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 1Suppose , then the integral (1) is absolutely convergent and is bounded by . Thus is a bounded linear operator with norm .

*Proof:*

**Remark:** If , then (1) is also absolutely convergent. Let . Then

The function is called a **kernel function**. The above shows that

We have the following proposition.

**3. Spectral expansion of the kernel functions **

Let . Then is an orthonormal basis of . Formally we define

Proposition 3Suppose has a continuous second derivative satisfyingand . 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 4Suppose satisfies the conditions of Proposition 3. Suppose . Thenis absolutely convergent and is a bounded linear operator. Also

*Proof:* By the previous proposition, is bounded, say by .

This proves the absolutely convergence and the boundedness.

Let . Then by the previous proposition,

Thus (5) is valid for . Because is an orthonormal basis of and both sides of (5) are bounded linear operator, (5) is valid for any .

Suppose satisfies the conditions of Proposition 3. Comparing (3) and (5), for any ,

Hence

Because both and are continuous function,

This is the Poisson summation formula.