Introduction to Selberg Trace Formula.

Supriya Pisolkar

Abstract These are my notes of T.I.F.R. Student Seminar given on 30th Novem- ber 2012. In this talk we will first discuss Poisson summation formula and try to understand how Selberg trace formula is a generalisation of Poisson summation formula in the case where G is locally compact group and Γ is a co-compact lattice in G.

Poisson Summation formula

The group Z ,→ R ( via 1 → 1 or 1 → l) is the basic example of a . The space Z\R is a homogenoeous space of R (continuous and transitive action). Poisson summation formula is the most important consequence of the embedding Z ,→ R which relates Z\R as a manifold with representations of R on periodic functions on R. Poisson summation formula(PSF) - Let f be a rapidly decreasing function ˆ ˆ R ∞ −2πixy and f it’s defined by f(x) = −∞ e f(y)dy. Then Poisson summation formula states that X X f(n) = fˆ(2πn) n∈Z n∈Z To a given Riemannian manifold with a Riemannian metric, there is a canonical Laplace operator associated to it, which in the case of S1 is given by −d2/dx2. The Poisson summation formula (PSF) leads to a precise relation between eigen- values of the Laplace operator which are 4π2n2, n ∈ Z and lengths of closed geodesics on S1 = Z\R parameterized by Z. So PSF relates geometric data to spectral data. For the standard embedding Zn ,→ Rn, the lengths of closed geodesics on Zn\Rn are given by | v |, v ∈ Zn since the lifts to Rn are straight lines whose end points are indentified under translation by Zn. The eigenvalues of flat Riemannian manifold Zn\Rn are 4π2|v|2. For each general lattice Γ ∈ Rn there is a Poisson summation formula. The L.H.S of the formula is over the lattice representing lengths of geodesics in the flat Riemannian manifold Γ\Rn and the R.H.S. is over the dual lattice representing the eigenvalues.

1 There is also another interpretation of the Poisson summation formula where the Laplace spectrum will be replaced by representation spectrum. For e.g. if Γ is a cocompact lattice in Rn then on R.H.S, the points of the dual lattice will correspond to the irreducible unitary representations of Rn on the space L2(Γ\Rn). We can see this as follows. As a representation space of the abelian group Rn, L2(Γ\Rn) decomposes as a discrete sum of one dimensional subspaces. Let χ : Rn → S1 be a unitary character. As R is a universal cover of S1, by lifting criterion, the continuous group homomorphism χ extends to a continuous group homomorphism Rn → R. This map is in fact R-linear (easy to verify! ) ∗ ∗ and thus belongs to the dual space Rn . Similarly, any element of the space Rn when composed with a covering map R → S1 will give a unitary character on Rn. ∗ The unitary characters of Γ\Rn are precisely the elements {f ∈ Rn | f(Γ) ⊂ Z}. ∗ As Γ\Rn is compact, Γ∗ is a latice in Rn . Thus the points of the dual lattice correspond to the irreducible unitary representations of Rn. For the case n = 1 one can summarise this as the Fouier analysis for the circle group S1 can be studied in two ways: either as expanding a function in terms of eigenvalues of the Laplace operator, or via the characters of the topological group S1.

The Γ\H Trace formula is a general identity that relates X X {geometric terms} = {spectral terms}

The natural generalisation of Z is Zn and they share many properties for e.g the Poisson summation formula and they both give rise to compact flat Rieman- nian manifolds. To get different manifolds and new results one needs to go to nonabelian generalisation which is the SL2(Z). It is discrete sub- group of SL2(R). Note that SL2(R)/SO2(R) is non-compact and to avoid the problem of non-simply connectedness consider SL2(R)/SO2(R). This is identified with the upper half space

H = {x + iy | y > o}

2 dx2+dy2 The Poincar´emetric ds = y2 on H is the unique metric of constant negetive curveture upto a scalar multiplication. This metric is also SL2(R) invariant. The quotient space SL2(Z)\H is non-compact and has finite area There is canonical map called j-invariant of elliptic curves which identifies SL2(Z)\H to C. Modular form of weight 2 is a holomorphic function f(z) on H such that the product f(z)dz descends to the holomorphic 1-form on the Riemann surface

2 Γ\H. Cusps forms are rapidly decreasing objects at infinity, which represents holomorphic eigenfunctions of ∆ and are square integrable on Γ\Γ. STF is a non-abelian generalisation of PSF. When Γ\H is compact it is ob- tained by computing the trace of certain integral operator in two different ways: one is directly in terms of the original function and other in terms of the spectral decomposition of the function. This gives equality between spectral side involv- ing eigenvalues and the geometric side involving lengths of closed geodesics of the hyperbolic surface Γ\H. It is known that the spectrum of the Laplace operator in compact hyperbolic surfaces have only discrete spectrum i.e. discrete eigenvalues with finite multiplicities. When Γ\G is non-compact, then there is a continuous spectra of ∆ which is represented by Eisenstein series. The presence of continuous spectra suggests that the integral operator mentioned above is not of “trace class”.

General STF

Locally symmetric spaces are those, whose universal cover is of the form G/K where G is a connected, semisimple (or reductive) and K is a maxi- mal compact subgroup of G. Selberg generalised trace formula for Γ\H to any compact locally symmetric space. Langlands, inspired by the work of Selberg, studied the Eisenstein series (the continuous spectrum) for general locally sym- ∞ metric spaces. When G is a reductive group over Q, any function f ∈ Cc (G(A)) provides a convolution operator R(f) on the Hilbert space H = L2(G(Q)\G(A)) , which in turn has an orthogonal decomposition R(f) = Rdisc(f) ⊕ Rcont(f) relative to the discrete and continuous spectra. The trace formula in this case is due to work of Arthur and is thus called Arthur-Selberg trace formula.

A particular case of STF in detail

We explain here the STF in the case when G is a locally compact topological group and Γ is a co-compact lattice in G.

Let us first revise some needed meaure theory on locally compact topological groups. Let G be a locally compact topological group. Let µ be a Borel measure on G. Then µ is called translation invariant if for any Borel subset E of G, µ(sE) = µ(E) for all s ∈ G. A left Haar measure on G is a left translation invariant, nonzero Radon measureµ on G. The folowing holds:

−1 1. µ is a left Haar measure iff µe(E) = µ(E ). 2. µ is a left Haar measure implies µ(U) > 0 for any non-empty open subsets R of G. and for all f ∈ Cc(G), G fdµ > 0

3 3. µ is a left Haar measure then µ(K) < ∞ if and only if K is compact.

Theorem 0.1. Every locally compact topological group admitts a unique (upto scalar multiplication) a left Haar measure.

Fix a Haar measure µ on G. Let φ : G → G be a topological group automor- phism and ν(E) = µ(φ(E)). Then ν is also a left haar measure on G. Thus by uniqueness, there exists a positive no. c(φ) associated to each such φ given by ν = c(φ)µ, c(φ) ∈ R. For an element x ∈ G consider the inner automorphism −1 Innx : g → xgx and let c(x) be the constant associated to Innx. Then G is called unimodular if c(G) = 1 i.e. c(x) = 1 for all x ∈ G.

Theorem 0.2. Any connected semisimple Lie group, abelian group, , compact group is unimodular.

Theorem 0.3. Let G be a locally compact topological group, H be a closed sub- group of G. Then there exists a non-zero Haar measure on H\G if and only if c(G) |H ≡ c(H). Selberg trace formula for compact quotients:- Let G be a connected semisimple Lie group, Γ be a torsion free abelian subgroup of G and µ denote the Haar measure on G. Let 2 RΓ : G → L (Γ\G) be a right regular unitary representation of G given by

g → (RΓ(g)φ)(x) = φ(xg).

In this case STF relates representation spectrum of L2(Γ\G) with the geometry of conjugacy classes of elements of Γ. ∞ 2 For any f ∈ Cc (G) define the convolution operator on L (Γ\G) by, Z RΓ(f)(φ)(x) = f(y)φ(xy)dµ(y) G by left invariance of Haar measure we get, Z −1 RΓ(φ)(x) = f(x y)φ(y)dµ(y) G Let µ0 denote the normalised G-invariant Haar measure on the quotient Γ\G), which certainly exists by above theorems and assumptions on G and Γ. Since f is compactly supported, one can use Fubini’s theorem applied to Γ × Γ\G. Z X −1 0 RΓ(f)(φ)(x) = f(x γx)φ(y)dµ (y) γ∈Γ Γ\G

4 " # Z X = f(x−1γx) φ(y)dµ0(y) Γ\G γ∈Γ P −1 Let κ(x, y) = γ∈Γ f(x γy)

The sum over Γ is finite for any x and y since it may be taken over the intersection of the discrete subgroup Γ with the compact subset xSupp(f)y−1 of Γ. The function κ satisfies the following: Lemma 0.4. 1. κ(γx, γy) = κ(x, y) for all γ ∈ Γ.

2. It is differentiable on Γ\G × Γ\G. Thus, RΓ(f) is a integral operator with the kernel function κ(x, y). Lemma 0.5. Let M be a compact smooth manifold and h is a smooth function 2 R on M × M. Then an operator T on L (M) given by φ 7→ M φ(g)h(x, y)dy is of trace class ans it’s trace class is given by, Z tr(T ) = h(x, x)dx. M Z tr(T ) = κ(x, x)dµ0(x) Γ\G

Let Γγ = Gγ ∩ Γ, which is a lattice in Gγ and Γγ\Gγ is compact. Let us fix some notations:

1. [γ]G(resp. [γ]Γ) will denote conjugacy class of γ in G (resp. in Γ).

2. [Γ] (resp. [Γ]G) will denote the set of conjugacy classes in Γ (resp. G- conjugacy classes of elements if Γ).

3. Gγ will denote the centraliser of γ in G and Γγ = Gγ ∩ Γ.

Then Γγ is a lattice on Gγ and the quotient Γγ\Gγ is compact. As Gγ is unimod- ular, Gγ\G has a G-invariant Haar measure, say dγx. Normalise the measure on Gγ and Gγ\G so that   Z X X −1 −1 tr(RΓf) =  f(x δ γδx) dµ Γ\G γ∈[Γ] δ∈Γγ \Γ Z P −1 = f(x γx)dγx γ∈[Γ] Since the space Γ \G is ZΓγ \G Z γ P −1 −1 = γ∈[Γ] f(x u γux)du dγx Gγ \G Γγ \GγZ P −1 = γ∈[Γ] Vol(Γγ\Gγ) f(x γx)dγx Gγ \G

5 homeomorphic to the orbit of γ under conjugation, we denote Z −1 Oγ(f) := f(x γx)dγx Gγ \G to be the orbital integral of f along the conjugacy orbit of γ and , X aγ(Γ) := Vol(Γγ0 \Gγ0 ) 0 [γ ]Γ⊂[γ]G and thus, X tr(RΓ(f) = aγ(Γ)Oγ(f) [γ]∈[G]

Since γ\g is compact it is known that RΓ decomposes discretely as direct sum of irreducible unitary representations of G occuring with finite multiplicities. Since the operator RΓ(f) is of trace class, it is given by a absolutely convergent series X tr(RΓ(f) = m(π, Γ)χπ(f) π∈Gˆ where Gˆ denote the set of all irreducible unitary representations of G, m(π, Γ) the multiplicity of π in RΓ, χπ the character of π. Hence by equating tr(RΓ(f) we get the Selberg Trace Formula X X m(π, Γ)χπ(f) = aγ(Γ)Oγ(f) π∈Gˆ [γ]∈[G]

This is the Selberg trace formula for compact quotient Γ\G.

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