Arthur–Selberg trace formula

In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL₂ to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of Template:Math on the discrete part Template:Math of Template:Math in terms of geometric data, where Template:Math is a reductive algebraic group defined over a global field Template:Math and Template:Math is the ring of adeles of F.

There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications. The simple trace formula Template:Harv is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.

Notation

• F is a global field, such as the field of rational numbers.
• A is the ring of adeles of F.
• G is a reductive algebraic group defined over F.

The compact case

In the case when Template:Math is compact the representation splits as a direct sum of irreducible representations, and the trace formula is similar to the Frobenius formula for the character of the representation induced from the trivial representation of a subgroup of finite index.

In the compact case, which is essentially due to Selberg, the groups G(F) and G(A) can be replaced by any discrete subgroup Template:Math of a locally compact group Template:Math with Template:Math compact. The group Template:Math acts on the space of functions on Template:Math by the right regular representation Template:Math, and this extends to an action of the group ring of Template:Math, considered as the ring of functions Template:Math on Template:Math. The character of this representation is given by a generalization of the Frobenius formula as follows. The action of a function Template:Math on a function Template:Math on Template:Math is given by

$R(f)(\varphi )(x)=\underset{G}{\int }f(y)\varphi (xy)dy=\underset{\mathrm{\Gamma }\mathrm{\setminus }G}{\int }\sum _{\gamma \in \mathrm{\Gamma }}f(x^{-1}\gamma y)\varphi (y)dy.$

In other words, Template:Math is an integral operator on Template:Math (the space of functions on Template:Math) with kernel

$K_f(x,y)=\sum _{\gamma \in \mathrm{\Gamma }}f(x^{-1}\gamma y).$

Therefore, the trace of Template:Math is given by

$\mathrm{Tr}(R(f))=\underset{\mathrm{\Gamma }\mathrm{\setminus }G}{\int }{K}_f(x,x)dx.$

The kernel K can be written as

$K_f(x,y)=\sum _{o\in O}{K}_o(x,y)$

where Template:Math is the set of conjugacy classes in Template:Math, and

$K_o(x,y)=\sum _{\gamma \in o}f(x^{-1}\gamma y)=\sum _{\delta \in {\mathrm{\Gamma }}_{\gamma }\mathrm{\setminus }\mathrm{\Gamma }}f(x^{-1}{\delta }^{-1}\gamma \delta y)$

where $\gamma$ is an element of the conjugacy class $o$, and ${\mathrm{\Gamma }}_{\gamma }$ is its centralizer in $\mathrm{\Gamma }$.

On the other hand, the trace is also given by

$\mathrm{Tr}(R(f))=\sum _{\pi }m(\pi )\mathrm{Tr}(f(\pi ))$

where $m(\pi )$ is the multiplicity of the irreducible unitary representation $\pi$ of $G$ in $L^2(\mathrm{\Gamma }\mathrm{\setminus }G)$ and $f(\pi )$ is the operator on the space of $\pi$ given by $\underset{G}{\int }f(y)\pi (y)dy$.

Examples

• If Template:Math and Template:Math are both finite, the trace formula is equivalent to the Frobenius formula for the character of an induced representation.
• If Template:Math is the group Template:Math of real numbers and Template:Math the subgroup Template:Math of integers, then the trace formula becomes the Poisson summation formula.

Difficulties in the non-compact case

In most cases of the Arthur–Selberg trace formula, the quotient Template:Math is not compact, which causes the following (closely related) problems:

• The representation on Template:Math contains not only discrete components, but also continuous components.
• The kernel is no longer integrable over the diagonal, and the operators Template:Math are no longer of trace class.

Arthur dealt with these problems by truncating the kernel at cusps in such a way that the truncated kernel is integrable over the diagonal. This truncation process causes many problems; for example, the truncated terms are no longer invariant under conjugation. By manipulating the terms further, Arthur was able to produce an invariant trace formula whose terms are invariant.

The original Selberg trace formula studied a discrete subgroup Template:Math of a real Lie group Template:Math (usually Template:Math). In higher rank it is more convenient to replace the Lie group with an adelic group Template:Math. One reason for this that the discrete group can be taken as the group of points Template:Math for Template:Math a (global) field, which is easier to work with than discrete subgroups of Lie groups. It also makes Hecke operators easier to work with.

The trace formula in the non-compact case

One version of the trace formula Template:Harv asserts the equality of two distributions on Template:Math:

$\sum _{o\in O}{J}_o^T=\sum _{\chi \in X}{J}_{\chi }^T.$

The left hand side is the geometric side of the trace formula, and is a sum over equivalence classes in the group of rational points Template:Math of Template:Math, while the right hand side is the spectral side of the trace formula and is a sum over certain representations of subgroups of Template:Math.

The invariant trace formula

The version of the trace formula above is not particularly easy to use in practice, one of the problems being that the terms in it are not invariant under conjugation. Template:Harvtxt found a modification in which the terms are invariant.

The invariant trace formula states

$\sum _M\frac{|W_0^M|}{|W_0^G|}\sum _{\gamma \in (M(Q))}{a}^M(\gamma )I_M(\gamma ,f)=\sum _M\frac{|W_0^M|}{|W_0^G|}\underset{\mathrm{\Pi }(M)}{\int }{a}^M(\pi )I_M(\pi ,f)d\pi$

where

• Template:Math is a test function on Template:Math
• Template:Math ranges over a finite set of rational Levi subgroups of Template:Math
• Template:Math is the set of conjugacy classes of Template:Math
• Template:Math is the set of irreducible unitary representations of Template:Math
• Template:Math is related to the volume of Template:Math
• Template:Math is related to the multiplicity of the irreducible representation Template:Math in Template:Math
• $I_M(\gamma ,f)$ is related to $\underset{M(A,\gamma )\mathrm{\setminus }M(A)}{\int }f(x^{-1}\gamma x)dx$
• $I_M(\pi ,f)$ is related to trace $\underset{M(A)}{\int }f(x)\pi (x)dx$
• Template:Math is the Weyl group of M.

Stable trace formula

Template:Harvtxt suggested the possibility a stable refinement of the trace formula that can be used to compare the trace formula for two different groups. Such a stable trace formula was found and proved by Template:Harvtxt.

Two elements of a group Template:Math are called stably conjugate if they are conjugate over the algebraic closure of the field Template:Math. The point is that when one compares elements in two different groups, related for example by inner twisting, one does not usually get a good correspondence between conjugacy classes, but only between stable conjugacy classes. So to compare the geometric terms in the trace formulas for two different groups, one would like the terms to be not just invariant under conjugacy, but also to be well behaved on stable conjugacy classes; these are called stable distributions.

The stable trace formula writes the terms in the trace formula of a group Template:Math in terms of stable distributions. However these stable distributions are not distributions on the group Template:Math, but are distributions on a family of quasisplit groups called the endoscopic groups of Template:Math. Unstable orbital integrals on the group Template:Math correspond to stable orbital integrals on its endoscopic groups Template:Math.

Simple trace formula

There are several simple forms of the trace formula, which restrict the compactly supported test functions f in some way Template:Harv. The advantage of this is that the trace formula and its proof become much easier, and the disadvantage is that the resulting formula is less powerful.

For example, if the functions f are cuspidal, which means that

$\underset{n\in N(A)}{\int }f(xny)dn=0$

for any unipotent radical Template:Math of a proper parabolic subgroup (defined over Template:Math) and any x, y in Template:Math, then the operator Template:Math has image in the space of cusp forms so is compact.

Applications

Template:Harvtxt used the Selberg trace formula to prove the Jacquet–Langlands correspondence between automorphic forms on Template:Math and its twisted forms. The Arthur–Selberg trace formula can be used to study similar correspondences on higher rank groups. It can also be used to prove several other special cases of Langlands functoriality, such as base change, for some groups.

Template:Harvtxt used the Arthur–Selberg trace formula to prove the Weil conjecture on Tamagawa numbers.

Template:Harvtxt described how the trace formula is used in his proof of the Langlands conjecture for general linear groups over function fields.

See also

• Maass wave form
• Harmonic Maass form
• Arthur's conjectures

References

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External links

• Works of James Arthur Template:Webarchive at the Clay institute
• Archive of Collected Works of James Arthur at the University of Toronto Department of Mathematics