POSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA


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1 POSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA 1. Kernel and the trace formula Beginning from this lecture, we will discuss the approach to Langlands functoriality conjecture based on Arthur s trace formula. The techniques used are very different from that based on the use of Lfunctions and converse theorem, as we will see. The most impressive result in this direction is the recent works of Arthur, which besides establishing the functorial lift from classical groups to general linear groups, he also obtained the complete description of the discrete automorphic spectrum of classical groups, in terms of the automorphic spectrum of general linear groups. As before, G is a connected reductive group over the global field F. We fix a Haar measure dg on GpAq. Denote by G Q Res F {Q G the Weil restriction of scalars, and A GQ the Qsplit component of the centre of G Q. We put A`8 : A GQ prq 0 Then the quotient GpF qa`8zgpaq has finite volume with respect to dg. be We fix a unitary character ω of A`8, and consider the L 2 space of functions: L 2 pgpf qzgpaq, ωq where the translation action of A`8 is given by ω. The space L 2 pgpf qzgpaq, ωq support the right regular representation R of GpAq: for φ P L 2 pgpf qzgpaq, ωq prpgqφqphq φphgq, g, h P GpAq For a test function f P Cc 8 pgpaq, ω 1 q, the regular representation R defines the operator Rpfq on L 2 pgpf qzgpaq, ωq: ż prpf qφqphq f pgqφphgqdg GpAq which is an integral operator, whose kernel is given by the function: K f pg 1, g 2 q ÿ fpg 1 1 γg 2q γpgpf q 1
2 2 LECTURE 4 In a naive sense, given a test function f, the trace tr Rpfq of the operator Rpfq on L 2 pgpf qzgpaq, ωq is given by integrating the kernel funcction K f along the diagonal: ż (1.0.1) tr Rpfq K f pg, gqdg GpF qa`8zgpaq and the trace formula is obtained by equating the spectral expansion of the left hand side of (1.0.1), with the geometric expansion of the right hand side of (1.0.1). The main problem with this sketch is of course that, in general, the operator Rpfq is not of trace class, and hence tr Rpfq is not defined, and similarly the integral on the right hand side of (1.0.1) diverges in general. However, in the case where the quotient GpF qa`8zgpaq is compact (this is equivalent to the condition that G has no proper parabolic subgroups over F ), the operator Rpfq is of trace class, with no continuous spectrum on L 2 pgpf qzgpaq, ωq, and the right hand side of (1.0.1) converges, and in this way one gets the trace formula for compact quotient which was already derived by Selberg (c.f. page 89 of [Arthur]). For the most important case where for G is (quasi)split for instance, the automorphic quotient is no longer compact, and one must do highly nontrivial work in order to establish a trace formula. One has to truncate both the left and right hand side of (1.0.1) to obtain meaningful expressions. Even after that there are considerable difficulties. For the spectral side we have continuous components in the spectral decomposition of the L 2 space coming from Eisenstein series, which complicates the spectral side of the theory. For the geometric side, we have, in contrast to the case where the automorphic quotient is compact, elements in GpF q that are not semisimple, which complicates the geometric side of the theory. Without going into the details, let us simply say that, remarkably in a single handed manner, Arthur was able to develop the general trace formula (over a period of ten years). A good exposition is the paper [Arthur]. The form of the trace formula that Arthur finally obtained is nowadays called the Invariant trace formula. This forms the basis of the trace formula approach to Langlands functoriality, among many other applications. We first define the Hecke space HpGpAq, ω 1 q to which the trace formula applies. First we fix a maximal compact subgroup K ź v K v of GpAq.
3 LECTURE 4 3 Then the Hecke space HpGpAq, ω 1 q (with respect to K) is the subspace of C 8 c pgpaq, ω 1 q consisting of finite linear combination of functions: f ź v f v such that f v is K v finite for all archimedean v, and such that for almost all nonarchimedean v, the function f v is the characteristic function of K v. To simplify the exposition we assume that the character ω is trivial, and we omit the reference to ω in the notation. The general case can be treated with simple modification. The invariant trace formula is thus a linear functional Ipfq on the Hecke space of function f P HpGpAqq: Ipfq I spec pfq I geom pfq where I spec pfq is called the spectral expansion of the trace formula, while I geom pfq is called the geometric expansion of the trace formula. Here we note that the trace formula is called invariant, becasue all the terms appearing in both the spectral and geometric expansions are invariant under conjugation by GpAq. Remark The above description is of course a gross simplification; in reality the geometric expansion of the Invariant trace formula contains spectral terms as well; see the discussion of section of [Arthur]. In the next two section we describe the simplest terms occuring in the geometric and spectral expansion of the trace formula. In the case where the automorphic quotient is compact these are the only terms that appear. 2. Elliptic semisimple terms in the geometric expansion First we setup some notation. For a semisimple element γ P GpF q, we denote G γ,` the centralizer of γ in G, and by G γ the identity component of G γ,`, which is again a connected reductive group over F. We put ιpγq rg γ,` : G γ s Remark In this connection, a theorem of Steinberg states that if the derived group of G is simply connected, then G γ,` is already connected, i.e. G γ,` G γ for any semisimple γ. We define a semisimple element γ P GpF q to be ellipitc if A Gγ A G (recall that for a general connected reductive group H over F, we denote by
4 4 LECTURE 4 A H the F split component of the centre of H). For an elliptic semisimple element γ P GpF q we put a G 1 ellpγq : ιpγq volpg γpf qa`8zg γ paq, dg γ q The quantity a G ell pγq is the coefficient of the orbital integral f Gpγq associated to the GpAqconjugacy class of γ, for a test Hecke function f P HpGpAqq. It is defined as: ż f G pγq : fpg 1 γgqpdg{dg γ q G γpaqzgpaq We then put for, for a test function f as above: ÿ I ell pfq a G ell pγqf Gpγq γpgpf q{ conj Here the sum is over GpF qconjugacy classes of elements of GpF q. It gives the simplest, but in a sense the most significant, terms appearing in the geometric expansion of the trace formula. If the automorphic quotient is compact, we have I geom I ell. In general however, there will be contribution from orbital integrals of nonsemisimple elements, together with contributions of weighted orbital integrals from the Levi subgroups of G; see section 11 and 18 of [Arthur]. 3. Discrete spectral terms and the discrete component of the trace formula The discrete L 2 spectrum: L 2 discpgpf qa`8zgpaqq is by definition the largest GpAqsubspace of L 2 pgpf qa`8zgpaqq that decomposes discretely under the action of GpAq. Thus we have by definition a Hilbert space direct sum decomposition: L 2 disc pgpf qa`8zgpaqq xà π mpπqπ with π runs over irreducible admissible representations of GpAq, and mpπq ě 0 is a nonnegative integer, called the multiplicity of π in the discrete automorphic spectrum (of course we only need to include those π for which mpπq ě 1). The classical result of GelfandPiatetskiShapiro is that the cuspidal automorphic representations are in the discrete automorphic spectrum. We put, for f P HpGpAqq a test Hecke function: I L 2 disc pfq : ÿ mpπq tr πpfq π This represents the simplest, but in some sense the most significant, terms occuring in the spectral expansion I spec pfq of the trace formula. The other
5 LECTURE 4 5 terms involve more complicated expressions, namely weighted characters and intertwining operators coming from the continuous spectrum, which is described by Langlands theory of Eisenstein series. In the case where the automorphic quotient is compact, we have I spec pfq I L 2 discpfq, and thus in this case the trace formula for compact quotient is simply: I L 2 discpfq I ell pfq Back to the general case, we also define the discrete component of the trace formula I disc pfq for a Hecke test function f, as the expression: (3.0.2) I disc pfq ÿ MPL W0 M ÿ 1 W0 G detpw 1q wpw pmq reg a G trpm P pwqi P pfqq M The explanation of these terms is as follows: L is the set of standard Levi subgroups M of G; here by standard we mean the following: we fix a minimal Levi subgroup M 0 of G over F. Then a Levi subgroup M of G over F is called standard if M contains M 0. W pmq NormpA M, Gq{M is the relative Weyl group of G with respect to M. Here A M is the maximal F split component of the centre of M. Similarly W0 G and W 0 M denote the Weyl groups of G and M with respect ot M 0 respectively. a M (resp. a G ) is the real vector space Hom Z px F pmq, Rq (resp. Hom Z px F pgq, Rq). The canonical complement of a G in a M is noted as a G M. W pmq reg is the set of regular elements of W pmq, i.e. the set of elements w P W pmq such that detpw 1q a G 0. M P P PpMq is a parabolic subgroup of G with M as Levi component. I P is the representation of GpAq parabolically induced from the discrete automorphic spectrum: L 2 discpmpf qa` M,8 zmpaqq of MpAq. Here we denote by A` M,8 the identity component of A MQ prq M P pwq is a certain intertwining operator on I P associated to w that occurs in the functional equation of Eisenstein series; see page 34 of [Arthur]. Note that I L 2 discpfq corresponds to the summand with M G in (3.0.2). Even though the discrete component I disc pfq contains more complicated terms than I L 2 discpfq, nevertheless from the point of view of Arthur s theory of trace formula (and also from the point of view of Langlands functoriality) that it is more natural to consider I disc pfq rather than I L 2 discpfq. In any case the discrete component I disc forms the most important terms in the spectral
6 6 LECTURE 4 expansion I spec ; to define the other terms one would need the notion of weighted characters; see section 15 and 21 of [Arthur]. References [Arthur] J.Arthur, An Introduction to the Trace Formula. In Harmonic analysis, the trace formula and Shimura varieties. Clay Mathematics PRoceedings, vol.4 (2005).
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