AUTOMORPHIC FORMS AND L-FUNCTIONS
CONTENTS DISCRETE GROUP ACTIONS ACTION OF A GROUP ON A TOPOLOGICAL SPACE IWASAWA DECOMPOSITION SIEGEL SETS HAAR MEASURE INVARIANT MEASURE ON COSET SPACES VOLUME OF SL(N, Z)\SL(N, R)/SO(N, R) INVARIANT DIFFERENTIAL OPERATORS LIE ALGEBRAS UNIVERSAL ENVELOPING ALGEBRA OF GL(N, R) THE CENTER OF THE UNIVERSAL ENVELOPING ALGEBRA OF GL(N, R) EIGENFUNCTIONS OF INVARIANT DIFFERENTIAL OPERATORS AUTOMORPHIC FORMS AND L–FUNCTIONS FOR SL(2, Z) EISENSTEIN SERIES HYPERBOLIC FOURIER EXPANSION OF EISENSTEIN SERIES MAASS FORMS WHITTAKER EXPANSIONS AND MULTIPLICITY ONE FOR GL(2, R) FOURIER–WHITTAKER EXPANSIONS ON GL(2, R) RAMANUJAN–PETERSSON CONJECTURE SELBERG EIGENVALUE CONJECTURE FINITE DIMENSIONALITY OF THE EIGENSPACES EVEN AND ODD MAASS FORMS 3 10 HECKE OPERATORS 3 11 HERMITE AND SMITH NORMAL FORMS 3 13 L–FUNCTIONS ASSOCIATED TO MAASS FORMS 3 14 L-FUNCTIONS ASSOCIATED TO EISENSTEIN SERIES 3 15 CONVERSE THEOREMS FOR SL(2, Z) 3 16 THE SELBERG SPECTRAL DECOMPOSITION EXISTENCE OF MAASS FORMS THE IN?NITUDE OF ODD MAASS FORMS FOR SL(2, Z) INTEGRAL OPERATORS THE ENDOMORPHISM ? HOW TO INTERPRET ?: AN EXPLICIT OPERATOR WITH PURELY CUSPIDAL IMAGE THERE EXIST IN?NITELY MANY EVEN CUSP FORMS FOR SL(2, Z) A WEAK WEYL LAW INTERPRETATION VIA WAVE EQUATION AND THE ROLE OF ?NITE PROPAGATION SPEED INTERPRETATION VIA WAVE EQUATION: HIGHER RANK CASE MAASS FORMS AND WHITTAKER FUNCTIONS FOR SL(N, Z) WHITTAKER FUNCTIONS ASSOCIATED TO MAASS FORMS FOURIER EXPANSIONS ON SL(N, Z)\HN WHITTAKER FUNCTIONS FOR SL(N, R) JACQUET’S WHITTAKER FUNCTION THE EXTERIOR POWER OF A VECTOR SPACE CONSTRUCTION OF THE I? FUNCTION USING WEDGE PRODUCTS CONVERGENCE OF JACQUET’S WHITTAKER FUNCTION FUNCTIONAL EQUATIONS OF JACQUET’S WHITTAKER FUNCTION 5 10 DEGENERATE WHITTAKER FUNCTIONS AUTOMORPHIC FORMS AND L-FUNCTIONS FOR SL(3, Z) WHITTAKER FUNCTIONS AND MULTIPLICITY ONE FOR SL(3, Z) MAASS FORMS FOR SL(3, Z) THE DUAL AND SYMMETRIC MAASS FORMS HECKE OPERATORS FOR SL(3, Z) THE GODEMENT–JACQUET L-FUNCTION BUMP’S DOUBLE DIRICHLET SERIES THE GELBART–JACQUET LIFT CONVERSE THEOREM FOR SL(3, Z) RANKIN–SELBERG CONVOLUTION FOR GL(2) STATEMENT AND PROOF OF THE GELBART–JACQUET LIFT RANKIN–SELBERG CONVOLUTION FOR GL(3) BOUNDS FOR L-FUNCTIONS AND SIEGEL ZEROS THE SELBERG CLASS CONVEXITY BOUNDS FOR THE SELBERG CLASS APPROXIMATE FUNCTIONAL EQUATIONS SIEGEL ZEROS IN THE SELBERG CLASS SIEGEL’S THEOREM THE SIEGEL ZERO LEMMA NON-EXISTENCE OF SIEGEL ZEROS FOR GELBART–JACQUET LIFTS NON-EXISTENCE OF SIEGEL ZEROS ON GL(N) MAASS FORMS FOR SL(N, Z) HECKE OPERATORS FOR SL(N, Z) LANGLANDS EISENSTEIN SERIES 10 1 PARABOLIC SUBGROUPS 10 2 LANGLANDS DECOMPOSITION OF PARABOLIC SUBGROUPS 10 3 BRUHAT DECOMPOSITION 10 4 MINIMAL, MAXIMAL, AND GENERAL PARABOLIC EISENSTEIN SERIES 10 5 EISENSTEIN SERIES TWISTED BY MAASS FORMS 10 6 FOURIER EXPANSION OF MINIMAL PARABOLIC EISENSTEIN SERIES 10 7 MEROMORPHIC CONTINUATION AND FUNCTIONAL EQUATION OF MAXIMAL PARABOLIC EISENSTEIN SERIES 10 8 THE L-FUNCTION ASSOCIATED TO A MINIMAL PARABOLIC EISENSTEIN SERIES 10 9 FOURIER COEF?CIENTS OF EISENSTEIN SERIES TWISTED BY MAASS FORMS 10 10 THE CONSTANT TERM 10 11 THE CONSTANT TERM OF SL(3, Z) EISENSTEIN SERIES TWISTED BY SL(2, Z)-MAASS FORMS 10 12 AN APPLICATION OF THE THEORY OF EISENSTEIN SERIES TO THE NON-VANISHING OF L-FUNCTIONS ON THE LINE ?(S) = POINCAR´E SERIES AND KLOOSTERMAN SUMS 11 1 POINCAR´E SERIES FOR SL(N, Z) 11 2 KLOOSTERMAN SUMS 11 3 PL¨UCKER COORDINATES AND THE EVALUATION OF KLOOSTERMAN SUMS 11 4 PROPERTIES OF KLOOSTERMAN SUMS 11 5 FOURIER EXPANSION OF POINCAR´E SERIES 11 6 KUZNETSOV’S TRACE FORMULA FOR SL(N, Z) RANKIN–SELBERG CONVOLUTIONS 12 1 THE GL(N) × GL(N) CONVOLUTION 12 2 THE GL(N) × GL(N + 1) CONVOLUTION 12 3 THE GL(N) × GL(N?) CONVOLUTION WITH N < N? 12 4 GENERALIZED RAMANUJAN CONJECTURE 12 5 THE LUO–RUDNICK–SARNAK BOUND FOR THE GENERALIZED RAMANUJAN CONJECTURE 12 6 STRONG MULTIPLICITY ONE THEOREM LANGLANDS CONJECTURES 13 1 ARTIN L-FUNCTIONS 13 2 LANGLANDS FUNCTORIALITY N?S, (?(S) > 1), AND OBTAINED ITS MEROMORPHIC CONTINUATION AND FUNCTIONAL EQUATION ??S/2? S ?(S) = ??(1?S)/2? 1 ?S ?(1 ?S), ?(S) = ? N?Z ˆF (NY?1), WHERE F IS A FUNCTION WITH RAPID DECAY AS Y ??AND ˆF (Y) = ? ?? F (T)E?2?ITY DT, IS THE FOURIER TRANSFORM OF F THIS IS PROVED BY EXPANDING THE PERIODIC FUNCTION F(X) = N?Z F (X + N) 2(Y ˆF (0) ?F (0)), FROM WHICH IT FOLLOWS THAT FOR ?(S) > 1, ?(S) ? F (Y)YS DY Y = ? F (NY)YS DY Y = ? F (NY)YS + F (NY?1)Y?S DY Y = ? F (0) S + ˆF (0) 1 ?S IF F (Y) AND ˆF (Y) HAVE SUF?CIENT DECAY AS Y ??, THEN THE INTEGRAL ABOVE CONVERGES ABSOLUTELY FOR ALL COMPLEX S AND, THEREFORE, DE?NES AN ENTIRE FUNCTION OF S LET ˜F (S) = ? F (Y)YS DY Y DENOTE THE MELLIN TRANSFORM OF F , THEN WE SEE FROM THE ABOVE INTEGRAL REP- RESENTATION AND THE FACT THAT ˆˆF (Y) = F (?Y) = F (Y) (FOR AN EVEN FUNCTION F ) THAT ?(S) ˜F (S) = ?(1 ?S) ˜ˆF (1 ?S) CHOOSING F (Y) = E??Y2, A FUNCTION WITH THE PROPERTY THAT IT IS INVARIANT UNDER FOURIER TRANSFORM, WE OBTAIN RIEMANN’S ORIGINAL FORM OF THE FUNCTIONAL EQUA- TION THIS IDEA OF INTRODUCING AN ARBITRARY TEST FUNCTION F IN THE PROOF OF THE FUNCTIONAL EQUATION ?RST APPEARED IN TATE’S THESIS (TATE, 1950) A MORE PROFOUND UNDERSTANDING OF THE ABOVE PROOF DID NOT EMERGE UNTIL MUCH LATER IF WE CHOOSE F (Y) = E??Y2 IN THE POISSON SUMMATION FORMULA, THEN SINCE ˆF (Y) = F (Y), ONE OBSERVES THAT FOR Y > 0, ? N=?? E??N2Y = ?Y ? N=?? E??N2/Y THIS IDENTITY IS AT THE HEART OF THE FUNCTIONAL EQUATION OF THE RIEMANN ZETA FUNCTION, AND IS A KNOWN TRANSFORMATION FORMULA FOR JACOBI’S THETA FUNCTION ?(Z) = ? N=?? E2?IN2Z, 1 1 ACTION OF A GROUP ON A TOPOLOGICAL SPACE D ?C(D)(CZ + D) 2 ?(Z) HERE ?C IS THE PRIMITIVE CHARACTER OF ORDER ?2 CORRESPONDING TO THE ?ELD EXTEN- SION Q(C 2 )/Q, ?D = (MOD 4), AND (CZ + D) LET G DENOTE THE ADDITIVE GROUP OF INTEGERS Z THEN IT IS EASY TO VERIFY THAT THE GROUP Z ACTS CONTINUOUSLY ON THE REAL NUMBERS R WITH GROUP LET G = GL(2, R)+ DENOTE THE GROUP OF 2 × 2 MATRICES A B C D WITH A, B, C, D ?R AND DETERMINANT AD ?BC > 0 LET H := X + IY X ?R, Y > LET A GROUP G ACT CONTINUOUSLY ON A TOPOLOGICAL SPACE X WE SAY A SUBSET ? ?G IS DISCRETE IF FOR ANY TWO COMPACT SUBSETS A, B ?X, THERE ARE ONLY ?NITELY MANY G ?? SUCH THAT (G ?A) ?B ?= ?, WHERE ? DENOTES THE EMPTY SET 1 1 ACTION OF A GROUP ON A TOPOLOGICAL SPACE THE DISCRETE SUBGROUP SL(2, Z) LET ? = SL(2, Z) := A B C D A, B, C, D ?Z, AD ?BC = , AND LET ??:= M ?Z BE THE SUBGROUP OF ? WHICH ?XES ? NOTE THAT ??\? IS JUST A SET OF COSET REPRESENTATIVES OF THE FORM A B C D WHERE FOR EACH PAIR OF RELATIVELY PRIME INTEGERS (C, D) = 1 WE CHOOSE A UNIQUE A, B SATISFYING AD ?BC = 1 THIS FOL- LOWS IMMEDIATELY FROM THE IDENTITY 2 , WE HAVE Y (CX + D)2 < ? IF THE FOLLOWING INEQUALITIES HOLD: |D| > |C|R + (Y??1) 2 (R + 1), AND THE TOTAL NUMBER OF SUCH PAIRS (NOT COUNTING (C, D) = (0, ±1), (±1, 0)) IS AT MOST 4(??)?1(R + 1) ? DISCRETE GROUP ACTIONS IT FOLLOWS FROM LEMMA 1 1 6 THAT ? = SL(2, Z) IS A DISCRETE SUBGROUP OF SL(2, R) THIS IS BECAUSE: (1) IT IS ENOUGH TO SHOW THAT FOR ANY COMPACT SUBSET A ?H THERE ARE ONLY ?NITELY MANY G ?SL(2, Z) SUCH THAT (G ?A) ?A ?= ?; (2) EVERY COMPACT SUBSET OF A ?H IS CONTAINED IN A RECTANGLE RR,? FOR SOME R > 0 AND 0 < ? < ??1; (3) ((?G) ?RR,?) ?RR,? = ?, EXCEPT FOR ?NITELY MANY ? ???, G ???\? TO PROVE (3), NOTE THAT LEMMA 1 1 6 IMPLIES THAT (G ?RR,?) ?RR,? = ? EXCEPT FOR ?NITELY MANY G ???\? LET S ???\? DENOTE THIS ?NITE SET OF SUCH ELE- MENTS G IF G ??S, THEN LEMMA 1 1 6 TELLS US THAT IT IS BECAUSE IM(GZ) < ? FOR ALL Z ?RR,? SINCE IM(?GZ) = IM(GZ) FOR ? ???, IT IS ENOUGH TO SHOW THAT FOR EACH G ?S, THERE ARE ONLY ?NITELY MANY ? ???SUCH THAT ((?G) ?RR,?) ?RR,? ?= ? THIS LAST STATEMENT FOLLOWS FROM THE FACT THAT G ?RR,? ITSELF LIES IN SOME OTHER RECTANGLE RR?,??, AND EVERY ? ???IS OF THE FORM ? = , WITH CONGRUENT BOUNDARY POINTS SYMMETRIC WITH RESPECT TO THE IMAGINARY AXIS 1 1 ACTION OF A GROUP ON A TOPOLOGICAL SPACE 2 + IY Y ? ? IS EQUIVALENT TO THE VERTICAL LINE V := 2 ?RE(Z) < 0, |Z| = 2, |Z| = , UNDER THE TRANSFORMATION Z ??1/Z TO SHOW THAT D IS A FUNDAMENTAL DOMAIN, WE MUST PROVE: (1) FOR ANY Z ?H, THERE EXISTS G ?SL(2, Z) SUCH THAT G ?Z ?D; (2) IF TWO DISTINCT POINTS Z, Z? ?D ARE CONGRUENT (MOD SL(2, Z)) THEN RE(Z) = ± 2, ? ?IM(W) LET B? DENOTE THE ?NITE SET OF SUCH G ?SL(2, Z) CLEARLY, FOR SUF?CIENTLY SMALL ?, THE SET B? CONTAINS AT LEAST ONE ELEMENT WE WILL SHOW THAT THERE IS AT LEAST ONE G ?B? SUCH THAT G ?Z ?D AMONG THESE ?NITELY MANY G ?B?, CHOOSE ONE SUCH THAT IM(G ?Z) IS MAXIMAL IN D? IF |G ?Z| < 1, THEN FOR S = DISCRETE GROUP ACTIONS T = > IM(G ?Z) THIS IS A CONTRADICTION BECAUSE WE CAN ALWAYS CHOOSE M SO THAT T MSG ?Z ?D? SO IN FACT, G ?Z MUST BE IN D TO COMPLETE THE VERI?CATION THAT D IS A FUNDAMENTAL DOMAIN, IT ONLY REMAINS TO PROVE THE ASSERTION (2) LET Z ?D, G = A B C D ?SL(2, Z), AND ASSUME THAT G ?Z ?D WITHOUT LOSS OF GENERALITY, WE MAY ASSUME THAT IM(G ?Z) = Y |CZ + D|2 ?IM(Z), (OTHERWISE JUST INTERCHANGE Z AND G ?Z AND USE G?1) THIS IMPLIES THAT |CZ + D| ?1 WHICH IMPLIES THAT 1 ?|CY| ? ? 2 |C| THIS IS CLEARLY IMPOSSI- BLE IF |C| ?2 SO WE ONLY HAVE TO CONSIDER THE CASES C = 0, ±1 IF C = 2, THIS IMPLIES THAT EITHER B = 0 AND Z = G ?Z OR ELSE B = ±1 AND RE(Z) = ± Z IT FOLLOWS THAT A = 0 IF Z = E2?I/3 AND D = 1, THEN WE MUST HAVE A ?B = 1 IT FOLLOWS THAT G ?E2?I/3 = A ? IWASAWA DECOMPOSITION THIS MONOGRAPH FOCUSSES ON THE GENERAL LINEAR GROUP GL(N, R) WITH N ?2 THIS IS THE MULTIPLICATIVE GROUP OF ALL N × N MATRICES WITH COEF?CIENTS IN R AND NON-ZERO DETERMINANT WE WILL SHOW THAT EVERY MATRIX IN GL(N, R) CAN BE WRITTEN AS AN UPPER TRIANGULAR MATRIX TIMES AN ORTHOGONAL MATRIX THIS IS CALLED THE IWASAWA DECOMPOSITION (IWASAWA, 1949) THE IWASAWA DECOMPOSITION, IN THE SPECIAL CASE OF GL(2, R), STATES THAT EVERY G ?GL(2, R) CAN BE WRITTEN IN THE FORM: G = Y X ? ? ? ? D D (1 2 1) 1 2 IWASAWA DECOMPOSITION WHERE Y > 0, X, D ?R WITH D ?= 0 AND ? ? ? ? ?O(2, R), WHERE O(N, R) = G ?GL(N, R) G TG = I IS THE ORTHOGONAL GROUP HERE I DENOTES THE IDENTITY MATRIX ON GL(N, R) AND TG DENOTES THE TRANSPOSE OF THE MATRIX G THE MATRIX Y X IN THE DECOMPOSITION (1 2 1) IS ACTUALLY UNIQUELY DETERMINED FURTHERMORE, THE MATRICES ? ? ? ? AND D NOTE THAT EXPLICITLY, O(2, R) = ± COS T ?SIN T ± SIN T COS T 0 ?T ?2? WE SHALL SHORTLY GIVE A DETAILED PROOF OF (1 2 1) FOR GL(N, R) WITH N ?2 THE DECOMPOSITION (1 2 1) ALLOWS US TO REALIZE THE UPPER HALF-PLANE H = X + IY X ?R, Y > AS THE SET OF TWO BY TWO MATRICES OF TYPE Y X X ?R, Y > , (1 2 2) WHERE ZN = ? ?? ?? ? ?? D D ? ?? D ?R, D ?= ? ?? ?? IS THE CENTER OF GL(N, R), AND ?O(2, R), Z2?DENOTES THE GROUP GENERATED BY O(2, R) AND Z2 THE ISOMORPHISM (1 2 2) IS THE STARTING POINT FOR GENERALIZING THE CLASSICAL THEORY OF MODULAR FORMS ON GL(2, R) TO GL(N, R) WITH N > 2 ACCORDINGLY, WE DE?NE THE GENERALIZED UPPER HALF-PLANE HN ASSOCIATED TO GL(N, R) LET N ?2 THE GENERALIZED UPPER HALF-PLANE HN ASSOCIATED TO GL(N, R) IS DE?NED TO BE THE SET OF ALL N × N MATRICES OF THE FORM Z = X Y WHERE X = ? ?????? X1,N X2,N XN?1,N ? ?????? , WITH YI > 0 FOR 1 ?I ?N ?1 NOTE THAT THIS CAN ALWAYS BE DONE SINCE Y? I ?= THE GENERALIZED UPPER HALF PLANE H3 IS THE SET OF ALL MATRICES Z = X Y WITH X = ? ? THE GENERALIZED UPPER HALF-PLANE H3 DOES NOT HAVE A COM- PLEX STRUCTURE THUS H3 IS QUITE DIFFERENT FROM H2, WHICH DOES HAVE A COMPLEX STRUCTURE 1 2 IWASAWA DECOMPOSITION IF WE SET U = , THEN U SATIS?ES (1 2 7) IF ? ? , SO THAT WE MAY TAKE T = (?AC ?BD)/(C2 + D2) MORE GENERALLY, THE UPPER TRIANGULAR MATRIX U WILL HAVE N(N ?1)/2 FREE VARIABLES, AND WE WILL HAVE TO N TO ARRANGE THE BOTTOM RIGHT ENTRY OF (AU)?1 TO BE 1 IT ONLY REMAINS TO SHOW THE UNIQUENESS OF THE IWASAWA DECOMPOSITION SUPPOSE THAT ZKD = Z?K?D? WITH Z, Z? ?HN, K, K? ?O(N, R), D, D? ?ZN THEN, SINCE THE ONLY MATRICES IN HN AND O(N, R) WHICH LIE IN ZN ARE ±I WHERE I IS THE IDENTITYMATRIX,ITFOLLOWSTHATD? = ±D FURTHER,THEONLYMATRIXINHN ?O(N, R) IS I CONSEQUENTLY Z = Z? AND K = ±K? ? WE SHALL NOW WORK OUT SOME IMPORTANT INSTANCES OF THE IWASAWA DECOMPO- SITION WHICH WILL BE USEFUL LATER 1 2 IWASAWA DECOMPOSITION LET I DENOTE THE IDENTITY MATRIX ON GL(N, R), AND FOR EVERY 1 ?J < I ?N, LET EI, J DENOTE THE MATRIX WITH A 1 AT THE {I, J}TH POSITION AND ZEROS ELSEWHERE THEN, FOR AN ARBITRARY REAL NUMBER T, WE HAVE I + T EI, J = ? ?????????????? (T2 + 1) 2 T (T2 + 1) ? ?????????????? MOD (O(N, R) R×) , WHERE, IN THE ABOVE MATRIX, 2 OCCURS AT POSITION { J, J}, (T2 + 1) 2 OCCURS AT POSITION {I, I}, ALL OTHER DIAGONAL ENTRIES ARE ONES, T (T2 + 1) T2 + 1 E J, J, U?1 = I + T T2 + 1 E J,I, A?1 = I + ? T2 + EI,I THEREFORE, U?1A?1 = I + EI,I + T ? T2 + (MOD (O(N, R), R×)) ? LET N ?2, AND LET Z = XY ?HN HAVE THE FORM GIVEN IN DE?NITION 1 2 3 FOR I = 1, 2, , N ?1, DE?NE ?I = ? ?????????? ? ?????????? , TO BE THE N × N IDENTITY MATRIX EXCEPT FOR THE ITH AND (I + 1)TH ROWS WHERE WE HAVE ON THE DIAGONAL THEN ?IZ ? ? ?????? X? 1,N X? 2,N X? N?1,N THE GROUP GL(N, Z) ACTS ON HN PROOF RECALL THE DE?NITION OF A GROUP ACTING ON A TOPOLOGICAL SPACE GIVEN IN DE?NITION 1 1 1 THE FACT THAT GL(N, Z) ACTS ON GL(N, R) FOLLOWS IMMEDIATELY FROMTHEFACTTHAT GL(N, Z)ACTSONTHELEFTON GL(N, R)BYMATRIXMULTIPLICATION AND THAT WE HAVE THE REALIZATION HN = GL(N, R)/(O(N, R) R×), AS A SET OF COSETS, BY THE IWASAWA DECOMPOSITION GIVEN IN PROPOSITION 1 2 6 ? 1 3 SIEGEL SETS LET A, B ?0 BE ?XED WE DE?NE THE SIEGEL SET A,B ?HN TO BE THE SET OF ALL ? ?????? ? ?????? WITH |XI, J| ?B FOR 1 ?I < J ?N, AND YI > A FOR 1 ?I ?N ?1 LET ?N = GL(N, Z) AND ?N ???N DENOTE THE SUBGROUP OF UPPER TRIANGULAR MATRICES WITH 1S ON THE DIAGONAL WE HAVE SHOWN IN PROPOSITION 1 2 10 THAT ?N ACTS ON HN FOR G ??N AND Z ?HN, WE SHALL DENOTE THIS ACTION BY G ?Z THE FOLLOWING PROPOSITION PROVES THAT THE ACTION IS DISCRETE AND THAT ? 2 , FIX AN INTEGER N ?2 FOR ANY Z ?HN THERE ARE ONLY ?NITELY MANY G ??N SUCH THAT G ?Z ? ? 2 FURTHERMORE, GL(N, R) = ( G??N G ? ? 2 (1 3 3) REMARKS THE BOUND ? IN ORDER TO PROVE (1 3 3), IT IS ENOUGH TO SHOW THAT SL(N, R) = ( G?SL(N,Z) G ??? 2 , (1 3 4) WHERE ? T,U DENOTES THE SUBSET OF MATRICES T,U ZN WHICH HAVE DETERMINANT AND ?DENOTES THE ACTION OF SL(N, Z) ON ? 0,? NOTE THAT EVERY ELEMENT IN ? A,B DISCRETE GROUP ACTIONS IS OF THE FORM ? ?????? N,2 + + G2N,N CLAIM THE FUNCTION ? IS WELL DE?NED ON THE QUOTIENT SPACE SL(N, R)/SO(N, R) 1 3 SIEGEL SETS LET Z ?? 0,? THEN THE MINIMUM OF ? ON SL(N, Z) ?Z IS ACHIEVED AT A POINT OF ?? PROOF IT IS ENOUGH TO PROVE THAT THE MINIMUM OF ? IS ACHIEVED AT A POINT OF ?? 2 ,?BECAUSE WE CAN ALWAYS TRANSLATE BY AN UPPER TRIANGULAR MATRIX U = ? ?????? U1,N U2,N UN?1,N ? ?????? ?SL(N, Z) TO ARRANGE THAT THE MINIMUM OF ? LIES IN ?? THIS DOES NOT CHANGE THE VALUE OF ? BECAUSE OF THE IDENTITY ?(U Z) = ||EN U Z|| = ||EN Z|| WE SHALL USE INDUCTION ON N WE HAVE ALREADY PROVED A STRONGER STATEMENT FOR N = IN EXAMPLE 1 1 9 FIX ? ?SL(N, Z) SUCH THAT ?(? ?Z) IS MINIMIZED WE SET ? ?Z = X Y WITH X = ? ?????? 1 X1,2 X1,3 X1,N X2,3 X2,N 1 XN?1,N D ? ?????? , I=1 YN?I I )?1/N AS BEFORE WE MUST SHOW YI ? ? 2 FOR I = 1, 2, , N ?1 THE PROOF PROCEEDS IN 3 STEPS STEP Y1 ? ? 1 + THIS IMPLIES THAT Y1 ? ? 2 STEP LET G? ?SL(N ?1, Z), G = G? THEN ?(G? Z) = ?(? Z) THIS FOLLOWS IMMEDIATELY FROM THE FACT THAT EN G = EN STEP YI ? ? A SUITABLE DIAGONAL MATRIX BY INDUCTION, THERE EXISTS G? ?SL(N ?1, Z) SUCH THAT G? ?Z? = X? Y? ??? A J ? ? FOR J = 1, 2, , N ?2 (1 3 9) DE?NE G := G? ?SL(N, Z) THEN G ?? ?Z = G? ? Z? D? ? D = G? ?Z? D? ? D = X?? Y??, 1 4 HAAR MEASURE WHERE Y?? = Y?D? D , X?? = X? ? THE INEQUALITIES (1 3 9) APPLIED TO Y?? = Y?D? D = ? ???? Y1Y2 YN?1D Y1D D ? ????, IMPLY THAT YI ? ? 2 FOR I = 2, 3, , N ?1 STEP 2 INSURES THAT MULTIPLYING BY G ON THE LEFT DOES NOT CHANGE THE VALUE OF ?(? Z) STEP 1 GIVES Y1 ? ? THE GENERAL LINEAR GROUP GL(N, R) IS A LOCALLY COMPACT HAUSDORFF TOPOLOGICAL GROUP LET GL(N, R) DENOTE THE LIE ALGEBRA OF GL(N, R) VIEWED AS A SET, GL(N, R) IS JUST THE SET OF ALL N × N MATRICES WITH COEF?CIENTS IN R WE ASSIGN A TOPOLOGY