Yin and Yang are important concept in the Chinese philosophy. They usually describe two opposite things which are not only complement of each other but also have deep interrelationship. In mathematics, we called them duality. For example (1) primes vs zeros of Riemann zeta-functions, (2) in the theory of finite group representation, irreducible characters vs conjugacy classes, (3) singular homology vs deRham cohomology, (4) length of closed geodesic vs eigenvalues of Laplacian, (5) particles vs wave. See Arthur, Harmonic Analysis and Group Representation.
Poisson summation formula can be viewed as a formula that links the values related to and values related to the representations of . Let be a discrete subgroup of a measurable, topological group . Trace formulas are formulas relate the representations of (spectral side) and a summation over (geometric side). The formulas have many application to number theory, for example, it can be used to prove some special cases of Langlands functoriality.
In this note, we give another proof of the Poisson summation formula (See Lecture 2 Theorem 5 ) by using the trace formula.
1. Group representations
Let be a group and a linear vector space over . A group representation of on is a group homomorphism , where is the set of invertible endomorphism of . Equivalently, for and , such that and .
2. Right regular representations and kernel functions
Recall that is a topological group isomorphic to the multiplicative group by .
The right regular representation is a representation of on defined by
It can be easily shown that is a representation. Also
Let be a measurable function on . Formally define a linear operator on by
is like a weighted sum of with weight . To justify the well-definedness, we need to check the absolutely convergence of the above integral. For example
Proposition 1 Suppose , then the integral (1) is absolutely convergent and is bounded by . Thus is a bounded linear operator with norm .
Remark: If , then (1) is also absolutely convergent. Let . Then
We have the following proposition.
3. Spectral expansion of the kernel functions
and . Then
and is a continuous function.
Proof: Because and is continuous, . By Lecture 1, lemma 2 (i) for , . Thus for , . The proposition follows easily.
Proposition 4 Suppose satisfies the conditions of Proposition 3. Suppose . Then
Proof: By the previous proposition, is bounded, say by .
This proves the absolutely convergence and the boundedness.
Let . Then by the previous proposition,
Because both and are continuous function,
Letting , we obtain
This is the Poisson summation formula.