UDC 519.46

MATHEMATICS

A. I. SHTERN

COMPLETELY IRREDUCIBLE REPRESENTATIONS OF CLASS I OF REAL SEMISIMPLE LIE GROUPS

(Presented by Academician L. S. Pontryagin, January 30, 1969)

1. Let \(G\) be a connected semisimple real Lie group with finite center, and let \(U\) be its maximal compact subgroup. A weakly continuous representation of the group \(G\) by continuous operators in a quasicomplete locally convex linear topological space \(E\) will be called completely irreducible if the weakly closed algebra of operators in \(E\) containing all the operators of the representation \(T(g)\) contains all continuous operators in \(E\). We shall call a completely irreducible representation \(T\) a representation of class I (relative to \(U\)) if in \(E\) there is a vector \(\xi \ne 0\) such that \(T(u)\xi=\xi\) for all \(u \in U\).

Let \(E'\) be the space of continuous linear functionals on \(E\), endowed with the topology of uniform convergence on compact convex subsets of \(E\) (the Mackey topology); let \([T(g)]'\) be the operator in \(E'\) adjoint to \(T(g)\). The representation \(T' : g \to [T(g^{-1})]'\) will be called the representation adjoint to \(T\).

Completely irreducible representations \(T, T_1\) in the spaces \(E, F\), respectively, are called weakly equivalent if there exists an operator \(A: E \to F\) and a formally adjoint operator \(A^*: F' \to E'\), carrying a dense invariant set into a dense invariant set, and such that
\[ AT(g)=T_1(g)A,\qquad A^*T_1'(g)=T'(g)A^* \]
on the domains of definition of \(A, A^*\), respectively.

1. The topological conditions imposed on the representation space make it possible to pass to a representation of the group algebra \(X\) of the group \(G\) (\(X\) is the collection of finite, infinitely differentiable functions, with the usual linear operations and topology and with multiplication-convolution) by the formula
   \[ T(f)=\int f(g)T(g)\,dg. \]
   The definitions of complete irreducibility and weak equivalence carry over verbatim to representations of the group algebra \(X\). The following proposition was essentially proved in \({}^{(1)}\) (see also \({}^{(2)}\)):

Lemma 1. Let \(T\) be a completely irreducible representation of \(G\). Then:
1) in \(E\) there are no nontrivial closed subspaces invariant with respect to \(T(g)\);
2) the adjoint representation is completely irreducible;
3) the representation \(g \to T(g)\) is completely irreducible if and only if the corresponding representation of the group algebra is completely irreducible;
4) the representation \(g \to T(g)\) is weakly equivalent to the representation \(g \to T_1(g)\) if and only if the corresponding representations of the group algebra are weakly equivalent.

1. Let \(T\) be a completely irreducible representation of class I in the space \(E\). The formula
   \[ P=\int_U T(u)\,du \]
   defines in \(E\) a projector onto the subspace generated by the vectors invariant with respect to \(U\). Consider in \(X\) the subalgebra \(Y\),
   \[ Y=\{f\in X:\ f(ugv)=f(g),\ u,v\in U\}. \]
   The subspace \(PE\) is invariant with respect to \(T(f)\), \(f\in Y\). Introduce
   \[ A_T(f)=T(f)\big|_{PE}. \]
   As in \({}^{(1,2)}\), one proves

Lemma 2. 1) If the representation \(g \to T(g)\) of the group \(G\) is completely irreducible, then the representation \(f \to A_T(f)\) of the algebra \(Y\) is completely irreducible.
2) If \(g \to T(g)\), \(g \to T_1(g)\) are completely irreducible representations

groups \(G\) of class I, then the weak equivalence of the representations \(f \to A_T(f)\), \(f \to A_{T_1}(f)\) implies the weak equivalence of the representations \(T, T_1\).

1. The following construction makes it possible to describe, up to weak equivalence, all completely irreducible representations of the group \(G\) of class I with respect to \(U\).

Let \(\mathscr L_2(U)\) be the Hilbert space of functions \(\xi(u)\) on the group \(U\), square-summable with respect to the invariant measure on \(U\). Consider the Iwasawa decomposition of the group \(G\): \(G=UAN\). Let \(M\) be the centralizer of \(A\) in \(G\); \(\Gamma=M\cap U\) the centralizer of \(A\) in \(U\). Put \(\mathscr L_2^{(0)}=\{\xi\in \mathscr L_2(U): \xi(u\gamma)=\xi(u), \gamma\in\Gamma\}\). Let \(\mathfrak G_0\) be the Lie algebra of the group \(G\); \(\mathfrak h_{\mathbb R_0}\) the Abelian subalgebra of \(\mathfrak G_0\) corresponding to the subgroup \(A\); \(\nu\) a complex-valued linear form on \(\mathfrak h_{\mathbb R_0}\). Let \(\mathfrak h_0\) be a maximal Abelian subalgebra in \(\mathfrak G_0\) containing \(\mathfrak h_{\mathbb R_0}\); \(\mathfrak G\) the complexification of \(\mathfrak G_0\), \(\mathfrak h\) the complexification of \(\mathfrak h_0\); \(\rho\) the half-sum of the positive roots of the algebra \(\mathfrak G\) with respect to \(\mathfrak h\); \(W\) the Weyl group (the group of permutations of the positive roots). For \(u\in U\), \(g\in G\) put
\(g^{-1}u=v(g^{-1}u)\cdot a(g^{-1}u)\cdot n(g^{-1}u)\); \(v\in U\), \(a\in A\), \(n\in N\). Consider in \(\mathscr L_2^{(0)}(U)\) the representation \(\widetilde T_\nu\), defined by the formula

\[ [\widetilde T_\nu(g)\xi](u)=e^{(i\nu-\rho)(\log a(g^{-1}u))}\xi\bigl(v(g^{-1}u)\bigr). \tag{1} \]

The operators \(\widetilde T_\nu(g)\) are bounded. The representation \(\widetilde T_{\bar\nu}\) is conjugate to \(\widetilde T_\nu\).

Let \(\widetilde{\mathfrak M}_\nu\) be the closure of the cyclic hull of \(\xi_0\), \(\xi_0(u)\equiv 1\), in \(\mathscr L_2^{(0)}(U)\) with respect to \(\widetilde T_\nu\); analogously, \(\widetilde{\mathfrak M}_{\bar\nu}\) is the closure of the cyclic hull of \(\xi_0\), \(\xi_0(u)\equiv 1\), with respect to \(\widetilde T_{\bar\nu}\).

Let \(\mathfrak M_\nu=\widetilde{\mathfrak M}_\nu\cap \widetilde{\mathfrak M}_{\bar\nu}\), \(P_\nu\) be the projector in \(\mathscr L_2^{(0)}(U)\) onto \(\mathfrak M_\nu\), and \(T_\nu\) the representation in the space \(\mathfrak M_\nu\) acting by the formula

\[ [T_\nu(g)\xi](u)=P_\nu\{[\widetilde T_\nu(g)\xi](u)\}. \tag{2} \]

1. Lemma 3. The representations \(T_\nu\) are completely irreducible.

Proof (cf. (1)). Let \(\{e_i\}\), \(i=1,2,\ldots\), be a complete set of irreducible representations of \(U\); \(Q_{e_i}\) the projector in \(\mathscr L_2(G)\) onto the maximal subspace in which the restriction of the regular representation of \(G\) to the subgroup \(U\) is a multiple of \(e_i\). Let \(Q_n=\sum_{i=1}^n Q_{e_i}\). Denote, for \(f\in\mathscr L_2(G)\): \(f^*(g)=\overline{f(g^{-1})}\); \(Q_n j Q_n=Q_n[(Q_n f^*)^*]\). Let \(Q_nXQ_n=X_n\). Clearly, \(X_n\subset X\). Let \(E_n\) be the maximal subspace of \(\mathfrak M_\nu\) in which the restriction of \(T_\nu\) to \(U\) is a sum of representations multiple of \(e_i\), \(i=1,\ldots,n\). Let \(P_n\) be the projector onto \(E_n\) in \(\mathfrak M_\nu\). Since \([\widetilde T_\nu(v)\xi](u)=\xi(v^{-1}u)\), \(v\in U\), the restriction of \(\widetilde T_\nu\) to \(U\) is a subrepresentation of the regular representation of the group \(U\). Therefore \(E_n\) is finite-dimensional. From the construction of \(T_\nu\) it is evident that \(g\to T_\nu(g)\) is irreducible (i.e. contains no nontrivial subrepresentations). Then \(f\to T_\nu(f)\), \(f\in X\), is irreducible in \(\mathfrak M_\nu\). Then \(f\to T_\nu(f)\), \(f\in X_n\), is irreducible in \(E_n\). But \(\{T_\nu(f), f\in X_n\}\) is an algebra. By Burnside’s theorem, for any continuous operator \(A\) in \(\mathfrak M_\nu\) there is an \(f\in X_n\) such that \(P_nT_\nu(f)P_n=P_nAP_n\) on \(E_n\), while on \(E_n^\perp\) the latter equality is obvious; since \(P_nAP_n\to A\), the lemma is proved.

1. Theorem. 1) Every completely irreducible representation of the group \(G\) of class I with respect to \(U\) is weakly equivalent to one of the representations \(T_\nu\).

2) The representations \(T_\nu\), \(T_\lambda\) are weakly equivalent if and only if \(\nu\) and \(\lambda\) lie on the same orbit with respect to the Weyl group.

Proof. Let \(T\) be a completely irreducible representation of the group \(G\) of class I with respect to \(U\) in the space \(E\). Pass to the representation \(f\to A_T(f)\) of the subalgebra \(Y\subset X\) in the space \(PE\). The subalgebra \(Y\) is commutative (see \((^3)\)), hence any of its completely irreducible representations is one-dimensional (see \((^1)\)), i.e. \(A_T:f\to A_T(f)\) is a homomorphism

$Y$ in $C$. According to the definition of $A_T(f)$, we have
\[ A_T(f)\xi=\left[\int f(g)\eta(T(g)\xi)\,dg\right]\cdot \xi \]
for $\xi\in PE$, $\eta\in PE'$ such that $\eta(\xi)=1$. The function $\Phi_T(g)=\eta(T(g)\xi)$ is continuous and
\[ \eta(T(ugv)\xi)=\eta(T(u)T(g)T(v)\xi)=\eta(T(g)\xi), \]
i.e. $\Phi_T(g)$ is bi-invariant with respect to $U$. By (3) (see also (6), p. 445, Lemma 4.2), $\Phi_T(g)$ is an eigenfunction for every differential operator on $G$ invariant with respect to left translations from $G$ and right translations from $U$. According to (4), $\Phi_T(g)$ is equal to
\[ \Phi_\nu(g)=\int_U e^{(i\nu-\rho)(\log a(g^{-1}u))}\,du \]
for some complex-valued linear form $\nu$ on $\mathfrak{h}_{\mathfrak{p}_0}$. But
\[ \int f(g)\Phi_T(g)\,dg=\int f(g)\Phi_\nu(g)\,dg =\int f(g)(T_\nu(g)\xi_0,\xi_0)\,dg =(T_\nu(f)\xi_0,\xi_0), \]
where $\xi_0(u)\equiv 1$, i.e. the representation $T$ is weakly equivalent to $T_\nu$ by Lemma 2. The second part of the theorem follows from Lemma 2 and the relation for the functions $\Phi_\nu$ ((5), see also (6)): $\Phi_\nu\equiv\Phi_\lambda$ if and only if $\nu$ and $\lambda$ lie on one orbit with respect to the Weyl group.

1. In (7) a number of results are reported on representations $X^\lambda$, $\widetilde X^\lambda$ of the enveloping algebra $\mathfrak U$ of the Lie algebra $\mathfrak G$, corresponding to $\widetilde T_\lambda$ and $\widetilde T_\lambda|_{\mathfrak M_\lambda}$ ($X^\lambda$, $\widetilde X^\lambda$ act in the space of $U$-finite vectors in the spaces of the representations $\widetilde T_\lambda$, $\widetilde T_\lambda|_{\mathfrak M_\lambda}$, respectively). In particular, it is indicated that any algebraically irreducible representation of $\mathfrak U$ in a linear space $E$ with a one-dimensional subspace annihilated by $U$ is algebraically equivalent to one of the representations $\widetilde X^\lambda$; moreover, a Weyl chamber $C$ is singled out such that, for $\lambda\in C$, the representation $\widetilde X^\lambda$ is algebraically irreducible, and necessary and sufficient conditions on $\lambda$ are given under which $X^\lambda=\widetilde X^\lambda$.

2. Let $G$ be the group of $k$-rational points of a simple, simply connected, connected, quasisplit algebraic group over a finite extension $k$ of the field of $p$-adic numbers. Let $U$ be a maximal compact subgroup of the group $G$; let $S$ be a maximal torus in $G$ split over $k$; let $H$ be the centralizer of $S$, and suppose that the Iwasawa and Cartan decompositions of the group $G$ (see (8))
   \[ G=UHU=UHN \]
   ($N$ is a vector $k$-unipotent subgroup of $G$, normalized by $H$) satisfy Condition II in (9). Let $u$ be the group of units of the field $k$;
   \[ H^u=\{h\in H\mid \chi(h)\in u \text{ for all } k\text{-morphisms } \chi \text{ of the group } H \text{ into } k^*\}. \]
   Let $H^u\subset U$.

Let $a$ be a mapping of $H$ into $\mathbb C^*$; let $\mathcal H^\alpha$ be the Hilbert space of functions on $G$:
\[ f(ghn)=\alpha(h)f(g),\qquad g\in G,\quad h\in H,\quad n\in N; \]
\[ \|f\|^2=\int_U |f(v)|^2\,dv, \]
where $dv$ is Haar measure on $U$; the representation $T^\alpha$ is defined by the formula
\[ [T^\alpha(g_0)f](g)=f(g_0^{-1}g). \]
$T^\alpha$ is of class I with respect to $U$ (see item 1) if and only if $\alpha(H^u)=1$. Define $T^\alpha$ as in (2), item 4.

Using the arguments of items 2–6 and Theorem 2 in (9), we obtain that any completely irreducible representation of $G$ of class I with respect to $U$ is weakly equivalent to one of the representations $T^\alpha$ (one can also indicate a group $W$ acting in $\operatorname{Hom}(H,\mathbb C^*)$, to whose orbits the different spherical functions correspond one-to-one (see (9)), and hence also the inequivalent representations).

In conclusion, the author expresses gratitude to Prof. M. A. Naimark for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
27 XII 1969

References

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