Exercise 8, § 2

Suppose that G is a unimodular locally compact group. Let Γ be a discrete subgroup of G such that the quotient space $X = \Gamma \backslash G$ is compact. Let $\mu$ be a Haar measure on G and $\beta$ the counting measure on Γ; write $\widetilde{\mu}=\mu/\beta$.

Let $\varrho$ be the unitary representation of G on $L^2(X,\widetilde{\mu})$ defined in example 3 of V, p. 404.

For $\gamma \in \Gamma$, write $\Gamma_{\gamma}$ (resp. $G_{\gamma}$) for the centralizer of $\gamma$ in Γ (resp. G). a) The group $G_{\gamma}$ is unimodular.

b) Let $\varphi \in \mathscr{K}(G)$. For every Γ-automorphic unitary representation $\pi$, the endomorphism $\pi (\varphi )$ is of finite trace.

c) One has

Tr($\varrho (\varphi )$) $=\sum_{\pi\in\widehat{G}}m_{\pi}(\varrho )$ Tr($\pi (\varphi )$)

where $m_{\pi}(\varrho )$ is the multiplicity of $\pi$ in $\varrho$ ("spectral side of the trace formula").

d) Let $\Gamma^{\sharp}$ be the set of conjugacy classes of Γ. For $\gamma \in \Gamma^{\sharp}$, write $\beta_{\gamma}$ for the counting measure on $\Gamma_{\gamma}$. One has

Tr($\varrho (\varphi )$) $=\sum\int\varphi (x^{-1}\gamma x)d(\mu/\beta_{\gamma})(x)$.

$\gamma \in \Gamma$

e) Fix a Haar measure $\mu_{\gamma}$ on $G_{\gamma}$ for every $\gamma \in \Gamma$. One has

Tr($\varrho (\varphi )$) $=\sum\int\int\varphi (x^{-1}y^{-1}\gamma yx)d(\mu_{\gamma}/\beta_{\gamma})(y)d(\mu/\mu_{\gamma})(x)$.

$\in^{\sharp}G_{\gamma}\backslash G\Gamma_{\gamma}\backslash G_{\gamma}$ $\gamma$ Γ

f) For every $\gamma \in \Gamma$, the quotient measure $\mu_{\gamma}/\beta_{\gamma}$ of $\Gamma_{\gamma}\backslash G_{\gamma}$ is finite. g) One has

Tr($\varrho (\varphi )$) $=\sum(\mu_{\gamma}/\beta_{\gamma})(\Gamma_{\gamma}\backslash G_{\gamma})\int\varphi (x^{-1}\gamma x)d(\mu/\mu_{\gamma})(x)$

$\in^{\sharp}G_{\gamma}\backslash G$ $\gamma$ Γ

("geometric side of the trace formula"; the formula expressing the equality of this expression with that obtained in question c) is called the "trace formula" for Γ-automorphic representations).

h) Put $G =\mathbf{R}$ and $\Gamma =\mathbf{Z}$. Verify that the trace formula is equivalent in this case to Poisson's formula (cf. II, p. 239).