Analysis on arithmetic quotients 1

Analysis on arithmetic quotients 1 This talk will be found at http://www.math.ubc.ca/˜cass/sydney/talk2.pdf 2 Let be the group of real points on a reductive group defined over , a group of finite index in . The spectrum of is of great numbertheoretical interest. In this talk I’ll try to give some idea of some of the difficulties involved in the subject, by explaining . This talk will something of what happens for be largely a talk in analysis, since it is the analysis involved that most people find intimidating. I’ll say almost nothing of the connections with number theory. 3 Geometry 4 5 mod $ "# % ! ' & * ) ( ) ( ' & ) ( ' & - , -. , + * ' & ' & 1 2 0 / ) ( ) ( + . , , / 3 ) ! ( ! 6 8 7 654 9 7 : ; < 9 7 ; 7 5 = 8 5 = = @ A 654 > ? = @ 4 B 7 . is the stabilizer of The group D For in , and generate a lattice in . An eletakes this to the lattice spanned by ment in and , which span the same lattice. This is similar to the lattice spanned by and . C C K JI H GF E C O C L MN MP C O C L MN MP K I KQ I D S 8 R In fact, points of the quotient larity classes of lattices in . parametrize simi- T The fundamental domain of is the region U Vc ba` [ ` _ XW V ^]\ [Z Y XW V [ There is a simple reduction algorithm transforming an arbitrary to one in this region, a succession of transformations U Uf de g h [ [ U Uf V g [ W [ ei j The algorithm stops because in every cycle increases. It is also related to the problem of finding a relatively orthogonal basis for lattices. 9 Γ ∩ P \HY ,→ Γ\H (Y ≥ 1) y=Y 10 wxrm v t has height u srq n m po l k 11 : the image of . The problem 12 { y { y~ } | { yz { { { { } { { ~ y y~ } y { 13 is essentially self-adjoint The differential operator . on What is its spectrum? This is a special case of another question. Funcmay be identified with functions on tions on fixed on the right by . The operator is the of the Casimir element of , restriction to . which lies in the centre What is the decomposition of into eigenspaces of , or (equivalently) irreducible unitary representations of ? 14 The first step is to make a simple decomposition. is invariant under Any reasonable -invarant on integral horizontal translations, so | x The function is called cuspidal if the constant term vanishes. With mild conditions on , the differvanishes very rapidly at . The subspace ence ª ©§¨ ¥¦¤ ¡¢£ decomposes into finite-dimensional eigenspaces of (looks like of a compact Riemannian manifold). 15 ¬ « º ² º ¹¯ ³¸µ ·¶´µ ´° ¾ ¬ « ·¶´µ ´° ±³² ¿À ¬ « ±³² ¬ « ® ±¯° ® ¬ « ¶ ½¸¼ ³» ´µ ¯ ±¯° All three of these components are related to number theory. È ÇÆ ¿ Â Å Ä Â Á Ã ¬ « The spectrum of is , and we can describe the eigenfunctions contributing to it rather explicitly. ´µ ´° ±³² ±¯° ¬ « is a mystery. The few The decomposition of constituents explicitly known are related to representations of Galois groups. ½¸¼ ³» ´µ ¯ 16 are linked to- Ê É Í Î Õ ÓÕ ÔÒ ÔÏÓ and ÊËÉ The spaces gether. Ñ ÑÒ ÐÎÏ ÍÌ The continuous spectrum directly involves ß ÝÞØ Ü× Û â Ù Êã ä à Ù Ø Ü× Û Ú ß Ø Ù Ø× Ö å ë has a simple pole at is its residue there. Ü æ ß Ø× Ú Ù ß Ú ì Ù Ø× Ö ì We have on tion of the eigenvalues on on that line. winding of Ê É ïÌî Í¦í Î Õ Ï Ù Ø× Ö 17 éêè Ù Ø× çÙ Ø× á Ü æ å Ú Ù Ø× Û Ú Ù x Ø× Ö The function volume of and the and the distribuis related to the õ ôó ò ñð The principal tool is the trace formula, which still . holds many secrets, even for At any rate, understanding it requires understanding the continuous spectrum, which is spanned by Eisenstein series. 18 Eisenstein series 19 invari- ø For in the function is a function on and indeed . ant with respect to ö ÷ ûú ù ø ø ÿ Þþö ü ö ÷ ÷ ý The Eisenstein series is ø ü ÷ ý ý ø ÷ þÞö ø ÿ ö þ ö ü ý ý ü ÿ ö ø ý 20 ø . Since ÿ ö converges for For since ! " " ! ! this is also 3 2 6 3 ! " 4 5 4 1 , 0 / .-' # &%$ )+*'( # These series were first investigated by Maass, after some hints by Hecke. The hard theorems about them were first proven by Selberg. 21 8 9 8 : 7 ; 8 ; < 8 N J O M < KL J G IH AC F E D.C AB@ *>?= 22 X ZY \ _ _` ]^\ R U WVP S R T Q P a \ b [ edX ] c c m lk j i g f h U n 23 srq po The Laplacian is negative, which means that its spec. This corresponds to trum is contained in with as well as Eisenstein series on the line segment . t { Terra Incognita t <(s) = 1/2 q r ~ 24 s=1 w } | { wo v y x z u Here lieth zeros Convergence Theorem. The function continues meromorphically into all of . It satisfies the functional equation e Near e W In it has exactly one pole, which is a simple pole with residue the constant function . A consequence is that 25 The Plancherel theorem 26 is a continuous function of compact support , define the integral If on ® ¢ z ¬¦ ¡ £ «ª © ¨ § B¦¥ ¤ Then « « ¢ ® ¯ ¯ ¯ ¯ induces an isometry (up to a con- The map stant factor) ±° « ¹ ¸ µ Â Á ¿À ¿» ¼¾½ ¼º» . ¢ ·¶ ´ s³ ² ´ 27 « with The principal theorem is the decomposition Ä Ã Ä Ð Ä Ã Æ Ä Ã Å Î × ÕÓÔ ËÖ ÌÍ Ç Ó Ò ÕÌÔ Ë ÑÍ 28 ÏÎÌÍ ÌÈ ÉËÊ ÉÇÈ along with variants. Ü Ø ê ìâ ê ìâ eîå Ø ä ã Bâá à Ø ·ïÜ ·ïÜ ê ìâ ê è ë é é í ë çæ å Ü ë Þß ê Ý Ú êé è íë è ä ã Bâá à ë çæ å Ü êé Þß êé Ý Ú Ý Þß çæ å ä ã Bâá à Ú Ø Ù ð ï Ü ï·Ü ê ê è í ë é ë é óòë ê é è Ý ê ë é è Ú ë ï Ü Ú ë ê é ñ êè é ë ê é è í ñ ñ 29 Û Ú Ù Ø where ô In the rest of this talk, I’ll try to give some idea of how the proof of meromorphic continuation of goes. õ 30 The constant term 31 ú û Any function on is invariant under , and may therefore be expressed in a Fourier series ÷ ÷ ö ü ù ø û ö ö ý ÿ þý û ö ö þ ý ÿ þ ý ö ö ý 32 is called the constant term of If satisfies a certain mild growth condition on the , then is fundamental domain as . rapidly decreasing as The constant term controls the asymptotic behaviour of . The cuspidal component of is that of functions whose constant term vanishes, and cuspidal functions are of rapid decrease at . The cuspidal comof a compact Riemannian surponent looks like face. 33 Since and is the constant term of ! " ! # commutes with then , if "' %% " & $ # ! This has solutions * ) # ! , + - ( # # Since we already know that for of is the dominant term /.! * ) 01 #! # # ! " * ) 01 0 #! # , + - ./! # is square integrable if 34 rapidly decreasing stuff Analytic continuation by spectral analysis of the Laplacian 35 4 2 Fix . Let be a smooth function identically for large and for . 3 8 76 5 4 2 y −→ 9 : 7 7 5 The function may be identi. If is fied with one on satisfying any function on a mild growth condition then will be rapidly decreasing at . = ; < ; < = F D 5 E CB= A@= > ? G 36 J For in the region of convergence, . If then in O NEM KLJ I H will be V K H H TSR QP W U J K R R O N M I K J K U W U W U U W W J S O N M I W R K J K U J K K R O N M S W U W U X a function of compact support on , defined for all . Since is not in the spectrum of the self adis a bounded operator on joint operator , , and will be the unique function in satisfying this equation. This allows us to extend the definition to the whole region except the segment , since this region is in the complement of the spectrum. Y R H Z [ K R W U J O N M K J I X \ Q P W W Y X Y U Q P U /]W U _ ^ - V H V ` a U _-^ V 37 Setting i j k will define throughout this region compatibly with its definition by the series. b c Terra Incognita n m l n e h e h pqo h c m l E b g g e s=1 c c f f f d c b 38 Spectral complement Meromorphic continuation by Fredholm theory and Hecke operators 39 r A compactly supported function on that’s invariant under non-Euclidean rotation lifts to a function that’s left and right invariant with respect to on is a function on then multiplication by . If t s u s Aws { y | } | { r { z y v x t is again a function on . Convolution makes the set of all such functions into a ring (that’s for ). u | ~ u y 40 Geometrically speaking, this operator replaces the at a point by a weighted integral of its value of average values on non-Euclidean circles concentric at . The functions are what Selberg called pointpair invariants. If is the characteristic function of the unit disc replaces at any point around , the operator by the average value of over the unit disc centred at . 41 The operator commutes with the actions of and its Lie algebra on . The function is, up to scalar, unique with the two properties 42 Therefore there exists a homomorphism to such that from Define a truncated form of : This is a compact operator, hence by the Fredholm theory its resolvent ¢ £ ¡ ¡ ¤ ¤ ¦ ¥ is meromorphic. If convergence then in the region of §¤ ¡ ¤ ¤ ¤ § §- ¤ ¡ ¢£ ¤ ¤ « ©ª ¬ ¨ ¢£ ¤ ¤ ©ª ¤ ¬ ¨ « ¡ 43 ½ ¼ ² ² ¯ ¹º ² ·³ ¾ ·¸ ³ · ¶ ² ¯ ± ² ·³ · ¶ ´ ³ ² ± ¿ µ ² ¯ ® then we could set » ± µ ² ´-³ ² ± ¿ ¿ ² À ² Á ½ ¹º » ¼ ´-³ ´ µ ² ´ 44 ¯°® If we could solve To solve Æ Ä Å Æ Í ÌÇ ÌÂ Ë É L Ê Æ È-Ç Æ Å Ä°Ã Â Æ ÉÄ È È Ï Î Ð É Ê É we want to rewrite it as an equation with something on the right hand side. Set , in which gives ÌÇ Æ Å Æ Ë Ç Æ Ã Ã Ã Æ Ê ÌÇ ÌÂ ÌÇ Ã ÌÂ Ë Æ ÆÄ Ê ÌÇ ÌÂ Ë Ë Ë Ã Ã 45 ÌÇ ÌÂ ÌÂ Ä Â Â Ë Æ Æ È-Ç Æ Å È-Ç Æ Å É Ñ Ä Æ Ä É Ê É Ä Â and this right hand side has compact support. Ó Ô Ó Ò We can find as a meromorphic function of throughout , then find , and finally set Ö Õ Õ Ò Ó Ò Ó Ò Ö Ú × Õ Ò ßÞÝ Ü Û ÙØÕ Õà Ö Ú if we can somehow find the meromorphic continua. But tion of Ü Û ã ä Õ × Õ Ò â á ° Ü Û which means that Ó Ò ã Ö Ú ßÞÝ Õ Ü Û Ü Û Ü Õ â á ° â Õ Ö Ú Ü Û 46 × Õ Ó Ò ã â á ° Û so we can define as the ratio of the two sides. Summary: ê é - è /ïî ì íç è ë ê é - æ è ç - å Spectral analysis of allows us to analytically conto all but in . It also tinue allows us to get growth estimates, since there is a . classical bound on ó ô ñ ò ð ð For meromorphic continuation we use Hecke operators. ñ î è ç ö æ , hence . íç ö óô å ê é - è í ç õ è í å ø æ ù 47 õ î è í ç ö ÷ õ î÷ è õ î í The simplicity of the pole at . explicit formula for î íç ö õ æ å From the first, we get . Also on follows from an Completeness of the dis- ú ú 48 ú The Plancherel formula crete spectrum. References J. Bernstein, Meromorphic continuation of Eisenstein series, notes based on lectures, undated but probably sometime around 1985. A more elaborate and more general version of Selberg’s integral operator proof. ÿ-þ ý üû A. Borel, Automorphic Forms on , Cambridge Tracts in Mathematics 130, 1997. Follows Bernstein. Bill Casselman, On the Plancherel measure for the continuous spectrum for the modular group, pages 49 19–26 in Automorphic forms, automorphic representations, and arithmeric, Proceedings of Symposia in Pure Math. 66, A. M. S., 1997. Calculates the Plancherel measure without analytic continuation. Bill Casselman, Harmonic analysis of the Schwa, to appear in the proceedings of a trz space of conference in honor of Joe Shalika. Y. Colin de Verdière, Une nouvlle démonstration du prolongement méromorphe des séries dEisenstein, C. R. Acad. Sci. Paris 293 (1981), pages 361–363. The simplest version of continuation by spectral analysis of . 50 I. Y. Efrat, The Selberg Trace formula for , Memoirs of the American Mathematical Society 359, 1987. The simplest published form of Selberg’s proof of meromorphic continuation. E. Hecke, Eisensteinreihen höhere Stufe und ihre Anwendungen auf Funktionentheorie uns Arithmetic, Abh. Math. Sem. Hamburg, 5 (1927), 199-224. First mention of non-holomorphic automorphic forms. pendix F contains a proof of Selberg’s from which the proof given in Efrat’s book is derived. 51 , D. A. Hejhal, The Selberg Trace Formula for Lecture Notes in Math. 1001, Springer, 1983. Ap- T. Kubota, Elementary Theory of Eisenstein Series, Wiley, 1973. Mostly follows Selberg. R. P. Langlands, Eisenstein series, pages 235–252 in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics IX, A. M. S., 1966. Has his own proof od meromorphic continuation, aong with a sketch of what happens in higher rank. He also proves the Plancherel formula here, by analytic continuation. H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichen, Math. 52 Ann. 121 (1949), pages 141–183. First mention of the Eisenstein series , whose meromorphic continuation is proven by using known properties of applied term-by-term to its Fourier expansion. W. Roelcke, Über die Wellengleichung bei Grenzkreisgruppen erster Art, Sitzbericht Heidelberg Akad. Wiss., 1953–1955, 159–267. First attempt to use spectral analysis to prove continuation. W. Roelcke, Analytische Fortsetzung der Eisensteinreihen zu den parabolischen Spitzen von Grenzkreisgruppen erster Art, Math. Ann. 32 (1956), 121–129. Carries the program further, but inconclusively. 53 A. Selberg, Discontinuous groups and harmonic analysis, pages 177–189 in Proc. Int. Cong. Math., Stockholm, 1962. An announcement of the first complete proof of the meromorphic continuation relying only on spectral methods. Also the first proof of the spectral completeness of cusp forms and the residues of the Eisenstein series. A. Selberg, Introduction to the Göttingen lecture notes, in the Collected Works. A personal account of the subject. 54

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