In the case when 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 of a
locally compact group with compact. The group acts on the space of functions on by the right regular representation , and this extends to an action of the
group ring of , considered as the ring of functions on . The character of this representation is given by a generalization of the Frobenius formula as follows. The action of a function on a function on is given by :\displaystyle R(f)(\phi)(x) = \int_G f(y)\phi(xy) \,dy = \int_{\Gamma\backslash G}\sum_{\gamma\in \Gamma}f(x^{-1}\gamma y)\phi(y)\,dy. In other words, is an integral operator on (the space of functions on ) with kernel :\displaystyle K_f(x,y) = \sum_{\gamma\in \Gamma}f(x^{-1}\gamma y). Therefore, the trace of is given by :\displaystyle \operatorname{Tr}(R(f)) = \int_{\Gamma\backslash G}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 is the set of conjugacy classes in , and :K_o(x,y)= \sum_{\gamma\in o}f(x^{-1}\gamma y) = \sum_{\delta\in \Gamma_\gamma\backslash \Gamma}f(x^{-1}\delta^{-1}\gamma\delta y) where \gamma is an element of the conjugacy class o, and \Gamma_\gamma is its centralizer in \Gamma. On the other hand, the trace is also given by :\displaystyle \operatorname{Tr}(R(f)) = \sum_{\pi} m(\pi)\operatorname{Tr}(f(\pi)) where m(\pi) is the multiplicity of the irreducible unitary representation \pi of G in L^2(\Gamma \backslash G) and f(\pi) is the operator on the space of \pi given by \int_G f(y)\pi(y) dy.
Examples • If and are both finite, the trace formula is equivalent to the Frobenius formula for the character of an
induced representation. • If is the group of real numbers and the subgroup 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 is not compact, which causes the following (closely related) problems: • The representation on contains not only discrete components, but also continuous components. • The kernel is no longer integrable over the diagonal, and the operators 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 of a real
Lie group (usually ). In higher rank it is more convenient to replace the Lie group with an adelic group . One reason for this that the discrete group can be taken as the group of points for 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==