On Selberg's trace formula
Ramesh Gangolli, Garth Warner · Journal of the Mathematical Society of Japan · 1975
\S 1. Introduction.Let $G$ be a connected, non-compact, semisimple Lie group with finite center, and let $\Gamma$ be a discrete subgroup of $G$ such that the space $ G/\Gamma$ is compact.Fix a G-invariant measure $d\dot{x}$ on $ G/\Gamma$ , and denote by $L_{2}(G/\Gamma)$ the Hilbert space of measurable functions on $ G/\Gamma$ that are square-integrable with respect to this measure $d\dot{x}$ .We shall view an element of $L_{2}(G/\Gamma)$ as a function on $G$ , invariant under right translations by elements of $\Gamma$ .$G$ acts on $L_{2}(G/\Gamma)$ by the left regular representation $U$ .Thus for $f\in L_{2}(G/\Gamma),$ $x\in G,$ $(U(x)f)(y)$ $=f(x^{-1}y),$ $y\in G$ .$U$ is a unitary representation of $G$ , whose study is important in the theory of automorphic functions.Under the hypothesis that $ G/\Gamma$ is compact, it is well-known (see $e$ .$g$ .Gel- fand et al. [3]) that the representation $U$ decomposes into a discrete direct sum of irreducible unitary representations of $G$ , and, moreover, that the multi- plicity with which any given irreducible unitary representation of $G$ occurs in this decomposition is finite.Except in special cases, not much is known about which representations occur in $U$ , and what their multiplicities are.Now let $K$ be a maximal compact subgroup of $G$ .Let $U_{0},$ $U_{1},$ $\cdots$ be the inequivalent irreducible unitary representations of class one with respect to $K$ that occur in $U$ , and let $n_{0},$ $n_{1},$ $\cdots$ be their multiplicities.We can assume that $U_{0}$ is the trivial representation of $G$ , and so $n_{0}=1$ .Our object in the present paper is to get some information about the multiplicities $n_{i}(i=0, 1, )$ .Let $G=KAN$ be an Iwasawa decomposition of $G$ , and let $\mathfrak{a}=Lie$ algebra of $A$ .If $\mathfrak{F}$ is the space of complex valued linear functions on $\mathfrak{a}$ , then for every $\lambda\in \mathfrak{F}$ one has the elementary (zonal) spherical function $\varphi_{\lambda}$ on $G$ , defined by $\varphi_{\lambda}(x)=\int_{K}\exp(\lambda-\rho)(H(xk))dk(x\in G)$ , where $\rho$ is the half-sum of the positive roots of the pair $(\mathfrak{g}, \mathfrak{a})$ and $H(x)$ is the unique element of $\mathfrak{a}$ such that $ x\in$ $K$ exp $H(x)N$.If $W$ is the Weyl group of the pair $(\mathfrak{g}, \mathfrak{a})$ , then it is known that $\varphi_{\lambda^{\prime}}=\varphi_{\lambda^{ u}}$ if and only if $\lambda^{\prime}$ and $\lambda^{\prime}$ are conjugate under $W$ .Returning to the representation $U$ , let $\varphi_{0},$ $\varphi_{1},$ $\cdots$ be the positive definite elementary spherical functions that correspond to $U_{0},$ $U_{1},$ $\cdots$ etc.Then, by what we said above, we can find elements $\lambda_{j}\in \mathfrak{F}$ so that $\varphi_{j}=\varphi_{\lambda_{j}},$ $j=0,1,$ $\cdots$ , each