On the characters and equivalence of continuous series representations
Ronald L. Lipsman · Journal of the Mathematical Society of Japan · 1971
In this paper we shall compute some explicit formulas for the characters and infinitesimal characters of general continuous series representations.We then apply these results to deduce some facts concerning equivalence and disjointness among representations from various series.In more detail, let $G$ be a connected semisimple Lie group and $P$ a cuspidal parabolic subgroup of $G$ .Then $P$ has a Langland's decomposition $P=MAN$,Thenon-degenerate continuous series representations of $G$ (corresponding tc $P$ ) are obtained by inducing these " cuspidal " representations $\lambda$ from $P$ to $G$ .Let $\pi=Ind_{P}^{G}\lambda$ .It is known that for $f\in C_{0}^{\infty}(G),$ $\pi(f)=\int_{G}f(g)\pi(g)dg$ is a trace class operator.Moreover, there exists a locally integrable function $\theta_{\pi}$and $g_{1}^{-1}gg_{1}\in H$ for some $g_{1}\in G$ }.The main steps in the computation are as follows: (i) extend Harish-Chandra's results on the discrete series of connected semisimple Lie groups to connected reductive Lie groups (\S 3); (ii) employ Mackey's theory in order to compute the discrete series of the disconnected group $M$ (\S 4); (iii) develop an analog of the Weyl-Harish-Chandra integration formula for the group $M$ (\S 7); (iv) define an appropriate class function on $G$ (\S 8); and (v) combine various integral formulas to get the desired character formula (see 9.1).Although we do not evaluate $\theta_{\pi}$ on all of $G$ , we shall $*)$ Supported in part by the NSF through GP-13871.R. L. Ll P SMAN and set $K=the$ maximal compact subgroup of $G$ having $f$ as Lie algebra.Next let $P$ be a parabolic subgroup of $G$ .This means: $P$ is a closed subgroup of $G$ such that (i) if $\mathfrak{P}=LA(P)$ , then $P=N(\mathfrak{P})$ and (ii) $\mathfrak{P}_{c}$ contains a maximal solvable subalgebra of $\mathfrak{g}_{c}$ .Let $N=the$ maximal normal subgroupand the map $(m, a, n)\rightarrow man$ is an analytic diffeomorphism of $M\times A\times N$ onto $P$ .Suppose $P$ is cuspidal.By this we shall mean: there exists $\mathfrak{h}$ , a $\theta$ -stable Cartan subalgebra of $\mathfrak{g}$ , such that $\mathfrak{h}\cap \mathfrak{p}=\mathfrak{a}=LA(A)$ .Let $H=Z(\mathfrak{h})\cap G$ , a Cartan subgroup of $G$ .We call any such $H$ compatible with $P$ .Set $B=H\cap K$ .Then $H=BA$ is a direct product [le, p. 481].(Note : $H$ and $B$ are not necessarily connected or abelian [ld, p. 556].)DEFINITION.A Cartan subgroup of $M$ is the centralizer of a Cartan sub- algebra of $\mathfrak{m}$ .LEMMA 2.1.$B$ is a compact Cartan subgroup of $M$ .PROOF.$B$ is clearly a compact group.Set $b=\mathfrak{h}\cap f$ , so that $\mathfrak{h}=b\oplus \mathfrak{a}$ .Let $b\in B$ .Then $ b\in\Xi$ ; but the map $b\rightarrow|\chi(b)|,$ $\chi\in X(\Xi)$ , is a continuous homomorphism of $B$ into $R_{+}^{*}$ .Hence $B\subseteqq M$ , and so $b\subseteqq \mathfrak{m}$ .Moreover, it is clear that $b$ is a Cartan subalgebra of $\mathfrak{m}$ .Next let $\beta\in Z(\mathfrak{b})\cap M$ .Then $\beta\in H$, $\beta=ba,$ $b\in B,$ $a\in A$ .If $a eq 1$ , choose $\chi\in X(\Xi)$ so that $|\chi(a)| eq 1$ .Then since $\beta\in M,$ $b\in B\subseteqq M$ , we have $1=|\chi(\beta)|=|\chi(b)||\chi(a)| eq 1$ .Therefore $\beta=b\in B$ ;