ON THE TRACE FORMULA APPROACH TO DISCRETE SERIES MULTIPLICITIES (Automorphic forms, trace formulas and zeta functions)

Let $G$ be a connected reductive group over $\mathbb{Q}$ , and $\Gamma$ an arithmetic subgroup of $G(\mathbb{Q})$ . In this note we survey the trace formula method of Arthur for computing (sums of) multiplicities of discrete series representations in the spectrum of $L^{2}(\Gamma\backslash G(\mathbb{R}))$ . His formula expresses these multiplicities as a combination of p-adic orbital integrals, and asymptotic values of discrete series characters. We write down the most basic example of $GL_{2}$ in some detail. However we omit many details for general $G$ , giving references instead. We also state a conjecture which should refine Arthur $s$ formula to give the multiplicities directly. We conclude with an amusing application of the conjecture, together with known multiplicity formulas, towards computing p-adic orbital integrals and associated distributions. 1. NOTATION For $G$ an algebraic group defined over a field $F$ and $\gamma\in G(F)$ , write $G_{\gamma}$ for the centralizer in $G$ of $\gamma$ . If $T$ is a torus in a reductive group $G$ , write $T_{r}$ for the set of regular elements of $T$ . Write A for the adeles of $\mathbb{Q}$ , write $A_{f}$ for the finite adeles of $\mathbb{Q}$ , and $\hat{Z}$ for the integers in $A_{f}$ . 2. ASYMPTOTICS OF DISCRETE SERIES CHARACTERS 2.1. Case of $GL_{2}$ . Let $G=GL_{2}$ . Write $A$ for the subgroup of diagonal matrices in $G$ . We identify the character group $X^{*}(A)$ with $Z^{2}$ so that $(m, n)\in Z^{2}$ corresponds to the character $(\begin{array}{ll}a 00 b\end{array})\mapsto a^{m}b^{n}$ . We view the root $\alpha=(1, -1)$ as positive. Given $(m, n)\in X^{*}(A)$ , with $m\geq n$ , there is a unique irreducible finite-dimensional algebraic representation $E$ of $G(\mathbb{C})$ whose highest weight for $A$ is $(m, n)$ . In fact, (2.1) $E=$ Sym$m-nV\otimes\det^{n}$ , where $V=\mathbb{C}^{2}$ is the standard representation of $G(\mathbb{C})$ . Note that $E$ has dimension $m-n+1$ . There is a unique discrete series representation $\pi=\pi_{E}$ with the same central character and infinitesimal character as $E$ . Let us specify $\pi$ by its Harish-Chandra character $\Theta_{\pi}$ , viewed as a function defined on regular semisimple elements $\gamma\in G(\mathbb{R})$ . Then $\Theta_{\pi}$ is given by the following formulas: If $\gamma$ has complex eigenvalues $z\neq\overline{z}$ , then (2.2) . 数理解析研究所講究録 第 1767巻 2011年 1-13 1 STEVEN SPALLONE If $\gamma$ has real eigenvalues $\lambda_{1}\neq\lambda_{2}$ , with $|\lambda_{1}|\geq|\lambda_{2}|$ , then (2.3) $\Theta_{\pi}(\gamma)=\{\begin{array}{l}\frac{2\lambda_{1}^{n}\lambda_{2}^{m+1}}{|\lambda_{1}-\lambda_{2}|}, if \lambda_{1}\lambda_{2}>00, if \lambda_{1}\lambda_{2}<0\end{array}$ Let $\gamma$ be regular semisimple with nonreal eigenvalues. Write $T$ for the centralizer in $G$ of $\gamma$ , and $T_{r}$ for its reguIar eIements. The restriction of $\Theta_{\pi}$ to $T_{r}(\mathbb{R})$ extends continuously to alI of $T(\mathbb{R})$ . Moreover, it extends to the union of all conjugates of $T(\mathbb{R})$ . Write $\Phi_{G}(\gamma, \Theta_{\pi})$ for this extension. For $a\in \mathbb{R}^{\cross}$ , we have (2.4) $\Phi_{G}(aI, \Theta_{\pi})=-a^{m+n}(m-n+1)$ . We record $some$ more special values of $\Phi_{G}(\gamma, \Theta_{\pi})$ for later use. Put $\gamma_{4}=(\begin{array}{ll}0 -11 0\end{array})$