O. V. MANTUROV

ON HOMOGENEOUS RIEMANNIAN NONSYMMETRIC SPACES WITH IRREDUCIBLE GROUP OF ROTATIONS

(Presented by Academician I. G. Petrovskii, 12 VI 1961)

1°. Let (M=\mathfrak{G}/\mathfrak{H}) (the group (\mathfrak{H}) is compact) be a nonsymmetric homogeneous Riemannian space with irreducible group of rotations. Our problem is to find all such spaces when (\mathfrak{G}) is simple and of type (A_m) (if (\mathfrak{G}) is nonsimple or noncompact, the spaces sought do not exist). (\hat G) and (H) denote the Lie algebras of the groups (\mathfrak{G}) and (\mathfrak{H}). Define the action of (H) on (\hat G) by the formula

[
h(g)=[h,g],\qquad h\in H,\qquad g\in \hat G,
\tag{1}
]

where ([h,g]) is the commutator of the vectors in (\hat G). Formula (1) gives a representation of the algebra (H). The compact group (\mathfrak{H}) preserves some positive definite quadratic form (\gamma) in the space (\hat G). Therefore

[
\hat G=H+B,\qquad [H,B]\subset B,\qquad \gamma(h,b)=0,\qquad h\in H,\qquad b\in B.
\tag{2}
]

The space (B) is identified with the tangent space to (M). The infinitesimal transformations of (\mathfrak{H}) transform (B) in the following way:

[
h(b)=[h,b],\qquad h\in H,\qquad b\in B.
\tag{3}
]

By the property of the space (M), this representation of (H) on (B) must be irreducible. It is said that the pair (H\subset \hat G) has a symmetric structure if, in addition to formulas (2), the relation

[
[B,B]\subset H
\tag{4}
]

is satisfied.

Pairs of symmetric structure generate symmetric spaces.

2°. In what follows, by (\mathfrak{G}) we mean (SU) (the unitary unimodular group), and by (\mathfrak{G}^{+}) the complex unimodular group (SL^{+}). Then (\mathfrak{H}) is embedded in (\mathfrak{G}), and (\mathfrak{H}^{+}), the complex form of (\mathfrak{H}), is embedded in (\mathfrak{G}^{+}) by means of a certain linear representation (\varphi).

1. If the compact linear group (\mathfrak{H}) on the real space (B) is irreducible, then (\mathfrak{H}^{+}) on the complex (B^{+}) either is irreducible, or decomposes into two inequivalent mutually contragredient representations.

We shall call the problem of finding all pairs (H^{+}\subset G^{+}) ((H^{+}) is the complex envelope of a compact algebra, (G^{+}) is the algebra of complex matrices with trace zero) such that the conclusion of Theorem 1 is satisfied problem A.

1. If the compact linear group (\mathfrak{H}) on the (n)-dimensional complex space (B^{+}) is irreducible, or has only two invariant subspaces, the representations of (\mathfrak{H}) in which are inequivalent and mutually contragredient, then in (B^{+}) there exists a subset (B) which is a real invariant and irreducible space with respect to (\mathfrak{H}), of dimension (n).

The representation of the algebra (H^{+}) in the space (L) of all complex matrices by the formula

[
\psi(h)l=\varphi(h)l-l\varphi(h),\qquad h\in H^{+},\ l\in L,
]

is the Kronecker product of the representation (\varphi) by the representation (\tilde{\varphi}), contragredient to (\varphi). (L) is the direct sum of (G^{+}) and the one-dimensional space of scalar matrices. Thus, if (\varphi(H)\subset G) generates a pro-

space (M) with irreducible rotation group, then in the decomposition of the Kronecker product (\varphi(H^+)\times \widetilde{\varphi}(H^+)) into irreducibles, in addition to the adjoint representation and the zero representation acting in the space of scalar matrices, there will occur either a single irreducible representation, or only two irreducible non-equivalent mutually contragredient representations. Conversely, this is sufficient for the irreducibility of the rotation group in the space (\mathfrak{G}/\mathfrak{S}).

(3^\circ). An irreducible representation (\varphi) of a semisimple algebra is completely characterized by its highest weight (\Lambda). Any vector (P) of a Cartan subalgebra can be specified by a diagram, writing in the diagram of the algebra above each simple root (\alpha) the number
[
P_\alpha=\frac{2(P,\alpha)}{(\alpha,\alpha)}
]
(the numerical mark). In order that a vector (P) be the highest weight (\Lambda) of an irreducible representation, it is necessary and sufficient that all its numerical marks be integral and nonnegative.

Let us list the highest weights of the adjoint representations of simple groups:

The numbers under the diagrams will be called the numbers of the simple roots. The diagrams (\Lambda) and (\widetilde{\Lambda}) of mutually contragredient representations for the algebras (A_m), (D_m) ((m\geq 4)), (E_6) are related by the formulas:
[
\Lambda_{\alpha_i}=\widetilde{\Lambda}{\alpha A_m;}},\quad i=1,2,\ldots,m,\quad \text{for
]
[
\Lambda_{\alpha_i}=\widetilde{\Lambda}{\alpha,\quad i=1,2,\ldots,5;\quad}
\Lambda_{\alpha_6}=\widetilde{\Lambda}{\alpha_6}\quad \text{for } E_6;
]
[
\Lambda}=\widetilde{\Lambda{\alpha_i},\quad i=1,2,\ldots,m-2;\quad
\Lambda}}=\widetilde{\Lambda{\alpha_m},\quad
\Lambda}=\widetilde{\Lambda{\alpha D_m.}}\quad \text{for
]
All irreducible representations of the algebras (B_m, C_m, E_7, E_8, F_4, G_2) are self-contragredient.

Following E. B. Dynkin ((^1)), we call a chain of simple roots of a semisimple algebra a collection (\alpha_{i_1},\alpha_{i_2},\ldots,\alpha_{i_k}) of simple roots such that
[
(\alpha_{i_s},\alpha_{i_{s+1}})\ne 0,\quad s1.
\tag{5}
]

The chains of simple roots of the algebras (A_m, B_m, C_m, F_4, G_2) have the form
[
\alpha_r,\alpha_{r+1},\ldots,\alpha_{r+s},\quad r\geq 1,\ s\geq 0,\ r+s\leq m.
]
The chains of simple roots of the algebras (D_m) ((m\geq 4)), (E_6, E_7, E_8) are as follows:
[
D_m:\ \mathrm{I}.\ \alpha_r,\alpha_{r+1},\ldots,\alpha_{r+s},\ r\geq 1,\ s\geq 0,\ r+s\leq m-1.
]
[
\mathrm{II}.\ \alpha_r,\alpha_{r+1},\ldots,\alpha_{m-2},\alpha_m,\ 1\leq r\leq m-2.\qquad
\mathrm{III}.\ \alpha_{m-1},\alpha_{m-2},\alpha_m.
]
[
E_m\ (m=6,7,8):\ \mathrm{I}.\ \alpha_r,\alpha_{r+1},\ldots,\alpha_{r+s},\ r\geq 1,\ s\geq 0,\ r+s\leq m-1.
]
[
\mathrm{II}.\ \alpha_r,\alpha_{r+1},\ldots,\alpha_3,\alpha_m,\ 1\leq r\leq 3.\qquad
\mathrm{III}.\ \alpha_m,\alpha_3,\alpha_4,\ldots,\alpha_l,\ 3\leq l\leq m-1.
]

Theorem (E. B. Dynkin ((^1))). Let (\varphi) and (\psi) be two irreducible representations of a complex semisimple algebra (L) of highest weights (\Lambda) and (M), and let (\alpha_{i_1},\alpha_{i_2},\ldots,\alpha_{i_k}) be a chain of simple roots such that
[
(\Lambda,\alpha_{i_1})\ne 0;\qquad
(\Lambda,\alpha_{i_s})=0,\quad s>1;\qquad
(M,\alpha_{i_k})\ne 0,\quad (M,\alpha_{i_s})=0,\quad s<k.
\tag{6}
]
Then in the decomposition of the Kronecker product (\varphi\times\psi) there will occur a representation of highest weight (\Lambda+M-S), (S=\alpha_{i_1}+\alpha_{i_2}+\cdots+\alpha_{i_k}).

For two nonzero highest weights (\Lambda) and (M) of representations of simple algebras there always exists a chain with properties (6).

(4^\circ). Solution of problem A, if the embedding (\varphi(H^+)\subset G^+) is irreducible. In this case (H^+) is necessarily semisimple.

1. First consider the case of a simple algebra (H^+). The decomposition of the Kronecker product (\varphi\times \widetilde{\varphi}) ((\widetilde{\varphi}) contragredient to (\varphi)) contains the components (\Lambda+\widetilde{\Lambda}) ((\widetilde{\Lambda}) is the highest weight of (\widetilde{\varphi})), the adjoint representation (highest weight (T)), (\Lambda+\widetilde{\Lambda}-S), where (S) is a chain of simple roots for (\Lambda) and (\widetilde{\Lambda}) with properties (6).

If (\varphi(H^+)\subset G^+) is a solution of problem A, then one of the relations holds:
[
\Lambda+\widetilde{\Lambda}-S=T,
\tag{7}
]
[
\Lambda+\widetilde{\Lambda}=T.
\tag{8}
]

If (7) holds, then (T+S=\Lambda+\widetilde{\Lambda}) is self-contragredient and has nonnegative numerical labels. Among all the schemes of chains written out by us for (A_m, B_m, C_m, F_4, G_2), only the following satisfy this condition:

[
\begin{aligned}
A_m:\;& 1)\ r=1,\ r+s=m;\quad 2)\ r=2,\ r+s=m-1,\ m\geqslant 3.\
B_m:\;& 1)\ r=1,\ r+s=m;\quad 2)\ r=1,\ s=0,\ m\geqslant 3.\
C_2:\;& 1)\ r=2,\ s=0.
\end{aligned}
]

This gives for (T+S=\Lambda+\widetilde{\Lambda}) the following possible schemes:

[
\begin{array}{c}
\text{(diagram for }A_m\text{)}
\end{array}
\tag{9}
]

[
\begin{array}{c}
\text{(diagram for }A_m\text{)}
\end{array}
\tag{10}
]

[
\begin{array}{c}
\text{(diagram for }A_m\text{)}
\end{array}
\tag{11}
]

[
\begin{array}{c}
\text{(diagram for }B_m\text{)}
\end{array}
\tag{12}
]

[
\begin{array}{c}
\text{(diagram for }B_m\text{)}
\end{array}
\tag{13}
]

[
\begin{array}{c}
\text{(diagram for }C_2\text{)}
\end{array}
\tag{14}
]

Hence (\Lambda), up to contragredience, can only be the following:

[
\begin{array}{c}
\text{(diagram for }A_m,\ m\geqslant 1\text{)}
\end{array}
\tag{15}
]

[
\begin{array}{c}
\text{(diagram for }A_m,\ m\geqslant 3\text{)}
\end{array}
\tag{16}
]

[
\begin{array}{c}
\text{(diagram for }B_m,\ m\geqslant 2\text{)}
\end{array}
\tag{17}
]

[
\begin{array}{c}
\text{(diagram for }A_m,\ m\geqslant 2\text{)}
\end{array}
\tag{18}
]

The chains of simple roots of the algebras (D_m) ((m\geqslant 4)), (E_6, E_7, E_8, F) cannot satisfy relation (7), with the exceptions:
[
D_m\ (m>4):\ \alpha_1;\quad
D_4:\ 1)\ \alpha_1,\ 2)\ \alpha_3,\ 3)\ \alpha_4;\quad
D_5:\ 1)\ \alpha_4,\ \alpha_3,\ \alpha_5.
]
[
E_6:\ \alpha_1,\ \alpha_2,\ \alpha_3,\ \alpha_4,\ \alpha_5.
]

This leads to the following (\Lambda) (up to outer automorphisms):

[
\begin{array}{c}
\text{(three diagrams)}
\end{array}
\tag{19}
]

Equality (8) gives (up to contragredience):

[
\begin{array}{c}
\Lambda\ \text{(diagram)}
\end{array}
\tag{20}
]

or

[
\begin{array}{c}
\Lambda\ \text{(diagram)}
\end{array}
\tag{21}
]

By counting dimensions we find that the solutions of problem A are the representations (\varphi) specified by the diagrams (15), (16), (17), (19), (21).

1. Let (H^{+}=\sum_{i=1}^{k} H_i) be not simple, but semisimple. An irreducible representation (\varphi(H^{+})) is specified by a diagram ((\Lambda_1,\Lambda_2,\ldots,\Lambda_k)), where (\Lambda_i) ((i=1,2,\ldots,k)) is the diagram of an irreducible representation of (H_i).

In the decomposition (\varphi\times\widetilde{\varphi}) there enter the irreducible components of the adjoint representation of (H^{+}) (their diagrams are: ((0,0,\ldots,0,T_i,0,\ldots,0)), (0) is the diagram of the zero representation, (T_i) is the diagram of the adjoint representation of (H_i)) and the components

[
(\Lambda_1+\widetilde{\Lambda}1,\Lambda_2+\widetilde{\Lambda}_2,\ldots,
\Lambda}+\widetilde{\Lambda{i-1},\ \Lambda_i+\widetilde{\Lambda}_i-S_i,\Lambda,\ldots,}+\widetilde{\Lambda}_{i+1
]

[
\ldots,\Lambda_k+\widetilde{\Lambda}_k),\qquad i=1,2,\ldots,k.
]

Here (S_i) is a chain for (\Lambda_i) and (\widetilde{\Lambda}_i) with properties (6). Therefore, if (\varphi) is a solution of problem A, (k=2) and

[
\Lambda_i+\widetilde{\Lambda}_i-S_i=0,
]

i.e. (\varphi), up to contragredience, is specified by the diagrams

[
\left(
\begin{array}{c}
{}^{1}\circ-\circ-\cdots-\circ-\circ\
{}^{1}\circ-\circ-\cdots-\circ-\circ
\end{array}
\right)
\quad \text{or} \quad
\left(
\begin{array}{c}
{}^{1}\circ-\circ-\cdots-\circ-\circ\
\circ-\circ-\cdots-\circ-{}^{1}\circ
\end{array}
\right),
\qquad m_1>1,\ m_2>1 .
\tag{22}
]

(m_i) denotes the rank of (H_i), (i=1,2). The diagrams (22), as dimension counting shows, give a solution of problem A. If (\varphi) is reducible, the solutions of problem A possess a symmetric structure.

After eliminating the solutions of problem A of symmetric structure, we obtain the final result. All nonsymmetric homogeneous Riemannian spaces with irreducible rotation group and with motion group isomorphic to (SU(N)), up to local isomorphism, are described by the following table of highest weights (for the embedding (\varphi(\mathfrak{G})\subset SU(N))):

The author expresses his gratitude to P. K. Rashevskii, who posed the problem and supervised the work.

Moscow State University
named after M. V. Lomonosov

Received
1 VI 1961

References

1. E. B. Dynkin, Tr. Moskovsk. matem. obshch., 1, 39 (1952).