Back-stable Schubert calculus

Transcription

1 Back-stable Schubert calculus Thomas Lam, Seung-Jin Lee, and Mark Shimozono Virginia Tech Sage Days, ICERM July /53

2 Type A Dynkin diagram with vertex set Z: Fl: infinite flag ind-variety Back-stable Schubert calculus: study of H (Fl) 2/53

3 Type A Dynkin diagram with vertex set Z: Fl: infinite flag ind-variety Back-stable Schubert calculus: study of H (Fl) Apply to affine flag ind-varieties and other settings: algebra module nilhecke H G (Fl Fl) HT (Fl) K-nilHecke K G (Fl Fl) KT (Fl) affine Hecke K G C (T (Fl) N T (Fl)) KT C (Fl) [Su, Zhao, Zhong] [Aluffi, Mihalcea, Schuermann, Su] stable bases 2/53

4 Infinite flags Fl and Infinite Grassmannian Gr C [, ) = {(,c 1,c 0,c 1,...) c i C, c i = 0 for i 0} E k = i kce i standard flag E = ( E 1 E 0 E 1 ) Fl: set of flags F = (F i i Z) such that F i = E i for i 0 and i 0 dimf i /F i 1 = 1 for all i Z 3/53

5 Infinite flags Fl and Infinite Grassmannian Gr C [, ) = {(,c 1,c 0,c 1,...) c i C, c i = 0 for i 0} E k = i kce i standard flag E = ( E 1 E 0 E 1 ) Fl: set of flags F = (F i i Z) such that F i = E i for i 0 and i 0 dimf i /F i 1 = 1 for all i Z Gr: set of V such that F a V F b for some a 0 < b dim(f b /V) = b 3/53

6 Infinite flags Fl and Infinite Grassmannian Gr C [, ) = {(,c 1,c 0,c 1,...) c i C, c i = 0 for i 0} E k = i kce i standard flag E = ( E 1 E 0 E 1 ) Fl: set of flags F = (F i i Z) such that F i = E i for i 0 and i 0 dimf i /F i 1 = 1 for all i Z Gr: set of V such that F a V F b for some a 0 < b dim(f b /V) = b Fl + Fl: F i E i only for i > 0 Fl Fl: F i E i only for i < 0 3/53

7 Fibration algebra isomorphism! Fl Fl + Fl Gr x = (...,x 2,x 1,x 0 ) x + = (x 1,x 2,...) x = x x + Λ(x ): symmetric functions F F 0 H (Fl) = H (Gr) H (Fl ) H (Fl + ) = Λ(x ) Q[x ] Q[x + ] = Λ(x ) Q[x] Find Schubert basis by back-stable limit 4/53

8 Familiar example of back-stable limit: Schur functions Basis of cohomology: Schur polys indexed by min. coset reps. in S n /(S k S n k ) or partitions λ k (n k) rectangle H (Gr(k,n)) = λ k (n k) s 1,s 2,...,s k 1,s k,s k+1,...,s n 1 s 1 k,...,s 1,s 0,s 1,...,s n k 1 Qs λ (x 1 k,...,x 0 ) Omitted reflection has been shifted from k back to 0. 5/53

9 Familiar example of back-stable limit: Schur functions Basis of cohomology: Schur polys indexed by min. coset reps. in S n /(S k S n k ) or partitions λ k (n k) rectangle H (Gr(k,n)) = λ k (n k) s 1,s 2,...,s k 1,s k,s k+1,...,s n 1 s 1 k,...,s 1,s 0,s 1,...,s n k 1 Qs λ (x 1 k,...,x 0 ) Omitted reflection has been shifted from k back to 0. Take limit: Gr = Gr(k, n) k n k H (Gr) = Λ(x ) = λ YQs λ (x ) Y: set of partitions 5/53

10 Permutations s i = (i,i+1) S Z =...,s 1,s 0,s 1,... S + = s 1,s 2,... permutes Z >0 S =...,s 2,s 1 permutes Z 0 S 0 = S S + no s 0 S 0 Z = {w S Z ws i > w for all i 0} Y = partitions Grass. perms Y = S 0 Z λ w λ w λ/µ = w λ wµ 1 µ λ 6/53

11 Divided difference operators i on Q[x] i (f) = f(...,x i,x i+1,...) f(...,x i+1,x i,...,) x i x i+1 Lemma image( i ) = ker( i ) = s i -invariants. In particular 2 i = 0. i i+1 i = i+1 i i+1 and i j = j i for i j 2. 7/53

12 Schubert polynomials [Lascoux, Schützenberger] Theorem There is a unique family {S w w S + } Q[x + ] such that S id = 1 Moreover S w is homogeneous of degree l(w) i S w = { S wsi if ws i < w 0 otherwise. {S w w S + } is a Q-basis of Q[x + ] = H (Fl + ). S w is s i -symmetric if w(i) < w(i+1). S w (n) 0 = x n 1 1 x n 2 2 x 1 n 1 w (n) 0 S n longest element 8/53

13 Examples x 2 1x x 1 x 2 x x 1 x 1 +x s 1 s 2 s 1 s 2 s 1 s 2 s 2 s 1 s 1 s 2 s 1 s 1 s 2 s 1 s 2 1 S si = x 1 +x 2 + +x i. 9/53

14 Back-stable Schubert polynomials [Knutson] shift(s i ) = s i+1 shift(x k ) = x k+1 Definition For w S Z the back-stable Schubert polynomial S w is Sw = lim p shiftp 1 (S shift 1 p (w) ) Monomial expansion is well-defined (by e. g. Billey-Jockusch-Stanley formula) 10/53

15 Back-stable Schubert polynomials [Knutson] shift(s i ) = s i+1 shift(x k ) = x k+1 Definition For w S Z the back-stable Schubert polynomial S w is Sw = lim p shiftp 1 (S shift 1 p (w) ) Monomial expansion is well-defined (by e. g. Billey-Jockusch-Stanley formula) Sshift(w) = shift( S w ) 10/53

16 Examples Ss3 = +x 1 +x 0 +x 1 +x 2 +x 3 Ss 2 = +x 3 +x 2 11/53

17 Theorem (Lam, SJ Lee, S.) { S w w S Z } is the unique family of elements of R = Λ(x ) Q[x] such that 1 Sid = 1 2 Sw is homogeneous of degree l(w) 3 i Sw = { Swsi if ws i < w 0 otherwise. Moreover they form a Q-basis of R = H (Fl). 12/53

18 variety Dynkin nodes ring basis indexing set Sw Fl Z Λ(x ) Q[x] S Gr Z mod (Z 0) Λ(x ) s λ (x ) Y = SZ 0 Fl + Z >0 Q[x + ] S w S + Fl Z <0 Q[x ]??? S 13/53

19 Negative Schubert polynomials Definition S Z rev S Z rev(s i ) = s i rev(x k ) = x 1 k α i = x i x i+1 rev(α i ) = α i Q[x] rev Q[x] for all i Z for all k Z for all i Z For w S the negative Schubert polynomial S w Q[x ] is S w = rev(s rev(w) ) 14/53

20 Negative Schubert polynomials Definition S Z rev S Z rev(s i ) = s i rev(x k ) = x 1 k α i = x i x i+1 rev(α i ) = α i Q[x] rev Q[x] for all i Z for all k Z for all i Z For w S the negative Schubert polynomial S w Q[x ] is Examples S w = rev(s rev(w) ) S s 2 = rev(s s2 ) = rev(x 1 +x 2 ) = x 0 x 1 S s 3 s 2 s 1 = rev(s s3 s 2 s 1 ) = rev(x 3 1) = x /53

21 Basis Apply rev to Get H (Fl + ) = Q[x + ] = QS w w S + H (Fl ) = Q[x ] = QS w w S 15/53

22 Basis Apply rev to Get Combine: H (Fl + ) = Q[x + ] = H (Fl ) = Q[x ] = For w = w w + S 0 = S S + define w S + QS w w S QS w S w = S w S w+ Q[x ] Q[x + ] = Q[x] 15/53

23 Basis Apply rev to Get Combine: H (Fl + ) = Q[x + ] = H (Fl ) = Q[x ] = For w = w w + S 0 = S S + define w S + QS w w S QS w S w = S w S w+ Q[x ] Q[x + ] = Q[x] Basis: H (Fl Fl + ) = Q[x] = w S 0 QS w 15/53

24 Two bases for R = Λ(x ) Q[x] First basis: Back stable Schubs H (Fl) = R = Λ(x ) Q[x] = w S Z Q S w H (Gr) = Λ(x ) = λ (x ) λ YQs H (Fl Fl + ) = Q[x] = R = v S 0 QS v (λ,v) Y S 0 Qs λ (x )S v Second basis: Schur functions times Schubs. Change of basis coefficients are... 16/53

25 Two bases for R = Λ(x ) Q[x] First basis: Back stable Schubs H (Fl) = R = Λ(x ) Q[x] = w S Z Q S w H (Gr) = Λ(x ) = λ (x ) λ YQs H (Fl Fl + ) = Q[x] = R = v S 0 QS v (λ,v) Y S 0 Qs λ (x )S v Second basis: Schur functions times Schubs. Change of basis coefficients are... Edelman-Greene coefficients! Why? 16/53

26 Wrong way map and Stanley functions There is a Q-algebra map η 0 H (Fl Z ) η 0 H (Gr) = = Λ(x ) Q[x] Λ(x ) η 0 x k 0 p r (x ) p r (x ) 17/53

27 Wrong way map and Stanley functions There is a Q-algebra map η 0 H (Fl Z ) η 0 H (Gr) = Λ(x ) Q[x] Λ(x ) η 0 x k 0 p r (x ) p r (x ) Definition For w S Z the Stanley symmetric function is F w = η 0 ( S w ) Λ(x ). = 17/53

28 Wrong way map and Stanley functions There is a Q-algebra map η 0 H (Fl Z ) η 0 H (Gr) = Λ(x ) Q[x] Λ(x ) η 0 x k 0 p r (x ) p r (x ) Definition For w S Z the Stanley symmetric function is F w = η 0 ( S w ) Λ(x ). F wλ = S wλ = s λ (x ) = 17/53

29 Edelman-Greene coefficients j w λ Define j w λ Q by F w = λ Yj w λ s λ Theorem [Edelman-Greene] j w λ Z 0 with combinatorial formula. Other formulas: [Lascoux,Schützenberger], [Haiman], [Reiner, S.] 18/53

30 Edelman-Greene coefficients j w λ Define j w λ Q by F w = λ Yj w λ s λ Theorem [Edelman-Greene] j w λ Z 0 with combinatorial formula. Other formulas: [Lascoux,Schützenberger], [Haiman], [Reiner, S.] [Lam, SJ Lee, S.] New formula for j w λ using bumpless pipedreams 18/53

31 Hopf structure Λ(x ) is a Hopf algebra with primitive generators p r : (p r ) = p r 1+1 p r for all r 1. R = Λ(x ) Q[x] is a Λ(x )-comodule: Act by on Λ(x ) factor R id Q[x] Λ(x ) R = = Λ(x ) Q[x] id Q[x] Λ(x ) Λ(x ) Q[x] 19/53

32 Coproduct formulae [Lam, SJ Lee, S.] w. = uv means w = uv and l(w) = l(u)+l(v) Theorem For all w S Z ( id Q[x] )( S w ) = Sw = w =uv. F u (x ) Sv (u,v) S Z S Z w =uv. F u (x )S v (u,v) S Z S 0 in Λ(x ) R in R 20/53

33 Coproduct formulae [Lam, SJ Lee, S.] w. = uv means w = uv and l(w) = l(u)+l(v) Theorem For all w S Z ( id Q[x] )( S w ) = Sw = w =uv. F u (x ) Sv (u,v) S Z S Z w =uv. F u (x )S v (u,v) S Z S 0 in Λ(x ) R in R Corollary Sw = w. =uv v S 0 λ Y j u λ s λ(x )S v 20/53

34 S-structure constants For u,v,w S Z (resp. S + ) define c w uv Q (resp. c w uv) by Su Sv = w c w uv Sw S u S v = w c w uvs w. Example S 2 s1 = S s2 s 1 + S s0 s 1 S 2 s 1 = S s2 s 1 S 2 s2 = S s3 s 2 + S s1 s 2 S 2 s 2 = S s3 s 2 +S s1 s 2 21/53

35 Proposition (Lam, SJ Lee, S.) 1 For u,v,w S +, we have c w uv = c w uv. 2 Every back stable Schubert structure constant is a usual Schubert structure constant. 22/53

36 Theorem (Lam, SJ Lee, S.) Let u S m, v S n and λ Y. Let u v := ushift m (v) S m+n S +. Then j u v λ = w S Z c w uv j w λ. [1994 Reiner, S. unpublished] Explicit combinatorial conjecture for η 0 (S u S v ), s λ = η 0 ( w S + c w uvs w ), s λ = Tells difference between c w uv and c w uv. w S + c w uvf w, s λ = w S + c w uvj w λ 23/53

37 Summary of novel features of infinite setting Back-stable Schubert basis Wrong way algebra map H (Fl) H (Gr) gives natural definition of Stanley function Negative Schubert polynomials Coproduct formula New clue on product of Schubs New formula for Edelman-Greene coefficients 24/53

38 Equivariance H T(pt) = Q[a] = Q[a k k Z] H T(Fl) = H T(Gr) H T (pt) H T(Fl Fl + ) H T(Fl Fl + ) = H T(pt) H (Fl Fl + ) = Q[x,a] 25/53

39 Supersymmetric functions Let Λ(x a) be the Q[a]-algebra of supersymmetric functions: Λ(x a) = Q[a][p k (x a) k 1] p k (x a) = i 0(x k i a k i). It is a polynomial Hopf Q[a]-algebra generated by p k (x a) for k 1. The p k (x a) are primitive. 26/53

40 Supersymmetric functions Let Λ(x a) be the Q[a]-algebra of supersymmetric functions: Λ(x a) = Q[a][p k (x a) k 1] p k (x a) = i 0(x k i a k i). It is a polynomial Hopf Q[a]-algebra generated by p k (x a) for k 1. The p k (x a) are primitive. Superization: the Q[a]-algebra map Q[a] Λ(x ) Λ(x a) p k (x ) p k (x a) f(x) f(x/a) 26/53

41 Supersymmetric functions Let Λ(x a) be the Q[a]-algebra of supersymmetric functions: Λ(x a) = Q[a][p k (x a) k 1] p k (x a) = i 0(x k i a k i). It is a polynomial Hopf Q[a]-algebra generated by p k (x a) for k 1. The p k (x a) are primitive. Superization: the Q[a]-algebra map Q[a] Λ(x ) Λ(x a) p k (x ) p k (x a) f(x) f(x/a) H T(Gr) = Λ(x a) H T(Fl) = Λ(x a) Q[a] Q[x,a] =: R(x;a) 26/53

42 Localization S Z acts on all x variables in R(x;a) = Λ(x a) Q[a] Q[x,a]: 27/53

43 Localization S Z acts on all x variables in R(x;a) = Λ(x a) Q[a] Q[x,a]: Definition For f R(x;a) = Λ(x a) Q[a] Q[x,a] and w S Z define i w by H (Fl) i w H (pt) = = Λ(x a) Q[a] Q[x,a] Q[a] i w i w(f) = f(wa;a). Replace each x i by a w(i) in both tensor factors. 27/53

44 Double Schubert polynomials [Lascoux,Schützenberger] Definition For w S n S + the double Schubert polynomial S w (x;a) Q[x +,a + ] is defined by S (n) w (x;a) = (x i a j ) 0 i+j n It is well-defined for w S +. S w (x;a) = x i S wsi (x;a) if ws i > w 28/53

45 Double Schubert polynomials [Lascoux,Schützenberger] Definition For w S n S + the double Schubert polynomial S w (x;a) Q[x +,a + ] is defined by S (n) w (x;a) = (x i a j ) 0 i+j n It is well-defined for w S +. S w (x;a) = x i S wsi (x;a) if ws i > w S v (wa;a) is the localization of the v-th opposite Schubert class at the w-th T-fixed point. 28/53

46 Negative double Schubert polynomials rev(s i ) = s i rev(x k ) = x 1 k rev(a k ) = a 1 k. For w S define Example(s) S w (x ;a ) = rev(s rev(w) (x + ;a + )) S s3 s 2 s 1 = (x 1 a 1 )(x 1 a 2 )(x 1 a 3 ) S s 3 s 2 s 1 = ( x 0 +a 0 )( x 0 +a 1 )( x 0 +a 2 ) 29/53

47 Negative double Schubert polynomials rev(s i ) = s i rev(x k ) = x 1 k rev(a k ) = a 1 k. For w S define Example(s) S w (x ;a ) = rev(s rev(w) (x + ;a + )) S s3 s 2 s 1 = (x 1 a 1 )(x 1 a 2 )(x 1 a 3 ) S s 3 s 2 s 1 = ( x 0 +a 0 )( x 0 +a 1 )( x 0 +a 2 ) For w = w w + S S + = S 0 define S w (x;a) = S w S w+ H T(Fl Fl + ) = Q[x;a] = w S 0 Q[a]S w (x;a) 29/53

48 Back-stable double Schubert polynomials S w (x;a) = ( 1) l(u) S u 1(a)S v (x) w. =uv Since single Schubs back-stabilize, so do double Schubs. 30/53

49 Back-stable double Schubert polynomials Define the back-stable double Schubert polynomial S w (x;a) by Sw (x;a) = l(u) w =uv. ( 1) S u 1(a) S v (x). u,v S Z 31/53

50 Back-stable double Schubert polynomials Define the back-stable double Schubert polynomial S w (x;a) by Sw (x;a) = l(u) w =uv. ( 1) S u 1(a) S v (x). u,v S Z Sshift(w) (x;a) = shift( S w (x;a)) 31/53

51 Back-stable double Schubert polynomials Define the back-stable double Schubert polynomial S w (x;a) by Sw (x;a) = l(u) w =uv. ( 1) S u 1(a) S v (x). u,v S Z Sshift(w) (x;a) = shift( S w (x;a)) Proposition Sw (x;a) = w =uvz. ( 1) l(u) S u 1(a)F v (x/a)s z (x) u,z S 0 and in particular S w (x;a) R(x;a) = Λ(x a) Q[a] Q[x,a]. 31/53

52 Double Stanley function (new!) Wrong-way Q[a]-algebra map η H T(Fl) H T(Gr) Λ(x a) Q[x; a] η Λ(x a) f(x/a) g(x;a) f(x/a)g(a;a) Set x i to a i in the second tensor factor only. 32/53

53 Double Stanley function (new!) Wrong-way Q[a]-algebra map η H T(Fl) H T(Gr) Λ(x a) Q[x; a] η Λ(x a) f(x/a) g(x;a) f(x/a)g(a;a) Set x i to a i in the second tensor factor only. Definition The double Stanley function F w (x a) Λ(x a) is defined by F w (x a) = η( S w (x;a)) 32/53

54 Theorem (Lam, SJ Lee, S.) The equivariant class of Knutson s graph Schubert variety [G(w)] H T (Gr(n,2n)) is given by F w(x a) after truncation and setting a k a k n for all k. 33/53

55 Equivariant coproduct formulae acts on Λ(x a) factor Theorem (Lam, SJ Lee, S.) ( S w (x;a)) = w. =uv Sw (x;a) = F u (x a) S v (x;a) w =uv. F u (x a)s v (x;a) v S 0 F w (x a) = w =uvz. ( 1) l(u) S u 1(a)F v (x/a)s z (a). u,z S 0 34/53

56 F w (x a) = w =uvz. ( 1) l(u) S u 1(a)F v (x/a)s z (a). u,z S 0 Take w = w λ : Durfee square d(λ): Biggest d d λ Corollary [Molev] [Lam, SJ Lee, S.] s λ (x a) = ( 1) λ/µ S w 1 (a)s µ (x/a) λ/µ µ λ d(µ)=d(λ) s λ (x/a) = µ λ d(µ)=d(λ) S wλ/µ (a)s µ (x a) 35/53

57 Equivariant positivity, a la Peterson Define the double (factorial) Schur functions s λ (x a) Λ(x a) by s λ (x a) = S wλ (x;a) for λ Y. Define the double Edelman-Greene coefficients jλ w (a) Q[a] by F w (x a) = λ Yj w λ (a)s λ(x a). 36/53

58 Examples F sk+1 s k (x a) = s 2 (x a)+(a 1 a k+1 )s 1 (x a) F sk 1 s k (x a) = s 11 (x a)+(a k a 0 )s 1 (x a) for all k Z. Theorem (Lam, SJ Lee, S.) j w λ (a) Z 0[a i a j i j] where Problem Find a combinatorial formula for jλ w (a) that exhibits this positivity. By Peterson s quantum = affine theorem, these are some of the equivariant Gromov-Witten invariants for Fl n. 37/53

59 Summary for infinite equivariant setting Double Stanley functions and equivariant graph Schubert classes Equivariant positivity of Edelman-Greene coefficients Triple product formula for double Stanleys and back stable double Schuberts 38/53

60 Bumpless pipedreams For S n, tile n n square by unit square tiles 39/53

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66 Weight of box in row i and column j is x i a j Weight of pipedream is product of weights of boxes Theorem (Lam, SJ Lee, S.) The weighted sum of bumpless pipedreams for w S n inside the n n box is S w (x;a). For w S Z inside the plane: S w (x;a). For λ = l(w) the Edelman-Greene coefficient jλ w is the number of bumpless pipedreams for w of partition shape λ. 45/53

67 Weight of box in row i and column j is x i a j Weight of pipedream is product of weights of boxes Theorem (Lam, SJ Lee, S.) The weighted sum of bumpless pipedreams for w S n inside the n n box is S w (x;a). For w S Z inside the plane: S w (x;a). For λ = l(w) the Edelman-Greene coefficient jλ w is the number of bumpless pipedreams for w of partition shape λ. Different than [Billey, Bergeron] combinatorics 45/53

68 Homology H T (Gr), following [Molev] H T (Gr) = Λ(x a) and H T (Gr) are dual Hopf Q[a]-algebras Let ˆΛ(y a) be the completion of symm. funcs. in y = (...,y 1,y 0 ) with coefficients in Q[a]. Degree is allowed to be unbounded p r (x a) and p r (y) are primitive for r 1 H T (Gr) is isomorphic to a Q[a]-subalgebra of ˆΛ(y a). There is a Q[a]-bilinear pairing, : Λ(x a) Q[a] ˆΛ(y a) Q[a]: s λ (x/a), s µ (y) = δ λµ. That is, the pairing has reproducing kernel Ω[(x a)y] = i,j 0 (1 a i y j ) (1 x i y j ) Define Molev s dual Schur functions ŝ λ (y a) by s λ (x a), ŝ µ (y a) = δ λµ. 46/53

69 s λ (x a) = µ λ( 1) λ µ S wµw 1 (a)s µ (x/a) λ Corollary ŝ µ (y a) = S wλ w 1(a)s λ(y) µ λ µ d(λ)=d(µ) s µ (y) = ( 1) λ µ S wµw 1 (a)ŝ λ (y a) λ λ µ d(λ)=d(µ) Example ŝ 1 (y a) = ( a 0 ) q a p 1 s (p+1,1 q )(y) p,q 0 47/53

70 Left divided differences i a : divided difference on a variables. λ±i: partition λ with box added to (removed from) i-th diagonal a is λ (x a) = s λ i (x a) where the answer is 0 if λ does not have a removable box on the i-th diagonal. 48/53

71 Left divided differences i a : divided difference on a variables. λ±i: partition λ with box added to (removed from) i-th diagonal a is λ (x a) = s λ i (x a) where the answer is 0 if λ does not have a removable box on the i-th diagonal. Reason: { i a S si w(x;a) if s i w < w Sw (x;a) = 0 otherwise. 48/53

72 Homology divided differences Definition Ω[(x a)y] = i,j 0 [Naruse for type C ] [ Lam, SJ Lee, S.] (1 a i y j ) (1 x i y j ) δ i = Ω[(x a)y] a i Ω[(a x)y] If i 0, since a = a = (...,a 1,a 0 ), a i commutes with Ω[(a x)y] and we get δ i = a i if i 0 δ 0 = Ω[(x a)y](a 0 a 1 ) 1 (id s a 0)Ω[(a x)y] = α 1 0 Ω[ a 0y](id s a 0)Ω[a 0 y] = α 1 0 (id Ω[(a 1 a 0 )y]s a 0). Analogy: Ω[(a 1 a 0 )y] is a translation by highest coroot a 1 a 0 49/53

73 Theorem (Lam, SJ Lee, S.) a iŝ λ (y a) = ŝ λ+i (y a) where the answer is 0 if λ does not have an addable box on the i-th diagonal. In particular Example ŝ λ (y a) = δ wλ (1). ŝ 1 (y a) = δ 0 (1) = (a 0 a 1 ) 1 (id Ω[(a 1 a 0 )y]s 0 )(1) = (a 0 a 1 ) a 0 y k 1 a 1 y k k 0 = ( a 0 ) q a p 1 s (p+1,1 q )(y) p,q 0 50/53

74 Conjecture (Lam, SJ Lee, S.) H T (Gr) is isomorphic to the Q[a]-subalgebra of ˆΛ(y a) generated by elements of the form Ω[(a i a j )y] 1 a i a j for i j. For affine root systems [Bezrukavnikov, Finkelberg, Mirkovic] 51/53

75 Conjecture (Lam, SJ Lee, S.) H T (Gr) is isomorphic to the Q[a]-subalgebra of ˆΛ(y a) generated by elements of the form Ω[(a i a j )y] 1 a i a j for i j. For affine root systems [Bezrukavnikov, Finkelberg, Mirkovic] Similar operators work for affine type A, creating double k-schur functions. 51/53

76 Future directions Schubert bases in the Rees rings (T C ) Affine flag varieties. K-theory; K-analogue of Rees construction Affine Hecke analogues 52/53

77 Sage code Localization for H T (Fl) Symmetric function code for H T (Gr) Double Affine Hecke algebra 53/53