2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald Weyl Demazure. g, h Cartan., Q := i I Zα i h root lattice, Q + := i I Z 0α

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1 2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald 1.1. Weyl Demazure. g, h Cartan, Q := i I Zα i h root lattice, Q + := i I Z 0α i Q, P := i I Zϖ i h g weight lattice ;, ϖ i h, i I, g, P + := i I Z 0ϖ i P dominant weight, P ++ := i I Z >0ϖ i P regular dominant weight, W = r i i I GL(h ) g ( ) Weyl ;, r i GL(h ) simple reflection, l : W Z 0 length function, w W, q, t, N 2Z >0 P 1 Q ( ) N, K := Q(t)(q 1/N ) P A := K[P ] e ν, ν P,, W K[P ] we ν := e wν, w W, ν P,, A W A W - λ P + m λ := µ W λ eµ (orbit-sum), {m λ λ P + } A W K, + Q + g root, ρ := 1 2 α α = + i I ϖ i P., a ρ := w W ( 1) l(w) e wρ = e ρ, λ P + α + (1 e α ), a λ+ρ := s λ := a λ+ρ a ρ (Weyl ) ( 1) l(w) e w(λ+ρ), λ P +, {s λ λ P + } A W K, λ P + s λ = m λ +, λ µ λ µ Q + w W µ<λ,µ P + K λ,µ m µ, K λ,µ Z 0 ; (B): ( : 16H03920) 2010 Mathematics Subject Classification: 17B37; 33D52 extremal weight module, Demazure module, Macdonald polynomial naito@math.titech.ac.jp 131

2 , λ P +, s λ λ g L(λ) : s λ = ν λ(dim C L(λ) ν )e ν ;, L(λ) ν L(λ) ν-weight, b g ( + ) Borel, w W weight wλ extremal weight vector v wλ L(λ) wλ b- L w (λ) := U(b)v wλ Demazure, Demazure 1.2. Macdonald A W A = K[P ], Macdonald K {P λ (q, t) λ P + } : P λ (q, t) = m λ + a λ,µ m µ, a λ,µ K. µ<λ,µ P + Remark 1.1. [N], Macdonald pseudo-quantum Lakshmibai-Seshadri path (pqls path),, A = K[P ], Macdonald K {E µ (q, t) µ P } : E µ (q, t) = e µ + b µ,ν e ν, b µ,ν K. ν<µ,ν P ν < µ, µ + ν + Q + \{0}, µ + = ν + W Bruhat order v(µ) v(ν) ;, ν P ν + P + W ν P + = {ν + }, v(ν) W v(ν)ν = ν + (, [M].) 1.3. Macdonald (1) : q = t, λ P + P λ (q, q) = s λ (Weyl ), q, µ = wλ, w W, E µ (0, 0) Demazure L w (λ) (Demazure ) (2) : t = 1, λ P + P λ (q, 1) = m λ (orbit-sum), q, µ = wλ, w W, E µ (0, 1) = e µ (3) : q = 0 132

3 , λ P + P λ (0, t) Hall-Littlewood, : P λ (0, t) = 1 W λ (t) ( w w W, µ P +, E µ (0, t) = e µ (4) : t = q k, k R >0 ; q 1 ( t 1) J (k) λ e λ ) 1 te α, W 1 e α λ (t) = α + w W,wλ=λ t l(w)., λ P + P λ (q, t) Jacobi (Jack ) 2. Semi-infinite Lakshmibai-Seshadri paths g aff = ( C[z, z 1 ] C g ) Cc Cd (untwisted ), h aff = h Cc Cd Cartan, aff = { α + kδ α, k Z } ( ) ; g, h Cartan, h g ( ), W aff = r i i I aff = W Q ( ) Weyl., W = r i i I g ( ) Weyl, Q = i I Zα i ;, I aff = I {0} ( ) + aff, + aff = + { α+kδ α, k Z 1 } ; baff g aff ( + aff {kδ k Z 1} ) Borel 2.2. Semi-infinite Bruhat Definition 2.1. ( ) Weyl W aff = W Q semi-infinite length function l 2 : W aff Z, l 2 (wtξ ) := l(w) + 2 ρ, ξ, w W, ξ Q,, ρ := (1/2) α + α, l : W Z 0 ( ) Weyl W length function Definition 2.2 ([INS]). ( ) Weyl W aff semi-infinite Bruhat BG 2 (Waff ), W aff + aff, directed edges x β r β x, x W aff, β + aff, s.t. l 2 (rβ x) = l 2 (x) + 1 Definition 2.3 ([L], [Pe]). ( ) Weyl W aff semi-infinite Bruhat, W aff 2, β 1 β 2 β r x = x 0 x1 xr = x : x, x W aff, x 2 x 133

4 BG 2 (Waff ) directed path 2.3. Semi-infinite Lakshmibai-Seshadri paths., λ P ++ := i I Z 1ϖ i ;, ϖ i = Λ i a i Λ 0, i I, (Λ i, i I aff, g aff ).,,, λ P + := i I Z 0ϖ i ;, Definition 2.4. σ Q, 0 < σ 1, (BG 2 (Waff ) ) BG 2 λ,σ (W aff ), W aff + aff, directed edges BG 2 (Waff ) directed edges x β r β x σ xλ, β Z Remark 2.5. BG 2 λ,1 (W aff ) = BG 2 (Waff ) Definition 2.6 ([INS]). λ P ++ semi-infinite Lakshmibai-Seshadri (LS) path of shape λ, η = (x 1 > 2 > 2 x s ; 0 = σ 0 < σ 1 < < σ s = 1), x k W aff, σ k Q (1 k s),, 1 k s 1 x k+1 x k BG 2 λ,σk (W aff ) directed path (, x k+1 x k BG 2 (Waff ) directed path, x > 2 x k+1 ) Remark 2.7. η, σ k (x k+1 λ x k λ) i I aff Z 0 α i, 1 k s 1,, s 1 wt(η) := (σ k+1 σ k )x k+1 λ λ + Zα i i I aff k=0 λ P ++, B 2 (λ) semi-infinite LS path of shape λ., B 2 (λ) η = (x 1 > 2 > 2 x s ; 0 = σ 0 < σ 1 < < σ s = 1), ι(η) := x 1 (initial direction), κ(η) := x s (final direction) 2.4. Standard monomial theory for semi-infinite LS paths. λ, µ P ++ ( λ + µ P ++ ). B 2 (λ) B 2 (µ) S 2 (λ + µ) S 2 (λ + µ) := {π η B 2 (λ) B 2 (µ) κ(π) 2 ι(η)} 134

5 Theorem 2.8 ([KNS3]). λ, µ P ++, S 2 (λ + µ) B 2 (λ) B 2 (µ) subcrystal, crystal S 2 (λ + µ) = B 2 (λ + µ) 3. extremal Demazure extremal U q := U q (g aff ) g aff ( ) Definition 3.1 ([Kas1]). M U q -, 0 v M λ P aff := ( i I aff ZΛ i ) + Zδ v extremal, { v x }x W aff M v e = v : x W aff j I aff, E j v x = 0 and F (k) j v x = v rj x if k := xλ, α j 0, F j v x = 0 and E ( k) j v x = v rj x if k := xλ, α j 0;, E j, F j (j I aff ) U q Chevalley, E ( k) j, F (k) j divided power (j I aff ) 0 v M µ P aff extremal, j I aff S j v := { (k) F j v if k := µ, αj 0, E ( k) j v if k := µ, αj 0, M extremal W aff ; 0 v M λ P aff extremal, Definition 3.1, x W aff S x v = v x Remark 3.2. M extremal W aff, braid. Definition 3.3 ([Kas1]). λ P + = i I Z 0ϖ i ( λ ) extremal V (λ), 1 v λ U q -, v λ V (λ) λ extremal Theorem 3.4 ([INS]). λ P ++ ( λ ) extremal V (λ), semi-infinite LS path of shape λ B 2 (λ) 135

6 3.2. Demazure. λ P + = i I Z 0ϖ i, V (λ) ( λ ) extremal, U q := F j j I aff U q, U q Definition 3.5 ([Kas2]). x W aff, V (λ) Demazure V x (λ) V x (λ) := U q S x v λ V (λ) Theorem 3.6 ([NS9]). λ P ++, x W aff V (λ) Demazure V x (λ), Demazure crystal B 2 x (λ) := { η B 2 (λ) κ(η) 2 x} ; κ(η) W aff η final direction λ, µ P ++, x W aff, Theorem 2.8 (crystal ) B 2 (λ + µ) = S 2 (λ + µ) B 2 (λ) B 2 (µ), Demazure crystal B 2 x (λ + µ) B 2 (λ + µ), Theorem 3.7 ([KNS3]). λ, µ P ++, x W aff, (crystal ) B 2 x (λ + µ) = {π η S 2 (λ + µ) κ(η) 2 x} 3.3. Demazure, λ P + = i I Z 0ϖ i, w W W aff Demazure V w (λ) V (λ), g aff Cartan h aff : V w (λ) = β Q + aff V w (λ) wλ β ;, β Q + aff := i I aff Z 0 α i, V w (λ) wλ β wλ β ( ), β Q + aff k Z 0,, V w (λ) gch V w (λ) gch V w (λ) := γ Q, k Z 0 ( dim V w (λ) wλ+γ kδ ) e wλ+γ q k β = γ + kδ, γ Q, Remark 3.8. Demazure V w (λ) ( h aff - ), gch 136

7 4. Macdonald (t = 0, t = ) t = 0 Demazure. λ P ++, w W P λ (q, t) Macdonald, E wλ (q, t) Macdonald ([M], [RY], [OS] ). Remark 4.1. w W, t = 0, E w λ(q, 0) = P λ (q, 0) Theorem 4.2 ([NS9]). λ = i I m iϖ i P ++ (m i Z 1, i I) Demazure Ve (λ) gch Ve (λ), : ( m i 1 gch Ve (λ) = (1 q )) r E w λ(q 1, 0). i I r=1 w W, Demazure Vw (λ) ( ) U w(λ) := V w (λ)/ V w (λ) i I V t α i (λ), gch U w(λ) Macdonald t = 0 E w wλ(q 1, 0), Proposition 4.3 (see [LNS 3 4]; cf. [FM]). λ = i I m iϖ i P ++ (m i Z 1, i I), w W, : w ( gch U w (λ) ) = ( i I 1 (1 q )) r E w wλ(q 1, 0); m i 1 r=1, w W w e ν = e w ν, ν P, gch U w(λ) 4.2. t = Demazure. λ = i I m iϖ i P ++ (m i Z 1, i I), w W, Macdonald E wλ (q, t) t = ([CO], [OS] ). Remark 4.4. E w λ(q, ) P λ (q, ) = P λ (q 1, 0) Theorem 4.5 ([NNS1]). λ = i I m iϖ i P ++ (m i Z 1, i I) Demazure V w (λ) gch V w (λ), : ( m i 1 gch Vw (λ) = (1 q )) r E w λ(q, ). i I r=1 Remark 4.6. V w (λ) V e (λ) V (λ) 137

8 w W, Demazure V w (λ) (level-zero van der Kallen ) K w(λ) := V w (λ)/ ( z>w,z W V z (λ), gch K w(λ) Macdonald t = E wλ (q, ), Theorem 4.7 ([NS10]; cf. [Kat], [FKM]). λ = i I m iϖ i P ++ (m i Z 1, i I), w W, : ( m i +ϵ 1 i gch K w(λ) = (1 q )) r E wλ (q, );, i I ϵ i := r=1 { 1 (wsi > w), 0 (ws i < w) 5. Demazure 5.1. Borel-Weil-Bott. G, B G Borel, H B torus, X := G/B λ P + Hom(H, C ) L G/B (λ) λ (, λ ) G-, w W = N G (H)/H, X(w) := BwB/B G/B = X Schubert ; Bruhat G/B = w W BwB/B λ P + L G/B (λ) X(w) G/B L X(w) (λ), H i (X(w), L X(w) (λ)), i 0, B- (w = w X(w ) = G/B, H i (G/B, L G/B (λ)) G- ), ( ) Fact 5.1. λ P +, w W (1) G- : (2) B- : (3) i > 0, H 0 (G/B, L G/B (λ)) = L(λ). ) H 0 (X(w), L X(w) (λ)) = Lw (λ). H i (X(w), L X(w) (λ)) = {0}. 138

9 5.2. Semi-infinite N Borel B G unipotent radical, G/N quasi-affine G/U affine closure (, G/N := Specm C[G/N] );, ( ) Peter-Weyl, G- C[G] = λ P + (L(λ) L(λ)), G- C[G/N] = λ P + L(λ), Z := G/N \ (G/N) (affine ) G/N, C[z ]- Z [z ] G/N [z ], H (free) right quotient ( ) Q G := G/N [z ] \ Z [z ] /H semi-infinite ( ) ;, G/B G/N H (free) right quotient., z = 0 ev 0 : G[z ] G Borel B G I := ev 1 0 (B) G[z ] ( ) Q G W aff W 0 aff := W Q,+, Q,+ := i I Z 0αi, : Q G = O(x); x=wt ξ W 0 aff, I- O(x), x W 0 aff, Q G l 2 (x) = l(w) + 2 ρ, ξ, semi-infinite Bruhat I- 2, x W 0 aff, Q G(x) := O(x) Q G semi-infinite Schubert (, Q G = Q G (e) = O(e) ) 5.3. Semi-infinite Borel-Weil-Bott semi-infinite Q G, (Drinfeld-) Plücker ( ) P(L(ϖ i )[z ]) i I P(L(ϖ i)[z ]),, i I Q G G[z ]- O QG (ϖ i ) ; λ = i I m iϖ i P +, O QG (λ) := i I O Q G (ϖ i ) m i Q G G[z ]-, x W 0 aff, Q G(x) Q G I- O QG (x)(λ), H i (Q G (x), O QG (x)(λ)), i 0, I- (, I b (zc[z] C g) b aff ) I-, Borel-Weil Theorem 5.2 ([KNS3]). λ P +, x W 0 aff (1) : (2) i > 0, : gch H 0 (Q G (x), O QG (x)(λ)) = gch V x ( w λ). H i (Q G (x), O QG (x)(λ)) = {0}. 139

10 [BF1] [BF2] [BFP] [BN] A. Braverman and M. Finkelberg, Semi-infinite Schubert varieties and quantum K-theory of flag manifolds, J. Amer. Math. Soc. 27 (2014), A. Braverman and M. Finkelberg, Weyl modules and q-whittaker functions, Math. Ann. 359 (2014), F. Brenti, S. Fomin, and A. Postnikov, Mixed Bruhat operators and Yang-Baxter equations for Weyl groups. Int. Math. Res. Not. IMRN 1999, no. 8, J. Beck and H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), [CO] I. Cherednik and D. Orr, Nonsymmetric difference Whittaker functions, Math. Z. 279 (2015), [FF] [FKM] [FM] [FN] [INS] [Kas1] [Kas2] B. Feigin and E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), E. Feigin, S. Kato, and I. Makedonskyi, Representation theoretic realization of nonsymmetric Macdonald polynomials at infinity, preprint 2017, arxiv: E. Feigin and I. Makedonskyi, Generalized Weyl modules, alcove paths and Macdonald polynomials, Selecta Math. (N.S.) 23 (2017), N. Fujita and S. Naito, Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases, Math. Z. 285 (2017), M. Ishii, S. Naito, and D. Sagaki, Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras, Adv. Math. 290 (2016), M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), M. Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41 (2005), [Kat] S. Kato, Demazure character formula for semi-infinite flag manifolds, preprint 2016, arxiv: [KNS1] [KNS2] [KNS3] [L] [LNS 3 1] [LNS 3 2] [LNS 3 3] [LNS 3 4] S. Kato, S. Naito, and D. Sagaki, Polytopal estimate of Mirković-Vilonen polytopes lying in a Demazure crystal, Adv. Math. 226 (2011), S. Kato, S. Naito, and D. Sagaki, Tensor products and Minkowski sums of Mirković- Vilonen polytopes, Transform. Groups 17 (2012), S. Kato, S. Naito, and D. Sagaki, Equivariant K-theory of semi-infinite flag manifolds and Pieri-Chevalley formula, preprint 2017, arxiv: G. Lusztig, Hecke algebras and Jantzen s generic decomposition patterns, Adv. Math. 37 (1980), C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, Int. Math. Res. Not. IMRN 2015, no. 7, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, Quantum Lakshmibai-Seshadri paths and root operators, Adv. Stud. Pure Math. Vol. 71 (2016), C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals II: Alcove model, path model, and P = X, Int. Math. Res. Not. IMRN 2017, no. 14, C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at t = 0 and Demazure characters, Transform. Groups 22 (2017),

11 [LS] [M] [N] [NNS1] [NNS2] [NS1] [NS2] [NS3] [NS4] [NS5] [NS6] [NS7] [NS8] [NS9] [NS10] T. Lam and M. Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Math. 204 (2010), I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics Vol. 157, Cambridge University Press, Cambridge, UK, 2003., Macdonald t = 0 extremal weight, 60, S. Naito, F. Nomoto, and D. Sagaki, Specialization of nonsymmetric Macdonald polynomials at t = and Demazure submodules of level-zero extremal weight modules, Trans. Amer. Math. Soc. 370 (2018), S. Naito, F. Nomoto, and D. Sagaki, Representation-theoretic interpretation of Cherednik-Orr s recursion formula for the specialization of nonsymmetric Macdonald polynomials at t =, to appear in Transform. Groups, DOI: /s S. Naito and D. Sagaki, Path model for a level-zero extremal weight module over a quantum affine algebra, Int. Math. Res. Not. IMRN 2003, no. 32, S. Naito and D. Sagaki, Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra, Int. Math. Res. Not. IMRN 2005, no. 14, S. Naito and D. Sagaki, Path model for a level-zero extremal weight module over a quantum affine algebra. II, Adv. Math. 200 (2006), S. Naito and D. Sagaki, Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero shape, Proc. Lond. Math. Soc. (3) 96 (2008), S. Naito and D. Sagaki, Lakshmibai-Seshadri paths of level-zero weight shape and one-dimensional sums associated to level-zero fundamental representations, Compos. Math. 144 (2008), S. Naito and D. Sagaki, Mirković-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal, Adv. Math. 221 (2009), ,, extremal Littelmann,, 62 1 ( ), S. Naito and D. Sagaki, Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types B, C, and D, Trans. Amer. Math. Soc. 364 (2012), S. Naito and D. Sagaki, Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials, Math. Z. 283 (2016), S. Naito and D. Sagaki, Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at t =, preprint [OS] D. Orr and M. Shimozono, Specialization of nonsymmetric Macdonald- Koornwinder polynomials, to appear in J. Algebraic Combin., DOI: /s [Pe] D. Peterson, Quantum cohomology of G/P, Lecture notes, M.I.T., Spring [Po] [RY] A. Postnikov, Quantum Bruhat graph and Schubert polynomials, Proc. Amer. Math. Soc. 133 (2005), A. Ram and M. Yip, A combinatorial formula for Macdonald polynomials, Adv. Math. 226 (2011), [WN] H. Watanabe and S. Naito, A combinatorial formula expressing periodic R- polynomials, J. Combin. Theory Ser. A 148 (2017),

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