Catégories dérivées de blocs à défaut non abélien de GL(2,q)
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(42) . / %(@Z4{# 234u:! E ?@QZ . p!$# .3o@QQQ3Q p@Q P P 48.,B%(-:9 0. Q238?Gp! 9 97vc%! @2V*d-4-c!E->UV3@Q A ?@Q .3 A D %(4-uG' :NT%(4@Z-4 tsE@Qw .3@QQQ3Q ! A D %(4-uG' `@Q @.234u: H A 4-4U,3w4?@ D %(4-uG' ?@O%(-.3@f' .V3?@ A H A @.,AU,\%@QnD{@QP 23?@^9}L"-.3@Qf4 @ A ^%(8uQ?@m! 8Q-*8??@[@B.V[8?U, -*:%(4.V?@^9X0 E ?@Qa .3>J D %(4-uG' :N! .3> .3 D %(4-uG' cP\%(QQ 4-G:9 A ; 4 tsd@QO! .3O .3b@QQQ3Q C! D %(4-uG' C@Q W4{# 234u: 4l%a.ED !3%(.,U,\%@QnD{@QP 23_ E 9Ke. .3BA%(.V DA 4l%o^%(8uQ! 8Q-*8?wH! E A @ A ?@Q .3%(4-uG' [%@@dl%(Q-*:N$ D A 4l%^%(8uQ! 8Q-*8?! A @Q A ?@W .3 D %(4-uG' :N\.+%b4l%b?CPj:@Q-Q.+!$#k8?U, -*:%(4.3??@TQQl%(. u: 48??@W! ^%(8uQ?@ A ∞. ∞. ∞. ∗. ∗. ∞. ∗. ∞. ∞. ∞. DA. ∼. ∼. / DE. / D∞ E. ∼. / D∞ H ∗ E .. ; 43?@Qf_W.3fU,3X4l% A D=@QQ3Q "U,3X4{#k.! . .3"_ H E .]#k?@MP\%@M^%(.3. U,3:N 4-4c! 8Pj.3!>!E>B23nsZ!$# .3@?Q.+! 4l%b!E $8.VQ4-4O! E 9 VO?C.3?4Q@QGCB2\%(P -QbP\%(c! ?@WB%(P P)4@@Q W4?@ A D %(4-uG' ?@^N P @<R?#k8.3.3?4?@r8?@Q 4-B%(@ U,3}R# Q-4-@P\%(r4l%0@ -:N.OP\%(QQ 4-M! ?@ ~ QA 4?@ !E3?@X_ 9 u. 23->N,39 x,B%@Q23 +Tx)9V/wQ , 4Y*PjQQB%(.VH! "^%(4 47 .3 D=@QQQ3Q %! 8?U,\%(! H E P P Q88 "39 t9 A 3@@ %?*.3@! .3aUV3b4l%Z^%(8uQ[! 8Q-*8?a!E`' 4dCP Q-.3-P\%(4! (q) ?@Q8?UV -*:%(4.VI_I4l%c^%(8uQ0! 8Q-*8?0!$# .3 A D %(4-uG' A 9E7Q A D GL %(4-uG' ?@QWC-. -a%(4:N$:#k?@QSD _D=!E-bU,3A@B%[!E $8.VQ4-4C?@QW.V 4-4:9j; 4r@^#&%(u:-I! .3b!$# .3 %(4-uG' wu:B%!E38?`%@@dl%(Q-*w_o4l%U,34-41?@Q%RS 8?w .3 D=@QQ3Q : 9 0.3 4-4O@QQQ3Q ?@"4l%! . .38?O!$#&%(P P 4-^%(Q.3@ m P) T IA.VQI@QQQ., Pj:@Q-Q ~ n 9]a?.3@B%([U,3!3%(.3@A.3Qa@Q-Q\%(Q.??@O%(P P 4-^%(Q.3@b@.,.V 4-4?@ Pj T n > 5 4C "39 E9 t9 \#ktsdP 4--I?CP 4G.V"4l% D %(4-uG' W' .,3I!3%(.3@4?@7^%@X! (q) b! PGL (q) . @Q P P):@ %(.VbUV3A q ?@Qb?. u:Q _ &>d!E 4 <E9}PSL i6%1J %(4-uG' E A@.o234u: H E @.,!3%(.3@?[^%@cQG?@cPjQ-?@ 4{# 234u: ?@Qc! b!E-.3@Q.o?@QP)?Q-*., ! t9 ?a_[4{#k8?U, -*%(4.3?a@QP 4.3!E! ' .,3w!3%(.3@m41Q@QGw2\%(P -Q`. ?. .\%4 @QQQ3Q :9Xw?.3@QB%( %(4@WUV3!3%(.3@"??@W! EsZ^%@"-4}?@Q"P):@@-' 4c! B23@Q-" .3 D=@QQQ3Q 4-4 U,3I4?@ m @B.V"., 4@HPj n > 3 @?Q.3@ "39 "39 " t9 A Vbu8.38B%(4-@C.3@Q -C?O8?@Q 4-B%(%(`^%@! PGL (q) %?*? q -CP\%(-cU,34nD ?.3U,3:9Y3# Q-4-@BfPj 6??V4?@}QB%?*:%(Es! fvC9YKe!Ea%(. .]N 9 04-}/9^i$-.3B 4nD a%(. .]9E; 4@.,74l%@@ F 8"3@74?@7' 4,?@X_u: P)W! 0! 8 ~ %( X!E8?!EB%(4{Nd@BCnD=!E8?!EB%(4 7UV\%(Q. . U,3:N _c8?U, -*%(4.3?W! "/wQ-B%7! 8Q-*8?"P G?@Y9V; 4@! 8?Q-*.,XB2\%D .3m! [??@b4l%@@B?@A!$#k8?U, -*%(4.3?.Q!$#&%(4-uG' ?@O! m^%(U,3@^9 i]a' 4d P Q-.3-P\%(4"! PGL (q) 8B%(.Vm! Z! 8 ~ %( !E8?!EB%(4{NM??"P)Q! >^%(4 4 4l%a@QQQ3Q ! 4l% A D %(4-uG' ' .V3:9j.VQb%(4@"U,]# -4}?@Q"P):@@-' 4! B23@Q-H?Q0@QQQ3Q I! ~ % ?.Z_c?WU,3"3@74?@ m @B.VX.V 4@7Pj n > 3 9 ∗. ∞. ∞. . ∗. ∞. . 2. ∞. ∞. ∞. n. . . ∞. 2. 2. . ∗. . . . . n. 2. 2. ∞. n. ∞.
(43) .
(44) . 0 .Q *B%c.%(. .3tsE"L ! ?@MB%(P P)4@@Q M4HQ238?GW! ~ 3@Q.a!$#kL04-P)Q-. %(-.3@Q"U,3>@Q 4>?.,Q 41! >4l% ~ 3@Q.]9MKe. %(. .3ts +5 F u: .,[4?@aB%(' 4?@a! ^%(B%G?@! ?@cu: Pj?@ PSL (q) N PGL (q) N SL (q) N GL (q) N A S N]%(-.3@Q UV3I4 @"a%(QQ??@0! c! 8??CPj:@Q-Q.w"! AX%(QB%(.1.1^%(B%8Q@QQUV3 EN P @ 4l%b@QQQ3Q c! ?@Td!E 4?@HP (RQ?Q ~ @T-.3! 8??CP):@B%(' 4?@0Pj A S 9 2. 2. 2. 2. 4. 4. 4. 4.
(45)
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(47)
(48) . GL2(q). J %(.3@0?O2\%(P -Qb.33@W! 8 F . @@.3@W4?@W.3B%(Q.3@IUV @.VW Q-4-@8??@I!3%(.3@ c4l%aQ23G?@:N\0.33@0! . .3.3@I! ?@0P P Q88?@0u8.38B%(4?@I!Ewu: P) GL (q) 9 3@AB%(P P)4-4.3@a8uV%(4.VC4?@O8?@Q 4-B%(@b! a4l%1Q238?Q! mJW4--u:.3tD=i]3@)(?Q-u %(P P 4-UV38?%(Zu: Pj GL (q) 9 2. 2. !#"$%!$'&(*)+% ,.-/,.-/, 02131 jÄX@i%4j8\#V6587ÉRvk:9dj<;vX@k1j xd- ..3' +P C^970. F s Z . -;: 4
(49) ;!&" #5 O, k) N :#k?@QSD _D=!E-`U,3A4{#k.pB23@Q-c .o%(. .3^%(p! b*:%(4-\%(Q.o!E` @GC?C(K, P 4 O Nj! ?QP3@]8?@Q!E34 N?4@}UV3 @-]! f^%(B%8Q@QUV3f.V 4-4f ! M^%(B%8Q@QQU,3 .3.ED{.V 4-4 ` 9\0k.>.3 K 4Oc?QP3@T! ?@ ~ B%Q.3@T! O 9 k 0.Z.3 P 4{# ! 8^%(4]a%sE-a%(46! O N @Q$' .1UV3 P = `O k = O/P 9 0.`@Q P P):@BO! bP 4-3@U,3 ?@%@B@ (bu::@P) I3@I4?@Iu: Pj?@ F . @. P 8?@.3?:N . >= xd N\2\%(P -Q @? O9 xd- .o.3' mP Cb .?QP3@C%(4-u8' QU,3.VC4:@A! [^%(B%tD 8Q@QQUV3p p 96xd- q .3bP @@B%(.3?CF! p Nj.`.3 F 4C@B3@SD=?QP3@! F _ q 848.V@^N3 F 4@B3@SD=?QP3@"! F _ q 848.V@^9 0.>23@Q-" .Zu8.38B%( β !EZu: Pjd4-U,3 F ".ZP):@B α = β 9 L"-.3@Q α . u.3!E F 9 q} 0 T.VQ N @B- 4@3@QD{u: P)A 4-Q-P 4-^%(Q ~ ! ?.V.\%(., 4?@ x 4@ UV3 x = 1 9n0.b23µ@-} .Ou8.38B%( ζ ! µ NY! .3O .37B%-.3! 4{# . -8H!$#k!E q −1 ! O 9:0.bPj:@ ξ = ζ N! .3 ξ ?@Q .37B%-.3H! 74{# . -8 !$#k!E q − 1 ! O ξ . u.3!E µ 9VL"-.3@Q 4?@%@@Q-u:.\%(Q.3@ β 7→ ζ α 7→ ξ -.3!E @.V"! ?@"@QP 2 @Q?@0! Iu: P)?@ F −→ µ F −→ µ 9 . . 2. q2. q2. q. × q2. × q. q+1. ∗. n. n. 2. q 2 −1. q+1. q−1. . × q2. ∼. q 2 −1. × q. ∼. q−1.
(50) !.
(51) . . . GL2 (q). 0.b! 8 F . - a N b N u v ?C78B%(.Vr4?@} . UV3?@r.VQ@r4@rUV3 q−1 = ` u N N3 u v @.,TP C@0_ ` 9 q+1=` v 0.Pj:@ β = β N α = α N} β = β N}?[U,34{#k.P) O@238a%(Q@B ?CX@Q -^N =O .34--u:.37.,B%(.Vf! t _ t -.3! tsE8?7P\%( w @Q-u:. F XUV3 t = t < a. b. 0. uv. 0. u. 00. 1. v. 0. w. ???`a. u. α? ?? `b. β. 0. α0. ??`b β0. β 00. u. v. 0.1P):@B8uV%(4.V N 9357.1@ ^N\??@0.3B%(Q.3@ @.,TP 4-3@T@Q-CP 4?@0@Q ` ?@ζT!E= jζ8.,"ξ ! = Eξ9 ζ = ζ ,.- ,.- 7 55iqRUjk }3@4?@Au: Pj?@@.,.38?@A 4-Q-P 4-^%(Q-*.V^NM4 c . -8[@BB%.38? 9 x, ?@Qf .m.VQH.a.3B% 4Tu: Pj0E4-U,3"!$#k!E N 4Tu: Pj e ! ?@IPjnQ B%(Q.3@A! b4{#k.3@' 4C {1, . . . , n} N A 4C@3@SD{u: PjnCS!$# -.3!E? ! ?.V.\%(.V4?@ P)QA B%(Q.3@fP\%(-?@YNP @ D 47u: PjH!E8?!EB%(4\!$#k!E 2n N S ! 8 F . ]P\%("u8.38B%( @0H4l%(Q.3@0! 4l% ~ % ?.1@Q -*%(., < 0. uv. 0. u. 00. v. n. n. n. n. 2n. D2n = {r, s | r n = s2 = e, srs = r −1 }.. x, n ?@QHP\%(-^N3.+.3. Q2n. 4u: PjUV\%(Q. . U,3!$#k!E. H@Q n ?@Q7 . 4-Q-P 4! "O..3. 2n. ! 8 F . 6P\%( <. Q2n = {r, s | r n = e, s2 = r n/2 , srs−1 = r −1 }, SD2n. 4Wu: Pj0@CnD=!E8?!EB%(4{N ! 8 F . jP\%( <. SD2n = {r, s | r n = s2 = e, srs = r n/2−1 }.. ,x G ?@Q0 .wu: PjA H .w@B3@SD{u: P)O! G .33@W.3.3@ N (H) ?@QPj?Q-*.Va4[.3Qa%(4-@ %( O4?.,QB%(4-@B%( a! !3%(.3@ 9 C (H) 3@c.3.3@ Z(G) 4?.VQ[! G N] G 4a@3@SD{u: Pja. u.3H!E8P\%(A4G?@ d23@ ghg h = g h P\%(? ., G 960.`.3 G 4bu: P)CP Pj:@8 ! G N<:#k?@SD _D=!E-A4u: P)O! .VI4{#k.3@A' 4b@3@SD-R%?.VI?@Q0?4- ! G W4l% A 4-Q-P 4-^%(Q.w?@QT! 8 F . P\%( (g, h) 7→ hg 9 x, G ?@QX .u: P) F . {N .Z! 8?@Q-u:.3WP\%( O (G) 4WP 4-3@Xu:B%(.3!m@B3@SD{u: P) !E@QQ-. u:38b! G !$#k!EOP Cc_ ` N<0P\%( O (G) 4P 4-3@Wu:B%(.3!wUV3Q.,c! U, ]@-H . `D{u: Pj:9 G 0.%(P P)4-40@3@SD{u: Pj #5!.#,!$# .au: Pj G .m@3@SD{u: Pj"@QQQ7UV <?@Q 4.3Qa%(4-@ %( 0!$# . `D=@3@SD{u: Pj:9 G. G. −1 −1. 0. op. `0 `.
(52) . . . . ,.-/,.- Z 5 V8k Jb#&%(P G?@ = NQ238?G 'E9" ? NV! Esa^%(B%G?@ !$# .[u: P) G @B.V !3%(.3@H4 `D{' 4,! OG @]T@ 4.VW@Q].1χ% χ 0. ∀g ∈ G. =. χ0 (g)|Cl(g)| χ(g)|Cl(g)| ≡ χ(1) χ0 (1). (mod P).. ! 8?@Q-u:.3c4l%4l%@@! c?.RS uV%(@.1! g !3%(.3@ G .1%C! .3 |Cl(g)| = t9 0.p.3B% B (G) 4 `D{' 4,P Q-.3-P\%(4! OG 9]i6' 4dCP Q-.3-P\%(4?@QA?4- UV 7?.VQ.Va4[^%(B%GmQQ-*dl%(4{Nr! .3[ .^%(B%G χ ?@A!3%(.3@ B (G) @Q7 @ 4.V"@].1% . Cl(g) |G|/|CG (g)|. `. `. ∀g ∈ G. χ(g)|Cl(g)| ≡ |Cl(g)| χ(1). (mod P).. 0.b Q-4-@B%048?@ 4-B%( @ -*%(., <@Q η ?@Qr .37B%-.3H! 74{# . -87! O !$#k!E = ` .3c!E-*d@IP\%@ w N<%(4@T4l%b4l%@@! η d!E 4 P ?@QT!$#k!E w 9 `w ,.-/,. - 0
(53) Z ;37cj×b 55e 55Z/5 5WhgSidj xd- A .w%(. .3^%(]9<0.>.3 A 4{#&%(. .3^%(+P Pj:@8c! A 9 0..3 AD{,!a4l%c^%(8uQ0! ?@ AD{,!E 4?@X_IuV%(3B230! "=dP) F . {NV,!dD A 4l%1^%(8uQ+! ?@ AD{d!E 4?@_!E-Z! [ ,Pj F . {9f0..3 AD{P (RO4l%@3@SD ^%(8uQcP 4-.3! AD{d!>! .VW4?@"'ERQ@"@B.V"P (RQ?Q ~ @^9<x, B ?@Q" .w%( Q %(. .3^%(]N .Z.3 AD{d!dD B 4l%O^%(8uQc! ?@ (A, B)D{' -d!E 4?@"! I=dPj F . {9\x, @.,A! ?@%(4-uG' ?@O@Q .%(. .3^%(?C B%(Q ~ N$?Q^%(8uQ[?@Q A ^%(.3. BUV3.,"@QP 23O_O4l%C^%(8uQ (A ⊗ B )D{R,!69 0.Z.3 C(A) 4l%b^%(8uQ! ?@H?CP 4ts ?@"! AD{,!E 4?@0! I=dP) F . {9 i]?@ ?CP 4tsE?@W@.VW@Q P Pj:@8?@0_C!E $8.VQ4-4b@@ %(.V:Nj%(-.3@Q] .1?CP 4ts C ?@Q .>!El%(u:B%(C! AD{,! c. op. op. R. C:. di−1. / C i−1. / Ci. di. / C i+1. /. 4 U,3 d ◦ d = 0 Pj I I.,Q i 9jq} .VQ n ..3 C[n] 4 ?CP 4tsEb! 8?^%(48O! n B%(. um*@W4l%auV%(323b! C N)! 8 F . rP\%( C[n] = C 9 d = (−1) d 0. Q-4-@B%4?@I.3B%(Q.3@ Z (C) = ker d N B (C) = im d H (C) = 9 Z (C)/B (C) Ke.3@Q -Z..3 4l%^%(8uQZ23P U,3>! N ! .,4?@C'ERS@ @.V"?EsZ! C(A) HH(A) 4?@"QP 2 @Q?@"@B.V"?Es>! C(A) Ad!E 423P :9 i. i−1. i. i C[n]. in i C. i. i. n+i. i. i. i. i−1. i.
(54) !.
(55) . . . GL2 (q). 0 .`.3 D(A) 4l%^%(8uQ! 8Q-*8?C! A 96xd?@'ERQ@@.Vc?Es`! C(A) N$@?@ QP 2 @Q?@@.V! 8 F . @c.4d^%(4-@B%(.V4?@IQP 2 @Q?@! H(A) P\%(IB%(P P)Q %(EsCUV\%@nD{@QP 2 @Q?@^N/ d.C-.V*@B%(.,f4?@MU,\%@QnD{@QP 2 @Q?@f! H(A) *- = ? = ? Pj 4l%! 8 F . -Q.Q-u:-.34-4:N6`4?@A%(QQ4?@ =kv N$B2\%(P -Q ? N8=kv ? N =kv " ? t9 0.[!E-fUV]# .['ERSf! ! ?@QM'jQ.38H@^# -4\?@Qe@BQP 230_I . ?CP 4ts a'jQ.38:9r0..3[H(A) %(4@ C (A)D(A) N H (A) N6 D (A) 4?@O@3@SD=^%(8uQ?@ P 4-.3?@! :N?@QP)?Q-*.,^N C(A) N H(A) N7 D(A) ! .,m4?@m'ERQ@[@.,[4?@ 'ERQ@c'jQ.38?@^9}0.%>8?UV -*:%(4.3?a.,Q D (A) c4l%Z@B3@SD=^%(8uQP 4-.3[! ! .VZ4?@Z'ERQ@>@.,4?@>?CP 4tsE?@'jQ.38?@+_!E-:NT_ ?CPj:@B%(.V?@ H(A) P (RQ?Q-*?@^N30_O234u:')Q.38?A_OuV%(323:9 0.b!E- UV]# .b'ERS C ! D(A) ?@Q & ,@^# -4d?@Qr@QP 237_" .b?CP 4tsE 'jQ.38o_ ?CPj:@B%(.V?@+P (RS?Q-*?@^9 3@Z.3.3@ D{P\%( ~ 4l%@3@QD=^%(8uQ P 4-.3c! D(A) ! .VT4?@T'ERQ@T@.,TP\%( ~ %(-@^9 A x, ?@QX .Z'ERSXP\%( ~ %(-T! NE.Z!E-HUV]# -4$?@Q -8 &; "$X! D(A) @Q DA{P\%( ~ C?@QI4l%[P 4-3@IP)Q-b@3@QD=^%(D(A) 8uQCQQl%(. u: 48?bP 4-.3C! D(A) 4:@OP\%( ~ % "!E-?"?.,.\%(.V C 9 0.>!E-TU,]# .>'ERQ ! #5 -; -& . C@Q$P) ".,Q".3.ED{., 4 n .1C % D(A) . . . b. b. b. b. . . . HomDb (A) (C, C[n]) = 0.. xd-ra%(-.V.\%(., .3X%(4-uG' X%( (D{-.RS?Q-*:9:0.A.3 D=@B%('4l%T^%(8uQ Q@ B%(' 4O! A N):#k?@QSD _D=A!E-O4l%^%(8uQb! .VI4?@W'ERS@I@.,WA?Es1! AD{,!w 4?@HQP 2 @?@"@.,"! 8 F . @^N Pj X N Y 'ERS@0! AD{d!]NEP\%( HomAD=@QB%(' (X, Y ) = Hom (X, Y )/Hom (X, Y ) = Hom (X, Y ) ! 8?@Q-u:.34@3@SD=?@P\%?c! ?@HQP 2 @Q?@0! Hom (X, Y ) @ ~ %Q@B%(.V0P\%(H .>d!E 4P (RS?Q ~ 9 pr A. A. pr A. A. + "&( + )"$ &
(56) !+ GL2 0.`Pj:@ G = GL (F) 96W#k?@Q .`u: Pj%(4-u8' QUV3?. .3ts :960.o%(P Pj4-4 4{#k.3! QP 2 @QO@QB%(.3!3%(!>! '). -3@< F . . 2. F :. −→. G. ab. 7−→. cd. G aq cq. bq dq. . .. 0.P):@B 9r0. % <]4u: Pj G ?@Qc4{#k.3@A' 4m! ?@ Pj-.V@ F sE?@T! G G= @BGL 3@H(q) 4{#&%Q.+! GF 9 = G 2. F.
(57)
(58) . . . . GL2 (q). dx - T 4I@3@SD{u: PjW! ?@Xa%(QQ??@"!El%(u.\%(4?@T! G 93W#k?@QX .mIa%sEnD a%(4X! G Nr Ca%sE-a%(4T! G 4- T?@QC?.R u:38:9M; 4H?@Qb-.34-3@!3%(.3@b4 @3@SD{u: Pj>! >5X4 B ! ?@a%(QQ??@[QQl%(. u: 4l%(-?@[@Q Pj8Q ?@a[4l%P\%(- ?@QC@QB%(' 4P\%( F 9ei6u: P)Z! pd4 W ! G N ! 8 F . HP\%( W = (T , B) N?@Qf!$#k!E E9,0.a.3 e @B.a848.V7QQ-*,l%(4< s @B.[848.V N (T )/C (T ) .3.ZQQ-*,l%(4{9 xd- 4{#&%(P P 4-^%(Q.! +i6%(. u3N7! 8 F . +P\%( 9 JO#&%(P G?@f4LHQ:238?GG→"G! "i6%(. u . P\%(fts CP 4 =kJI/ EN @? L(g) = N =" ? tgN?FC(g) ?@QZ?. .3ts p%(4@ ?@QZ@Q RQ?Q-*:9Hqr Z NH.23@Q-Z . G P 8?@.VB%(., w˙ ! w !3L%(.3@ N (T ) NjP @ .%(.V8??8?! w.,∈ γW ! w˙ P\%( L 9$i6?@ ?@ D=@QB%(' 4?@! @.Va%(4@b4?@C?.RS u:38?@ 9 iM#k.3@' 41! ?@bP)-.,@ F sE?@HP\F%( F ! 2\% G.1!$#k.,QEsZ?@Q"%(P Pj48cT! G 9<0.+P):@ G. G. G. G. . . . . −1. G. w. G. γw. q} . w=e. N3.>B23@Q-"' .>@ s˙ =. 01 10. . T w = ( γw T ) F .. . e˙ = 1. 93qr . w=s q β β γ = γs = . 1 1. et. γe = 1. NE.+23@-. 0.>?CQQB%CP\%( ~ @T4{#&%(a%(4-uV%(.,Q s s˙ N .>.3B%(.V s = 9 0. ' Q.Vw! Es@3@QD{u: P)?@>! G <X4`@B3@SD{u: P) T ! ?@a%(QQ??@ !El%(u.\%(4?@O! G N6U,34{#k. %(P P)4-4* , #5 c! G Nr4a@B3@cu: Pj T N . u.3!E8P\%( %(P Pj48 . -; - ^%( s ?@Qr4{#k848.VM! T7?s ! W t9 7?@H! Esmu: Pj?@T@.,"%(')84-.3@YN T ?@X! ^%(!E-.\%(4 (q − 1) NE@QP 23c_ N3 T ?@QT! c^%(!E-.\%(4 q − 1 NE@QP 23O_ F 9 F ×F 3@.3.3@4 @7^%(B%G?@H%(-.3@Q <V@- ω 4"^%(B%GI! T ! 8 F . )P\%( 01 10. e. . . γ β 0 0 βq. . . s. . G. × q. × q. s. 2. e. 2. × q2. s. ω:γ. β 0 0 βq. . 7−→ ζ .. i]?@H^%(B%G?@H-Q8?!E3Q-' 4?@"! T @.,"%(4@X4?@ ω Pj i = 0 . . . q − 2 93xd- 4^%(B%G?@0! F ! 8 F . ]P\%( θ(α) = ξ 9 i]?@T^%(B%G?@"! T @.,"%(4@H4?@ θ NEPj i, j = 0 . . . q − 2 N\! 8 F . @TP\%( θ s. × q. i. 2. e. (i, j). θ(i, j) :. x0 0y. . 7−→ θ i (x)θ j (y).. + "& + GL2(q) i]bu: Pj ?@Q F . {N$! O^%(!E-.\%(4 960.` Q-4-@B 4{# -.RS?Q. GL G(q)=,→GLGL(q)(q ) % F .1!$#k8?Q-A?QB%((q-.3?−@01)4l%@Bq(q @?@T+! 1)GL (q) !3%(.3@ . . . 2. 2. 2. 2. 2. 2.
(59) ".
(60) . <6.o?.3@Q*B%Z 4C4. u+! ?btsEC4l%.3B%(Q. . %o%(4@48?@Q 4-B%(m@Q -*%(., M< @Q g = ?@a848.Vm! . . . GL2 (q). N%(4@ GL (F). GL2 (q 2 ). γ =. βq β 1 1. . Q@ ]T@ 4., g ∈ GL (q ) d = a c = b 9 g ∈ GL (q) i6?@^%(B%G?@7-Q8?!E3Q-' 4?@H! G @.V7! . .38?@7P\%(f4l%B%(' 4W5I9 P\%(u !!E9 ,.- '-/, jk 1 5877e_X@ZhW[kX´\]ji 7Ék 9djk \ 587cjk:9qj G 0.P):@B H = N (T ) 9rxd- W 4a@3@QD{u: P)!$#k!E >. u.3!E8aP\%( 9\0.1% N =>4{#&%Q.+! W @Q T ?@QT! . .38?P\%( s˙ = H =T oW ab cd. γ. 2. 2. 2. e. G. 01 10. e. e. q. 2. q. e. e. e. e. . x0 0y. s˙. s˙ −1 =. y0 0x. . e. .. 6i u: P) H ?@QH! .3c! ^%(!E-.\%(4 2(q − 1) NE@?@H^%(B%G?@T-Q8?!E3Q-' 4?@ @.,"! . .38?@TP\%("4l%OB%(' 4c5I9 ^ P\%(u !!E9 JI0I.[Pj:@ H = N (T ) 9 xd- W = W 9E0.Z% H = T o W Nd= 4{#&%Q.! W @Q T ?@c! . .38?CP\%( .p?.RS u:\%(.V3@4?@c848.V@P\%( 2. e. γ. . s. <. s. G. s. γ. s. . e. s. s. s. s. x 0 0 xq. s˙. . xq 0 0 x. s˙ −1 =. . .. i6u: P) ?@QH! .3c! ^%(!E-.\%(4 − 1) N @B?@H^%(B%G?@T-Q8?!E3Q-' 4?@ @.,"! . .38?@TP\%(H"4l%OB%(' 4c5I9 "P\%(u !!E2(q 9 ,.- '- jk-k 55idk 7 55iqRUjk 9dj3Q qZ 5 9dj G j8\-Z[ji 77k 1 5877e_X@ZhW[kX´\]ji 7Ék x, ` = p N! .3X@Q ` !E-*d@ q N:%(4@e4?@ `D=@3@SD{u: Pj?@e! Tx,,4.$! G @B.Ve4?@ @3@QD{u: P)?@I?.RS u:38?@I%(wu: PjA! Aa%(QQ??@! 4l% ~ Q = x ∈ F 9 ; 4@@.,c@QP 23?@b%(ou: Pj (F , +) 9r7Q[@Q-Q\%(Q.o.3@BB%ZP\%@8Q3!E8? !3%(.3@I?QOQ23G?@:Nj4-4b?@Q0QG?@I!E $8.VC! ?@%( Q?@^N).1P\%(QQ 4-.]#k.,Q P\%@!3%(.3@4b^%!EC! O4l%mQ238?QC! bJI4--u:.3tD=i$3)@ (?Q-u396i}%[?.RQ?Q ! b5738 %[88b! 8.VQ8?C!3%(.3@I?b^%@ <)/9<5738b%a' .,p! ?@0@B=dP ?@ =k5 N &J ? tN W9)0 d V%(a%[%a?.3@QQQ -0! ?@W8?UV -*:%(4.3??@W! 8Q-*8??@0Pj T4u: P) SL (q) = 0 ? t9 x, !E-*d@ NM%(4@ ?.VQ., . D=@3@SD{u: Pjma%sd-a%(4TU,3[4{#k. .3 P`9dW#k?@Qe q. −`D=1@B3@SD{u: TP)"! 0x,,4. $! `G @ ` 6= 2 N^%( G ?@QM! T^%(!E-.\%(4 9<0.1%C%(4@ N (P ) = N (T ) = H 9 q(q − 1) (q + 1) JICa@Q ` !E-*d@ q + 1 N]%(4@ T ?.VQ.,A . `D=@3@SD{u: PjCa%sE-a%(4 .38 P 9\W#k?@QH .Z@3@QD{u: P)! x,d4 $ ! G @Q ` 6= 2 9\0.+%b%(4@ N (P ) = 9 N (T ) = H 2. s. 1x 01. q. q. . . . 2. . e. e. 2. G. e. G. e. e. s. s. G. s. G. s. s.
(61) . . ` = 2 q. . . !&. . x, P P):@B.3@Xa%(-.V.\%(.VWUV3 ` = 2 XU,3 q ?@QX-CP\%(-^9\0.+%O%(4@ q − 1 = %?*? -CP\%(-@^N:f4{#k!ET!$# . D=@3@SD{u: PjH! "x,,4.$ 2 u q +1 = 2 v ! G ?@ 2 9dxd4.m4l%cu?. u:vQ3.3?I! q d!E 4 "3N,4{# `.[! ?@7.VQ@ a b ?@Q 8uV%(4e_ I4{#&%( QC?@Q@QQQ.,@Q Pj8Q _ 9ji]?@@3@SD{u: Pj?@ P P @.VH!$#k!EW?@QPj?Q ~ 2 2 N %(-.3@Q)4{# .Z! W??@X! Esmu: P)?@H?@QX!$# -.3!E? b!3%(.3@H .+@3@SD{u: Pjc! x,d4 $! G 9 x, q ?@QX?. u:Q+_ d!E 4 "3N %(4@ b = 1 NEX .@B3@SD{u: P)I! cx,d4 $! ? Q @ S = P o W 93;=4$.]#k?@HP\%@"%(')84-.]9 G xd- H = N (S ) 9)0.+' Q.V b. a. 2a+b. 2a. e. e. G. i]u: Pj. s. e. 0 e. He0. e. a+b. =. e. . x 0 0 xα0 n. . 0 x xα0 n 0. ,. ?@T! c^%(!E-.\%(4 2. He0. . x∈. a+1 (q. F× q ,n. − 1). . a. = 0...2 − 1 .. T.1%O .+@QP 2 @Q. He0 ' (Cq−1 × C2a ) o C2 ,. =4{#&%Q. ! Z4{#k848.,[.3.ED{QQ-*dl%(40! C @Q C × C ?@a! . .38?>P\%( . Q-4-@B%(.V4{# -.RQ?Q. ^%(.3. UV3 t9Hxd?@ (x, y) 7→ (xy, y ) ^%(B%G?@T-Q8?!E3Q-' 4?@W@.V"! . .38?@"P\%(H4l%bB%(' 4c5I9 &OP\%(Cu ! ,→ 9 C x, q ?@QT?. u:Q+_ d!E 4 "3N3%(4@ a = 1 H .>@3@SD{u: Pj! x,,4.$ ! ?@Q S = P o W 9 ;=4$.]#k?@Q"P\%@"%(')84-.+.3.>P 4-3@^9 G xd- H = N (S ) 9\0.+' Q.V s. s. q−1. 2a. . 2a. q−1. s. 0 s. G. Hs0. . i]u: Pj. 2. . −1. =. Hs0. s. γ. β 00 n 0 0 β 00 qn. . ,. 0. β 00 qn. β 00 n 0. ?@T! c^%(!E-.\%(4 2. . b. . n = 0 . . . 2 (q − 1) − 1 .. b+1 (q. − 1). N\.1%O .>@QP 2 @Q. Hs0 ' C2b (q−1) o C2. =O4{#&%Q.[! X4{#k848.VM.3.CQQ-*dl%(43! C @Q C ?@ ! . .38?TP\%( x 7→ x 9 xd?@T^%(B%G?@T-Q8?!E3Q-' 4?@W@.V0! . .38?@TP\%(H4l%bB%(' 4c5I9 'bP\%(u ! 9 0.>%b! .3I*,ZU,304?@ D=@3@SD{u: Pj?@H! x,,4.$ ! G .3I@.VHP\%@T%('j84-.3@ @Q]T@ 4.VW@Q q ?@QH-C`P\%(-" ` = 2 9 2. . . 2b (q−1). ! ` = 2 q
(62) & 0.>@Q P Pj:@! .3cU,3 ` = 2 "U,3 q ?@QH-CP\%(-^9 i]c^%(!E-.\%(46! G ?@Q . . |GL2 (q)| = (q − 1)2 q(q + 1) = 22a+b qu2 v,. q.
(63) !'.
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