Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta
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1 Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta theta ( ) Λ ( ) August 9, 2000 Theta lifting of representations and geometry of nilpotent orbits". 1 1
2 Bernstein ( ( ) ( ) Chen-bo Zhu (NUS) ) 1 G ( ) G = SL n ;SO n ;Sp n ( ) G g G y g ffn g = fx 2 g j ad x g x 2 g ad x g N g N g (a) N g P(g) PN g (b) J = C [g] G C [g] G J + augmentation ideal ( ) I(N g )=J + C [g] g==g = SpecC [g] G μ : g! g == Ad G N g = μ 1 (0) (c) G N g N g G #N g = Ad G<1. (d) PN g Weyl #W h intersect.#(n g ; h) PN g Q 1»i»l P Ng (t) = (1 tm i+1 ) (1 t) dim g fm 1 ;m 2 ;::: ;m l g (l = dim h = rank g) g ( W ) 2 2 C [g] G Q fm i +1g l i=1 g cohomology l Poincaré (1 + i=1 t2mi+1 ) 2
3 (e) g f(x) f(@) h 2 C [g] G f(x) 2 J + f(@)h =0 h G H C [g] 'HΩJ (J = C [g] G ) h Ω f h f G C[N g ] 'H C [N g ] 'H' M 2Q + m fi( ) (m = dim fi( ) T ) T Q Q + 2 Q + fi( ) G m =dimfi( ) T [12] Example 1.1 G = SL n (C ) () Jordan () n () n Young Young ( ) Young f0g : (). 9 >= >; n Ad G C A 3 () z } n
4 f1; ;n 1g P Ng (t) = Q n k=1 (1 tk ) (1 t) n2 deg N g = n! =#S n Weyl SL n (C ) Weyl n Young Springer Weyl G R ( ) K R G R G R K R gr; kr g; k G R G C k K C kr gr # # ±1 gr = kr Φ pr g = k Φ p N g R = N g gr G R G R N p = N g p (G C ;K C ) ([27] ) N p K C K C Example 2.1 G R SU(p; q);so(p; q);sp(p; q) (1) G R = U(p; q) 3 U(p; q) U(p) glp M K R = ; g = k Φ p = Φ p;q U(q) gl q M q;p M p;q = M p;q (C ) p q K C = GL p (C ) GL q (C ) p = M p;q Φ M q;p ' M p;q Φ M Λ p;q N p = f(x; Y ) 2 M p;q Φ M q;p j (XY ) k =0 (9k 2 N)g 3 U(p; q) SU(p; q) 4
5 (2) G R = O(p; q) 4 O(p; q) ρ O(p) sop X K R = ; g = k Φ p = Φ O(q) so t q X fi X 2 M p;qff K C = O p (C ) O q (C ) p ' M p;q (C ) X 2 M p;q (C ) p p X t X N p = fx 2 M p;q (C ) j (X t X) k =0 (9k 2 N)g (3) G R = Sp(n;R) G R = Sp(n;C ) U(n; n) 5 ρ g K R = t g ρ 1 A g = k Φ p = fi g 2 U(n) ff t A ' U(n); fi A 2 gl nff Φ Sym n Sym n Sym n (C ) n n K C ' GL n (C ) p ' Sym n (C ) Φ Sym n (C ) Λ N p = f(x; Y ) 2 Sym n (C ) Φ Sym n (C ) j (XY ) k =0 (9k 2 N)g (G C ;G R ) 6 ([27]) Kostant- Kostant- N g R N p (a) #N g R = Ad G R < 1; #N p = Ad K C < 1 (b) K # G C ( ) K # K C N p 7 K # C[p] K # + C [N p ] K # p K # C [N p ] ' X Φ fl m fl fl (m fl =dimfl M # ) 4 5 (GC ;KC) 6 gr igr 7 K # 2 4 ([26]) 5
6 3 3 fl K # pr ( ) ar M # = Z K# (ar) (ar K # ) fl M # fl M # ( ) (G C ;K C ) N p Kostant-Rallis [13] (c) Kostant- ([27]) N g R = Ad G R N p = Ad K C N g R = Ad G R bijective ψ! N p = Ad K C O R ψ! O # O R O # O C O C =AdG C (O R )=AdG C (O # ) O R O C ( dim R O R =dim C O C ) O C gr O # O C (2 dim C O # =dim C O C ) O C p (d) Kostant- KS- Kronheimer [14], Vergne [29] O R ψ! O # K R 8 Schmid-Vilonen [24] core K R C(O R ) ρo R C(O # ) ρo # (1) K R C(O R ) ' C(O # ) ; (2) O R ' T C(O R)O R ( ) O # ' T C(O# )O # 3 N g R N p 8 KR GR KC 6
7 G R ( ) K R Verma Borel-Weil Schur G R (ß; H) G R K R H K H K G R gr (K R ; g) g! g ( ) g U(g) (ß; H K ) U(g) ß : G R (1 ) =) (ß; H K ):(g;k R ) =) U(g) ß ß ß H K H Harish-Chandra Vogan, Schmid, Wallach g " g U(g) Definition 3.1 ( ) U(g) I (primitive ideal) U(g) M I = Ann M = fx 2 U(g) j Xm = 0 (8m 2 M)g G R ( ) ß K R U(g) H K I ß 9 Barbasch ([1]) 7
8 U(g) X 2 U(g) ( ) g G R S(g) ' C [g Λ ] I ß S(g) gr I ß gr I ß Theorem 3.2 I ß gr I ß G C V(gr I ß )=O C ß (9O C ß 2N g = Ad G) V(J) J Z(g) gr I ß S(g) G C + N g V(gr I ß ) I ß G C Borho-Brylinski, Joseph ([3], [4], [9]) 1980 G R g U(g) ß K C gr ß gr ß gr U(g) = S(g) Ann (gr ß) S(g) Definition 3.3 ( ) (ß; H) G R Ann (gr ß) K C AV (ß) = V(Ann (gr ß)) ß (associated variety) 10 gr Ann O C ß Theorem 3.4 G R (ß; H) N p K C O i (1» i» r) O i AV (ß) = r[ i=1 O i 10 V(gr I ß )=V(gr (Ann ß)) I ß U(g)=I ß 8
9 O i O i Ad G C (O i )=O C ß (8i) Vogan ([30], [31]) G R ( ) N p K C K C ( ) Definition 3.5 ( ) (ß; H) (g;k R ) M =grh A = S(g) AV (ß) = supp M = V(Ann M) = [ r i=1o i supp M O i P i M Pi P i M Pi A Pi m i AC (ß) = rx i=1 ß (associated cycle) m i [O i ] (m i = length APi M Pi ) Remark 3.6 2O i K C (K C ) m i (K C ) ( ) PM =0(P = P 1 ) m =dim C M =m( )M ( M 2O= O 1 m( ) ) (K C ) M =m( )M Example 3.7 (1) ß G R ß O C ß = f0g AV (ß) =f0g AC (ß) =dimß [f0g] (2) Q R ff ß =Ind G R QR ff O C ß =Ad(G C )n; AV (ß) =N p AC (ß) =dimff N p = 9 n =(q ) X O:max.dim dim ff [O]
10 (3) (G R ;K R ) Hermite K C p = p + Φp pr G R =K R p G R =K R G R ß G R =K R L 2 G R K R fi ß K 3.2 O C ß =Ad(G C )p + =Ad(G C )p AV (ß) =p + AC (ß) =dimfi [p + ] G R ß ß G R ( ) G R Harish-Chandra G R ß (wave front set) ß T Λ G R ß ß g Λ R Ad G R ( ) Ad Λ G R ß N g R ([7], [22]) WF (ß) Barbasch-Vogan ([2]) Fourier Fourier Fourier (asymptotic support) 11 AS (ß) AS (ß) = supp (Fourier-transf.( ß )) = Ad G C (O R i )=O C ß (8i) ([22]) r[ i=1 O R i (O R i 2N g R = Ad G R) Theorem 3.8 (Rossmann) G R WF (ß) =AS (ß) ß 11 Fourier (Rossmann [22]) 10
11 Kostant- Schmid-Vilonen [25] Theorem 3.9 (Schmid-Vilonen) ß G R AV (ß) AS (ß) Kostant- Fourier Schmid-Vilonen AC (ß) k P i m i[o i ] Kostant- ψ! ψ! Asympt.Cycle (ß) k P m i i[o R i ] 4 theta G R = Sp N (R) G R (G R ;G 0 R ) dual pair G R 12 G R ;G 0 R Lie G R G R = Sp N (R) [ [ G 0 R (G R ;G 0 R) : dual pair Example 4.1 dual pair G 0 R Hermite G R G R G 0 R Sp (p+q)n (R) Sp (p+q)(r+s) (R) Sp (p+q)2n (R) O(p; q) Sp n (R) U(p; q) U(r;s) Sp(p; q) O Λ (2n) G R = Sp N (R) Weil ( metaplectic Segal-Shale-Weil ) Ω 12 G R dual pair [23] 11
12 N p ( ) Ω=Ω + Φ Ω : ; O C Ω ± = O C min; AV (Ω ± )=O min Weil 13 (5.1) Weil [32], [10], [8], [21] [34] Definition 4.2 ß 2 c GR ;ß 0 2 c G 0 R theta Ω ß Ω ß 0 G R G 0 R ß theta corr. ψ! ß 0 () 9G R G 0 R-morphism : Ω ß Ω ß 0 ß = G 0 R!G R (ß0 )= (ß 0 ) ß ß 0 theta well-defined ß 0 ß theta Howe [8] p theta Howe duality correspondence, dual pair correspondence Example 4.3 (1) G 0 R G R G R Hermite ß 0 G 0 R ß = (ß0 ) (2) dual pair (G R ;G 0 R )=(Sp n(r);o(p)) ß = (ß 0 ) 2n <p ß n>2p ß theta? ß; ß 0 AC (ß 0 )= X i AC (ß) = X j m 0 i[o 0 i ] m j [O j ] O0 i 2N p 0=K 0 C O j 2N p =K C 13 Weil 12
13 Problem 4.4 ß; ß 0 theta ( ) 9 : O 0 i ψ! O i fm j g fm 0 ig? Przebinda, Trapa theta ([28], [6]) ß 0 (G 0 R ) theta 5 theta W = C n Hermite ( ; ) W R 2n W R W R W ( ; ) W R hu; vi =Im(u; v) (u; v 2 W R ) G R = Sp N (R) = Sp(W R ) G R Lie G Cartan G = K Φ P; P = P + Φ P ( 2.1 ) G R Cartan g = k Φ p (G 0 R ) p Φ p 0 ρ P Cartan p; p 0,! P P Λ! p Λ ; p 0Λ G AdG C P Λ ' P Weil : AV (Ω) = O min ρ P Λ. W=f±1g ' O min O min ρ Sym N (C ) ' P + ( 2.1 (3) ) O min = fx 2 Sym N (C ) j rank X =1gρP; O min = O min f0g (5.1) W O min W 3 w 7! w tw 2 O min ρ Sym N (C ) (w ) '; ψ 13
14 W A '? AA ψ O A min ff AU p p 0 O min! p P! p O min W = C N G R K R = U(N) K R K 0 R ρ K R K C K 0 C ρ GL N(C ) = K C W Lemma 5.1 ([18], [19]) '; ψ K C ;K 0 C ' : W! '(W ) ρ p K 0 C ψ : W! ψ(w ) ρ p0 K C Im ' ' W==K 0 C ; Im ψ ' W==K C Example 5.2 O(p; q) Sp n (R) ρ Sp (p+q)n (R) W = M p;n (C ) Φ M q;n (C ); p = M p;q (C ); p 0 =Sym n (C ) Φ Sym n (C ) W = M p;n (C ) Φ M q;n (C ) 3 (A; B) '(A; B) =A t B; ψ(a; B) =( t AA; t BB) W = M p;n Φ M q;n 3 (A; B) ' Ψ A t B 2 p = M Sym n Φ Sym n = p 0 3 ( t AA; t BB) dual pair (G R ;G 0 R) G 0 R Hermite (G R ;G 0 R ) stable range stable range stable range Hermite 3 14
15 G 0 R 3 G R G R G 0 R stable range Sp (p+q)n (R) O(p; q) Sp n (R) 2n <min(p; q) Sp (p+q)(r+s) (R) U(p; q) U(r;s) r + s<min(p; q) Sp (p+q)2n (R) Sp(p; q) O Λ (2n) n<min(p; q) Definition 5.3 ( theta ) O 0 N p 0 K 0 C N p K C ' ffi ψ 1 (O 0 )=O O ρ p O 0 ρ p 0 theta O = (O 0 ) ' ffi ψ 1 (O 0 ) = O O dual pair K C O ( ) ± = ψ 1 (O 0 ) ρ W K C KC 0 ' : W! '(W ) ( ) '(±) ' ±==KC 0 K C p K C O O'±==K 0 C ; C [O] =C [±] K0 C ; ±=ψ 1 (O 0 ) Example 5.4 U(2; 3)! U(5; 5) theta U(p; q) Young ( [5] ) Jordan ( Hermite (p; q) ) O(p; q) Sp(p; q) U(p; q) p + q + p q Young ± theta U(r;s)! U(p; q) (r;s) Young ( 0 ) (p + q) (r + s) 15
16 + +! theta x4 x5 G 0 R G R theta Theorem 6.1 ([16], [33], [17]) dual pair (G R ;G 0 R ) G R Hermite (G R ;G 0 R) stable range G 0 R ß 0 theta ß = (ß 0 ) theta O 1 = (f0g) AC (ß 0 ) = dim ß 0 [f0g]; AC (ß) =dimß 0 [O 1 ] Dim ß = dim O 1 ; Deg ß =dimß 0 deg O 1 Dim ß Gelfand-Kirillov Deg ß Bernstein Remark 6.2 G 0 R p ( ) G 0 R [16] [33] G 0 R ß0 ( ) ([17]) ß 0 stable range stable range stable range theta ([15] ) stable range dual pair (G R ;G 0 R) p theta Theorem 6.3 ([17], [20]) dual pair (G R ;G 0 R) p stable range G 0 R Hermite ß0 fi 0 ß 0 K 0 R ß = (ß0 ) theta O 0 ρ p
17 p 0+ = AV (ß 0 ) K 0 C Ohol = (O 0 ) theta AC (ß 0 ) = dim fi 0 [p 0+ ]; AC (ß) =dimfi 0 [O hol ] Dim ß = dim O hol ; Deg ß =dimfi 0 deg O hol Conjecture 6.4 dual pair (G R ;G 0 R ) stable range G0 R ß 0 O 0 AC (ß 0 )=m 0 [O 0 ] ß = (ß 0 ) AC ( (ß 0 )) = m 0 [ (O 0 )] Remark 6.5 U(p; q) U(r;s) ß = (ß 0 ) ([28]) theta open 7 G R 7.1 theta N p 0 KC 0 f0g theta (stable range ) theta O 1 2-step G 0 R stable range 2-step 15 (O(p; q);sp n (R)) C [O 1 ] [17], [18], [19], [20] Example 7.1 (G R ;G 0 R) = (O(p; q);sp n (R)) (2n < p; q) p = M p;q K C = O(p;C ) O(q;C ) O(p;C ) O(q;C ) ( 2.1 ) 15 17
18 O 1 = G 0!G (f0g) ρn p theta O 1 = fx 2 M p;q j rank X = n; X t X =0; t XX =0g; O 1 = fx 2 M p;q j rank X» n; X t X =0; t XX =0g dterminantal variety K C = O(p;C ) O(q;C ) C [O 1 ] ' X Φ 2P n ff p ( ) ff q ( ) P n = f =( 1 ;::: ; n ) j 1 n 0g n ff p ( ) O(p;C ) O 1 K C 7.2 theta O 0 ρ p 0 + ( ) theta ([17], [18]) p 0 + K0 C O0 O hol = (O 0 ) ρ p theta 3-step Example 7.2 (continued) p 0 + = Sym n (C ) K 0 C = GL n(c ) O 0 ρ Sym n (C ) O 0 = fx 2 Sym n (C ) j rank X = ng theta O hol ρ p = M p;q (C ) O hol = G 0!G (O 0 )=fx 2 M p;q j rank X = n; rank X t X = n; t XX =0g O hol = fx 2 M p;q j rank X» n; t XX =0g O hol K C = O(p;C ) O(q;C ) O hol O(p;C ) GL(q;C ) O hol ([20]) C [O hol ] ' X Φ 2P n ff p ( ) fi q ( ) fi q ( ) GL q (C ) O(p;C ) GL(q;C ) O hol K C [O hol ] 18
19 ([11], [20] ) Example 7.3 (continued) p dego hol = 1 V V = n! Y Z 0»i<j<n jd (x)d (x)j(x p+q 3n Dn An 1 1 x n ) dx; Ω n (j 2 i 2 )(j i) x =(x 1 ;::: ;x n ) D An 1 (x) = Y D Dn (x) = Ω n = fx j Y X Y 0»i<n»j<p=2 1»i<j»n 1»i<j»n 1»i»n (x i x j ); (j 2 i 2 ) (x i x j )(x i + x j ); x i» 1;x i 0(8i)g Y 0»i<n»j»q (j i) References [1] D. Barbasch, The unitary dual for complex classical Lie groups. Invent. Math. 96(1989), [2] D. Barbasch and D. A. Vogan, Jr., The local structure of characters. J. Funct. Anal. 37(1980), no. 1, [3] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, [4] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), no. 1, [5] David H. Collingwood and William M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold [6] A. Daszkiewicz, W. Kraśkiewicz and T. Przebinda, Nilpotent orbits and complex dual pairs. J. Algebra 190(1997), [7] R. Howe, Wave front sets of representations of Lie groups. In Automorphic forms, representation theory and arithmetic (Bombay, 1979), , Tata Inst. Fundamental Res., Bombay, Bombay,
20 [8] R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2(1989), no. 3, [9] A. Joseph, On the associated variety of a primitive ideal. J. Algebra 93(1985), [10] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, [11] Shohei Kato and Hiroyuki Ochiai, The degrees of orbits of the multiplicity-free actions. To appear in Astérisque. [12] B. Kostant, Lie group representations on polynomial rings. Amer. J. Math., 86(1963), [13] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces. Amer. J. Math., 93(1971), [14] P. B. Kronheimer, Instantons and the geometry of the nilpotent variety. J. Differential Geom. 32(1990), [15] J.-S. Li, Singular unitary representations of classical groups, Invent. Math., 97 (1989), [16] Kyo Nishiyama, Hiroyuki Ochiai and Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules Hermitian symmetric case. Kyushu University Preprint Series in Mathematics, To appear in Astérisque. [17] Kyo Nishiyama and Chen-bo Zhu, Theta lifting of holomorphic discrete series. The case of U(p; q) U(n; n). (submitted for publication). [18] Kyo Nishiyama, Theta lifting of two-step nilpotent orbits for the pair O(p; q) Sp(2n; R). In H. Heyer, T. Hirai and N. Obata (eds.), Infinite Dimensional Harmonic Analysis", Transactions of a Japanese-German Symposium held from September 20th to 24th, 1999 at Kyoto University, pp , Kyoto [19] Kyo Nishiyama, Multiplicity-free actions and the geometry of nilpotent orbits. To appear in Math. Ann. [20] Kyo Nishiyama, Theta lifting of holomorphic discrete series II. In preparation. [21] R. Ranga Rao, On some explicit formulas in the theory of Weil representation. Pacific J. Math. 157(1993), [22] W. Rossmann, Picard-Lefschetz theory and characters of a semisimple Lie group. Invent. Math. 121(1995), [23] H. Rubenthaler, Les paires duales dans les algébres de Lie réductives. Astérisque No. 219, (1994), 121 pp. [24] W. Schmid and K. Vilonen, On the geometry of nilpotent orbits. Asian J. Math. 3(1999),
21 [25] W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups. Ann. of Math. (2) 151(2000), [26] Jiro Sekiguchi, The nilpotent subvariety of the vector space associated to a symmetric pair. Publ. Res. Inst. Math. Sci. 20(1984), [27] Jiro Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair. J. Math. Soc. Japan, 39(1987), [28] P. Trapa, Annihilators, associated varieties, and the theta correspondence, preprint, November [29] M. Vergne, Instantons et correspondance de Kostant-Sekiguchi. C. R. Acad. Sci. Paris Sér. I Math. 320(1995), [30] D. A. Vogan, Jr., Associated varieties and unipotent representations, in Harmonic analysis on reductive groups (Brunswick, ME, 1989), , Progr. Math. 101, Birkhäuser, Boston, Boston, MA, [31] David A. Vogan, Jr. The method of coadjoint orbits for real reductive groups. In Representation theory of Lie groups" (Park City, UT, 1998), , IAS/Park City Math. Ser., 8, Amer. Math. Soc., Providence, RI, [32] A. Weil, Sur certains groupes d'opérateurs unitaires. Acta Math. 111(1964), [33] Hiroshi Yamashita, Cayley transform and generalized Whittaker models for irreducible highest weight modules. To appear in Astérisque. [34] 4 Weil,
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Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
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Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
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x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
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December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
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2014 x n 1 : : 2015 1 30 : 5510113 1 x n 1 n x 2 1 = (x 1)(x+1) x 3 1 = (x 1)(x 2 +x+1) x 4 1 = (x 1)(x + 1)(x 2 + 1) x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) 1, 1,0 n = 105 2 1 n x n 1 Maple 1, 1,0 n 2
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1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)
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1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p)
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l Wel (Yoch Meda) Graduate School of Mathematcal Scences, The Unversty of Tokyo 0 Galos ([M1], [M2]) Galos Langlands ([Ca]) K F F q l q K, F K, F Fr q Gal(F /F ) F Frobenus q Fr q Fr q Gal(F /F ) φ: Gal(K/K)
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January 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
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第5章 偏微分方程式の境界値問題
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