Arthur–Selberg trace formula
Open-knowledge reference entry
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 G(A) on the discrete part L20(G(F)\G(A)) of L2(G(F)\G(A)) in terms of geometric data, where G is a reductive algebraic group defined over a global field F and A 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 (Flicker & Kazhdan 1988) 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 G(F)\G(A) 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 Γ of a locally compact group G with Γ\G compact.
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