Chapter I . Linear groups of cohomogeneity 4

Closed subgroups of O(n) are called linear groups of degree n, and they will be

presented as pairs (G,I), where I: G -» O(n) is an almost faithful representation of a

compact Lie group. We shall identify representations whose corresponding subgroups in

O(n) are conjugate. The purpose of this chapter is to determine all connected linear

groups of cohomogeneity 4, that is

c(O) = dim [Rn/G 4

In the case c(O) = 1 all groups (G,a) are well known; they are listed in Table I for

convenience. In general, the nonspiitting linear groups are the "building blocks", see

(A) below. For c(O) = 2 and 3 these are tabulated in Table II and III, respectively.

Since c(O) is an additive function with respect to outer direct sum ©, we obtain all

pairs (G.O) satisfying c(O) 3 by suitably choosing at most three summands from the

tables. These are our linear models.

(A ) Generalities Two linear groups (Gj,Oj), i =1,2, are called c-equivalent if

they are both subgroups of the same linear group (G,0), i.e., Oj = o|Gj , and moreover

c(O) = c(Oj), i = 1, 2. For connected groups it is easy to see that c-equivalence means

that both groups (up to conjugacy in O(n)) have precisely the same orbits in

[Rn.

Furthermore, in each c-equivalence class there is a unique maximal group. Observe

that "maximal" has the same meaning as in [HL].

A linear group is called splitting if it can be represented as an outer direct sum

(G,0) = (G-j x G

2

,Oi © 0

2

), dim Oj 1.

Here Gj may be trivial. If such a decomposition is impossible, (G,0) is called

nonspiitting . On the other hand, the inner direct sum of the two G-representations Oj is

the restriction

(G,®1 + 0

2

) = ( 0 x 0 , 0 ! © 0

2

) I AG

where G is regarded as the diagonal subgroup AG of G x G. We have in general

( 1 ) c(0- | © I2 ) = c(O-j) + c ( 0 2 ) ,

c(0-j + J2 ) c(O-j) + c(X2).

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