety Double flag variety Definition (N-Ochiai [NO11]) G : reductive alg grp /C θ Aut G : involution K = G θ : symmetric subgrp ( C-fication of max cpt subgrp) Example type A : (G, K) = (GL 2n, Sp 2n ), (GL n, GL p GL q ) (n = p q) 2 type C : (G, K) = (Sp 2n, GL n ), (Sp 2n, Sp 2p Sp 2q ) 3 Group mfd: (G, K) = (G 1 G 1, diag G 1 ) G 1 : reductive grp P G : parabolic of G & Q K : parabolic of K Notation X P := G/P : partial flag var & Z Q := K/Q : pfv for K X P Z Q : double flag variety K acts diagonally Cf. Toshi s lecture Property (PP ) for minimal psg Nishiyama (AGU) Exotic Grassmannian 2016/03/18 5 / 27

6 Double flag variety for symmetric pair DFV of finite type Finiteness of orbits X P := G/P & Z Q := K/Q : partial flag for G & K respectively X P Z Q : double flag variety (= DFV) K acts diagonally Definition 3.2 #(X P Z Q )/K < def finite type Example 3.3 (Double flag var (DFV) of finite type [NO11]) Type AIII : G/K = GL n /GL p GL q (n = p q) P Q 1 Q 2 X P Z Q any mirabolic GL q X P P(C p ) any GL p mirabolic X P P(C q ) (λ 1, λ 2, λ 3 ) maximal maximal X P Gr k (C p ) Gr l (C q ) (λ 1, λ 2, 1) any any X P Z Q maximal any any Gr m (C n ) Z Q Nishiyama (AGU) Exotic Grassmannian 2016/03/18 6 / 27

7 Double flag variety for symmetric pair DFV of finite type Classification of DFV of finite type If (G, K) = (G 1 G 1, diag G 1 ), DFV becomes triple flag variety (TFV) ( ) G/P K/Q = (G 1 G 1 )/(P 1 P 2 ) (G 1 /P 3 ) = G 1 /P 1 G 1 /P 2 G 1 /P 3 classification of TFV of finite type Magyar-Weymann-Zelevinsky [MWZ99] [MWZ00] (see also [Mat13]) How about general DFV? Theorem 3.4 ([HNOO13]) Let B G G: Borel in G & B K K: Borel in K 1 Classification of X P Z BK of finite type (Q is Borel) K-spherical G/P 2 Classification of X BG Z Q of finite type (P is Borel) We will show the list of X P Z BK of finite type,... but very briefly (quoted from [HNOO13]) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 7 / 27

8 Double flag variety for symmetric pair Table : Q = B K is Borel g k Π \ J (P = P J) α 1 α 2 α n sl n1 sl n1 so n1 {α i}( i) sl 2m sp m {α i}( i), {α i, α i1}( i), 2m = n 1 {α 1, α i}( i), {α i, α n}( i), {α 1, α 2, α 3}, {α n2, α n1, α n}, {α 1, α 2, α n}, {α 1, α n1, α n} sl pq sl p sl q C {α i}( i), {α i, α i1}( i), p q = n p q {α 1, α i}( i), {α i, α n}( i), {α i, α j}( i, j) if p = 2, any subset of Π if p = 1 α 1 α 2 α n1 α n so 2n1 so pq so p so q {α 1}, {α n}, p q = 2n p q {α i}( i) if p = 2, any subset of Π if p = 1 α n1 so 2n α 1 α 2 α n2 α n so pq so p so q {α 1}, {α n1}, {α n}, p q = 2n 1 p q {α i}( i) if p = 2, n 4 {α i, α n1}( i) if p = 2, {α i, α n}( i) if p = 2, any subset of Π if p = 1 so 2n sl n C {α 1}, {α 2}, {α 3}, {α n1}, {α n}, n 4 {α 1, α 2}, {α 1, α n1}, {α 1, α n}, {α n1, α n}, {α 2, α 3} if n = 4 Nishiyama (AGU) Exotic Grassmannian 2016/03/18 8 / 27

9 Double flag variety for symmetric pair Table : Q = B K is Borel sp α 1 α n 2 α n1 α n sp n sl n C {α 1}, {α n} sp pq sp p sp q {α 1}, {α 2}, {α 3}, {α n}, {α 1, α 2}, p q = n 1 p q {α i}( i) if p 2, {α i, α j}( i, j) if p = 1 α 1 α 2 α 3 α 4 f 4 f 4 so 9 {α 1}, {α 2}, {α 3}, {α 4}, {α 1, α 4} α 2 α 6 α 1 α 3 e 6 α 4 α 5 e 6 sp 4 {α 1}, {α 6} e 6 sl 6 sl 2 {α 1}, {α 6} e 6 so 10 C {α 1}, {α 2}, {α 3}, {α 5}, {α 6}, {α 1, α 6} e 6 f 4 {α 1}, {α 2}, {α 3}, {α 5}, {α 6}, {α 1, α 2}, {α 2, α 6}, {α 1, α 3}, {α 5, α 6} α 2 α 6 α 1 α 3 e 7 α 4 α 5 α 7 e 7 sl 8 {α 7} e 7 so 12 sl 2 {α 7} e 7 e 6 C {α 1}, {α 2}, {α 7} Nishiyama (AGU) Exotic Grassmannian 2016/03/18 9 / 27

10 conormal variety for DFV Moment maps Moment maps X := X P Z Q K : diagonal K-action X P = {p p Ad G p} : collection of psg conj to p T X P = {(p, ξ) X P g ξ (p ) } G P u P Want to apply Steinberg theory to X /K: T X = T X P T Z Q µ X g k µ XP µ ZQ α ( (p, ξ), (q, η) ) (ξ, η) k ξ k η µ XP (T X P ) = G u P = O G P N (g) : Richardson orbit for P Nishiyama (AGU) Exotic Grassmannian 2016/03/18 10 / 27

11 conormal variety for DFV conormal variety conormal variety for DFV X = X P Z Q Y X := µ 1 X (0) = O X /K T O X : conormal variety Notation: x θ := 1 2 (x θ(x)) k g x ξ g x θ ξ k Consider diagram: T X ( (p 1, x), (q 1, y) ) ψ ψ 0 π φ T Z Q N (g) ( (q 1, y), x) φ 0 Z Q N (g) (q 1, x) Z Q g (q 1, x y) Put W := π(y X ) T (Z Q ) N (g) : closed subvariety Note : ((p 1, q 1 ), (x, y)) Y X = x y = x x θ =: x θ s Nishiyama (AGU) Exotic Grassmannian 2016/03/18 11 / 27

12 Exotic/Enhanced nilpotent variety Definition of exotic nullcone Exotic & enhanced nilpotent variety ψ ψ 0 Y X ((p 1, q 1 ), (x, y)) π φ W (q 1, (x, y)) φ 0 Z Q N (g) (q 1, x) Z Q s (q 1, x y = x θ ) Assumption 5.1 We assume E X := φ(y X ) Z Q N (s) in the following Definition E X = φ(y X ) Z Q N (s) is called exotic nilpotent variety 2 R := ψ(y X ) Z Q N (g) is called enhanced Richardson variety Nishiyama (AGU) Exotic Grassmannian 2016/03/18 12 / 27

13 Exotic/Enhanced nilpotent variety Definition of exotic nullcone Related works 1 Robinson-Schensted correspondence for mirabolic triple flag variety Finkelberg-Ginzburg-Travkin[FGT09], Travkin [Tra09] 2 Exotic nilpotent cone, Springer representations Syu Kato [Kat09], [Kat11] 3 Enhanced nilpotent cone Achar-Henderson [AH08] 4 Exotic character sheaves over GL 2n /Sp 2n 1 Exotic RS correspondence Henderson-Trapa [HT12] 2 Exotic character sheaves Shoji-Sorlin [SS13], [SS14a], [SS14b] 5 Shoji s recent works on exotic symmetric spaces, enhanced variety, multiple Kostka polynomials, Springer correspondence for enhanced varieties of higher level [Sho14], [SS15], [Sho15a], [Sho15b], [Sho15c] Nishiyama (AGU) Exotic Grassmannian 2016/03/18 13 / 27

14 Exotic/Enhanced nilpotent variety Finiteness of exotic nilpotent orbits Theorem 5.3 (Type AIII : G/K = GL n /GL p GL q (n = p q)) 1 In the following cases, Assumption 5.1 holds, i.e., E X Z Q N (s) (the exotic nilpotent variety is well-defined) P Q 1 Q 2 #E /K < any mirabolic GL q yes any GL p mirabolic yes (λ 1, λ 2, λ 3 ) maximal maximal??? (λ 1, λ 2, 1) maximal any??? (λ 1, λ 2, 1) any maximal??? maximal any any see below 2 If the pair (P, Q) is (a) (any, mirabolic) or (b) P and Q have abelian nilradicals, then #E X /K < holds Remark 5.4 For a large class of X = X P Z Q of finite type, it seems likely to hold E X Z Q N (s) and #E X /K < (but not always) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 14 / 27

15 Exotic orbit correspondence Abstract theory Orbit correspondence Assume that 1 DFV X = X P Z Q is of finite type: #X /K < 2 Exotic nullcone E X Z Q N (s) is well-defined 3 finiteness of exotic nilpotent K-orbits: #E X /K < φ : Y X E X : K-equiv map For O X /K, O E X /K s.t. φ(t O X ) = O This induces a map Φ : X /K Irr(Y X ) T O X O E X /K Definition 6.1 The correspondence Orbits on X Orbits on E X is called Exotic Robinson-Schensted correspondence Though the terminology is somewhat different from the original meaning... Nishiyama (AGU) Exotic Grassmannian 2016/03/18 15 / 27

16 Exotic orbit correspondence Exotic Grassmannian Exotic Grassmannian: setting 2n-dim vector space V := C 2n = C n C n =: V 1 V 2 (G, K) = (GL 2n, GL n GL n ) Hermitian symmetric pair of type AIII G = GL(V ), K = K 1 K 2 = GL(V 1 ) GL(V 2 ) 1 P = Stab G (V 1 ) : maximal parabolic stabilizing V 1 {( a b ) } = P (n,n) = a, d GLn, b M 0 d n G/P = Gr n (V ) : Grassmannian of n-spaces in V 2 Q = Q 1 GL n = Stab K (L 0 ) : mirabolic subgrp (stabilizer of a line) Q 1 = P (1,n1) GL n = K 1 : mirabolic in K 1 K/Q = P(V 1 ) : projective space (collection of lines) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 16 / 27

17 Exotic orbit correspondence Orbits on exotic Grassmannian Exercise on Exotic Grassmannian X = X P Z Q = GL 2n /P (n,n) GL n /P (1,n1) : DFV Gr n (V ) P(V 1 ) with GL n GL n -action Warming up exercise: What are K-orbits in Gr n (V ) through U V (n-dim subspace)? Answer: Dimensions (r, s) = (dim U V 1, dim U V 2 ) classify K-orbits Coding by Young tableaux; explaining by example n = Nishiyama (AGU) Exotic Grassmannian 2016/03/18 17 / 27

18 Exotic orbit correspondence Orbits on exotic Grassmannian Coding K-orbits on Gr n (V ) by Young tableaux Example n = T : 2-column Young tableaux filled with 0, 1, 2 Rules: T i : i-th column of the tableau T (i = 1, 2) We fill T with 0, 1, 2 in increasing order from top to bottom subject to T 1 contains 2 for r-times, T 2 contains 2 for s-times 1 appears in T 1 and T 2 in the same number The sum of figures in T is 2n Nishiyama (AGU) Exotic Grassmannian 2016/03/18 18 / 27

19 Exotic orbit correspondence Orbits on exotic Grassmannian Coding K-orbits in Exotic Grassmannian Gr n (V ) P(V 1 ) Marked tableaux : (tableau, mark) = (T, i) U V : n-dim subspace & L V 1 : a line K-orbits in Gr n (V ) through U T L marking on T 1 (a choice of 0, 1, 2 from T 1 ) Example 6.2 n = 3 ( 0 0, 0) ( 0 0, 2) ( 0 0, 0) ( 0 0, 2) (, 2) (, 0) ( 0 0 ( , 0) ( 0 0, 1) ( 0 0, 2) ( 1 0, 1) ( 1 0, 2) ( 0 1, 0), 1) ( 0 1, 0) ( 0 1, 1) ( 1 0, 1) ( 1 0, 2) (, 1) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 19 / 27

20 Exotic orbit correspondence Exotic nilpotent orbits Exotic nilpotent orbits striped signed Young diagram: classifying K-orbits in P(V 1 ) N (s) (Johnson[Joh10]) Example : (I), (I), (I), (III), (III), (III), (II), (I), (III), (II), (I), (III), (II), (II), (I), (III), (II), (II), type (I): appears & the rest are to the right of the mid-bar type (II): appears to the left of the mid-bar Nishiyama (AGU) Exotic Grassmannian 2016/03/18 20 / 27

21 Exotic RS correspondence Lagrangian Grassmannian Correspondence: (G, K) = (GL 6, GL 3 GL 3 ), X = Gr 3 (C 6 ) P(C 3 ) (, 2) ( 0 0, 2) ( 0 0, 2) ( 0 0, 0) ( 0 0, 0) (, 0) ( , 1) ( 1 0, 2) ( 0 0, 0) ( 0 0, 1) ( 0 0, 2) ( 0 1, 0) ( , 1) ( 1 0, 1) ( 1 0, 2) ( 0 1, 0) ( 0 1, 1) (, 1) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 21 / 27

22 Exotic RS correspondence Lagrangian Grassmannian Exotic Robinson-Schensted correspondence Recall the setting: V = C 2n = C n C n = V 1 V 2 Consider (G, K) = (GL 2n, GL n GL n ) Hermitian symm pair of type AIII X = G/P K/Q = Gr n (V ) P(V 1 ) : DFV on which K acts Exotic nullcone : E X = P(V 1 ) N 2 (s) : marked 2-step nilpotents Theorem 7.1 (Fresse-N) 1-to-1 correspondence X /K O O E X /K via moment map φ(t O X ) = O Remark 7.2 For 2-step nilpotents, the Spaltenstein variety is irreducible Nishiyama (AGU) Exotic Grassmannian 2016/03/18 22 / 27

23 Exotic RS correspondence Exotic Grassmannian in general General orbit correspondence General situation where V = C N = C p C q = V 1 V 2 (N = p q) (G, K) = (GL N, GL p GL q ) X = G/P K/Q = Gr k (V ) P(V 1 ) : DFV on which K acts Exotic nullcone : E X P(V 1 ) N 2 (s) : marked 2-step nilpotents Theorem 7.3 (Fresse-N) finite-to-1 map (at most 2-to-1) X /K O O E X /K via moment map φ(t O X ) = O We can describe the map concretely. Can extend this to type CI where (G, K) = (Sp 2n, GL n ) & expecting more. Let us give a simplest example... Nishiyama (AGU) Exotic Grassmannian 2016/03/18 23 / 27

24 Exotic RS correspondence Exotic Grassmannian in general Example: Further correspondence (G, K) = (GL 4, GL 3 GL 1 ) K acts on X = Gr k (C 4 ) P(C 3 ) (k = 1, 2) Orbit corresp. map ϕ : Θ k 2 = X /K Πk 2 = E /K ϕ 1 (T ) for k = 1 ϕ 1 (T ) for k = 2 T Π k 2 ( 0 0 0, 2 2 ( 0 1 1, 2 2 ) ( 0 0 ) ( 0 1 ) ( ) ( , 0 0 1, 1 0 0, ) ( 0 0 2, 2 ), ( 0 0 ) ( 2, 0 0, 2 ), ( 0 1 ) ( ) ( 0 1 1, 1 0, ), 0 ), 0 Nishiyama (AGU) Exotic Grassmannian 2016/03/18 24 / 27

25 Exotic RS correspondence Exotic Grassmannian in general References I [AH08] [FGT09] P.N. Achar and A. Henderson, Orbit closures in the enhanced nilpotent cone, Advances in Mathematics 219 (2008), no. 1, Michael Finkelberg, Victor Ginzburg, and Roman Travkin, Mirabolic affine Grassmannian and character sheaves, Selecta Math. (N.S.) 14 (2009), no. 3-4, MR MR [HNOO13] Xuhua He, Kyo Nishiyama, Hiroyuki Ochiai, and Yoshiki Oshima, On orbits in double flag varieties for symmetric pairs, Transform. Groups 18 (2013), no. 4, MR [HT12] [Joh10] [Kat09] Anthony Henderson and Peter E. Trapa, The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), MR C. P. Johnson, Enhanced Nilpotent Representations of a Cyclic Quiver, ArXiv e-prints (2010). Syu Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, MR (2010k:20013) [Kat11], Deformations of nilpotent cones and Springer correspondences, Amer. J. Math. 133 (2011), no. 2, MR (2012c:20121) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 25 / 27

26 Exotic RS correspondence Exotic Grassmannian in general References II [Mat13] Toshihiko Matsuki, An example of orthogonal triple flag variety of finite type, J. Algebra 375 (2013), MR [MWZ99] Peter Magyar, Jerzy Weyman, and Andrei Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, MR MR (99m:14095) [MWZ00], Symplectic multiple flag varieties of finite type, J. Algebra 230 (2000), no. 1, MR MR (2001i:14064) [NO11] [Sho14] [Sho15a] [Sho15b] Kyo Nishiyama and Hiroyuki Ochiai, Double flag varieties for a symmetric pair and finiteness of orbits, J. Lie Theory 21 (2011), no. 1, MR T. Shoji, Exotic symmetric spaces of higher level - Springer correspondence for complex reflection groups -, ArXiv e-prints (2014)., Enhanced variety of higher level and Kostka functions associated to complex reflection groups, ArXiv e-prints (2015)., Kostka functions associated to complex reflection groups, ArXiv e-prints (2015). [Sho15c], Springer correspndence for complex reflection groups, ArXiv e-prints (2015). Nishiyama (AGU) Exotic Grassmannian 2016/03/18 26 / 27

27 Exotic RS correspondence Exotic Grassmannian in general References III [SS13] T. Shoji and K. Sorlin, Exotic symmetric space over a finite field, I, Transform. Groups 18 (2013), no. 3, MR [SS14a], Exotic symmetric space over a finite field, II, Transform. Groups 19 (2014), no. 3, MR [SS14b], Exotic symmetric space over a finite field, III, Transform. Groups 19 (2014), no. 4, MR [SS15] [Tra09] L. Shiyuan and T. Shoji, Double Kostka polynomials and Hall bimodule, ArXiv e-prints (2015). Roman Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Math. (N.S.) 14 (2009), no. 3-4, MR (2011c:20091) Nishiyama (AGU) Exotic Grassmannian 2016/03/18 27 / 27