MATHEMATICS

A. A. Kirillov

REPRESENTATIONS OF THE ROTATION GROUP OF \(n\)-DIMENSIONAL EUCLIDEAN SPACE BY SPHERICAL VECTOR FIELDS

(Presented by Academician A. N. Kolmogorov on 2 IV 1957)

Let \(S_n\) be the sphere in \((n+1)\)-dimensional Euclidean space, defined by the equation \(\sum_{i=1}^{n+1} x_i^2 = 1\); \(R^{(n)}\) is the totality of continuous vector fields tangent to \(S_n\). The purpose of the present note is the decomposition of \(R^{(n)}\) into subspaces invariant and irreducible with respect to the group of all rotations \(S_n\), and the indication of the schemes of the corresponding representations.

Let \(R\) be an irreducible subspace of \(R^{(n)}\). Choose on the sphere a pole—the point \(P(0,0,\ldots,0,1)\)—and denote by \(R_0\) the totality of fields that vanish at the point \(P\). This is a subspace of \(R\), invariant with respect to rotations leaving the point \(P\) fixed. Let \(\widetilde R\) be the orthogonal complement to \(R_0\) in \(R\). (The scalar product of fields \(u\) and \(v\) from \(R^{(n)}\) is defined as \(\int_{S_n} (u(M), v(M))\,d\tau\), where \((u(M), v(M))\) is the scalar product of the vectors \(u(M)\) and \(v(M)\), and \(d\tau\) is a measure on \(S_n\) invariant with respect to rotations.)

Note that a field from \(\widetilde R\) is uniquely determined by its value at the point \(P\). Indeed, if \(v_1, v_2 \in \widetilde R\) and \(v_1(P)=v_2(P)\), then \(v_1(P)-v_2(P)=0\), \(v_1-v_2 \in R_0\). At the same time \(v_1-v_2 \in \widetilde R\). Consequently, \(v_1-v_2=0\); \(v_1=v_2\). The totality of vectors \(v(P)\), \(v\in \widetilde R\), forms a linear subspace in the space \(R_P\) of vectors tangent to \(S_n\) at the point \(P\). Since it is invariant with respect to all rotations \(R_P\), it either consists of zero alone or coincides with \(R_P\). In the first case, as is easily seen, all of \(R\) consists of zero. In the second case \(\dim \widetilde R=\dim R_P=n\).

Let \(\xi_1=(1,0,\ldots,0,0),\ldots,\xi_n=(0,0,\ldots,1,0)\) be the basis vectors in \(R_P\). The corresponding fields from \(\widetilde R\) we shall denote by \(v_1, v_2,\ldots,v_n\) and shall call zonal fields. Under all rotations about the point \(P\), zonal fields transform as the coordinates \(x_1,x_2,\ldots,x_n\).

Now consider the section of \(S_n\) by the hyperplane \(x_{n+1}=\mathrm{const}\). This is an \((n-1)\)-dimensional sphere \(S_{n-1}\); the rotation group \(S_n\) leaving the point \(P\) fixed coincides on \(S_{n-1}\) with the group of all rotations \(S_{n-1}\). The set of zonal fields, if it is considered only on \(S_{n-1}\), generates a space of vector fields on \(S_{n-1}\), invariant with respect to all rotations. These fields will not, generally speaking, be tangent to \(S_{n-1}\). Decompose each field \(v_k\) into the sum \(v'_k+v''_k\), where \(v'_k\) is a field normal to \(S_{n-1}\) (and tangent to \(S_n\)), and \(v''_k\) is a field tangent to \(S_{n-1}\). At each point of \(S_{n-1}\) there exists only one, up to a factor, vector,

normal to \(S_{n-1}\) and tangent to \(S_n\). As such a vector we choose \(\operatorname{grad}_{S_n} x_{n+1}\). Then
\[ v'_k=f_k(x_1,\ldots,x_{n+1})\operatorname{grad}_{S_n}x_{n+1}. \]
Under rotations of \(S_{n-1}\) the collection of functions \(f_k\) is transformed as a collection of zonal fields, i.e., as a collection of coordinates. But two equivalent collections of spherical functions can differ only by a factor. Therefore \(f_k=\lambda x_k\), where \(\lambda\) is constant on \(S_{n-1}\), i.e., \(\lambda=\lambda(x_{n+1})\). Finally we obtain
\[ v'_k=\lambda(x_{n+1})x_k\operatorname{grad}_{S_n}x_{n+1}. \]

Returning to the collection of tangent components \(v''_k\), consider the space spanned by it. A zonal collection for this space consists of \(n-1\) fields. Consequently, the subspace \(R_0\) in this case is one-dimensional. Let \(u\in R_0\). As the pole on \(S_{n-1}\) we take the point \(P_1(0,0,\ldots,1,0)\). As above, decompose \(u\) into the sum \(u'+u''\), where \(u'\) has the form
\[ f(x_1,x_2,\ldots,x_n)\operatorname{grad}_{S_{n-1}}x_n, \]
and \(u''\) generates on \(S_{n-2}\) a one-dimensional space of vector fields invariant under all rotations. But on a sphere of order higher than 1 there are no such spaces. Therefore, for \(n>3\), \(u''=0\). Further, \(f(x_1,\ldots,x_n)\) generates on \(S_{n-2}\) a one-dimensional space of spherical functions. There is only one such space—the collection of constants. Therefore \(f(x_1,\ldots,x_n)\) is constant on \(S_{n-2}\), and
\[ u'=f(x_n)\operatorname{grad}_{S_{n-1}}x_n =\operatorname{grad}_{S_{n-1}}F(x_n), \]
where \(F(x_n)\) is an antiderivative of \(f(x_n)\). Thus, \(u=u'\) is a gradient field. It follows that the entire space spanned by \(\{v''_k\}\) consists of gradient fields. Let
\[ v''_k=\operatorname{grad}_{S_{n-1}}F_k; \]
the functions \(F_k\) form a collection of spherical functions on \(S_{n-1}\) equivalent to the coordinate collection. Therefore
\[ F_k=\lambda x_k,\qquad v''_k=\lambda\operatorname{grad}_{S_{n-1}}x_k, \]
where \(\lambda\) is constant on \(S_{n-1}\), i.e., \(\lambda=\lambda(x_{n+1})\). Finally:
\[ v_k=v'_k+v''_k =\lambda_1(x_{n+1})x_k\operatorname{grad}_{S_n}x_{n+1} +\lambda_2(x_{n+1})\operatorname{grad}_{S_{n-1}}x_k. \]
In coordinate notation \(v_k\) has the form
\[ p\cdot(0,0,\ldots,0,-x_{n+1},0,\ldots,0,x_k)+ \]
\[ +q\cdot(-x_1x_k,-x_2x_k,\ldots,-x_{k-1}x_k,1-x_k^2-x_{n+1}^2,-x_{k+1}x_k,\ldots,x_nx_k,0), \]
where
\[ p=(1-x_{n+1}^2)\lambda_1;\qquad q=\lambda_2+x_{n+1}\lambda_1. \]

Now let \(R\subset R_k^{(n)}\), where \(R_k^{(n)}\) is the collection of fields from \(R^{(n)}\) whose coordinates are polynomials in \(x_1,x_2,\ldots,x_{n+1}\) of degree not higher than \(k\). Then \(p\) and \(q\) in the expression for \(v_k\) must be polynomials of degrees not higher than \(k-1\) and \(k-2\), respectively. The number of linearly independent fields of this form is equal to \(2k-1\); and since linearly independent zonal fields correspond to linearly independent irreducible subspaces, \(R_k^{(n)}\) decomposes into no more than \(2k-1\) irreducible subspaces. In particular, \(R_1^{(n)}\) is irreducible. A basis in \(R_1^{(n)}\) is formed by fields of the form
\[ v_{ij}=(0,\ldots,x_j,\ldots,-x_i,\ldots,0). \]
Hence
\[ \dim R_1^{(n)}=C_{n+1}^2. \]
The representation of this dimension is unique (this is easily established using Table 30 in \((^1)\)). We shall denote its highest weight by \(M_n\). The highest weights of the representations realized in spherical functions on \(S_n\), as follows from Cartan’s results \((^2)\), form a one-dimensional lattice. We shall denote them by \(k\Lambda_n\).

We shall show that \(M_n\) is not equal to \(k\Lambda_n\) for any \(k\). Indeed, in the contrary case the representation of the rotation group of \(S_n\) realized in \(R_1^{(n)}\) would be equivalent to some representation in spherical functions, and therefore \(R_1^{(n)}\) would contain a field invariant with respect to-

subgroup leaving fixed some fixed point. Such a field, as we have seen, must be a gradient field. But the fields belonging to \(R_1^{(n)}\) are not gradient fields (if only because the basic fields \(v_{ij}\) have closed lines of force), which proves the assertion.

Further, the linear span of the fields \(x_i v_j,\ v_j \in R_{k-1}^{(n)}\), belongs to \(R_k^{(n)}\). Therefore, if \(R_{k-1}^{(n)}\) contains an irreducible component with highest weight \(N\), then \(R_k^{(n)}\) contains a component with highest weight \(N+\Lambda_n\).

It follows that \(R_k^{(n)}\) contains components with highest weights
\(M_n,\ M_n+\Lambda_n,\ldots,\ M_n+(k-1)\Lambda_n\). Moreover, since the gradients of spherical functions of degree \(k\) have coordinates of degree \(k+1\), \(R_k^{(n)}\) contains gradient components with highest weights
\(\Lambda_n,\ 2\Lambda_n,\ldots,(k-1)\Lambda_n\). But the number of components does not exceed \(2k-1\); therefore \(R_k^{(n)}\) decomposes into the direct sum of the components listed.

Thus, the space \(R_\infty^{(n)}\) of fields whose coordinates are polynomials decomposes into irreducible components with highest weights of the form \(k\Lambda_n\) and \(M_n+k\Lambda_n\). Now let \(R\) be any irreducible subspace of \(R^{(n)}\). The corresponding representation is equivalent to at most one of those found. Hence \(R\) is orthogonal to all the subspaces found, except possibly one. From this and from the fact that \(R_\infty^{(n)}\) is everywhere dense in \(R^{(n)}\), it follows that \(R\) coincides with one of the subspaces found, or consists of zero alone. This completes the consideration of the case \(n>3\).

The case \(n=3\) is investigated similarly and differs only in that on a sphere of order \(1\)—a circle—there exists a tangent field invariant under rotations. Therefore, in the decomposition \(u=u'+u''\) one has to take into account the tangent component \(u''\). This leads to the fact that \(R_k^{(3)}\) decomposes into \(3k-1\) components (instead of \(2k-1\) for \(n>3\)).

In particular, \(R_k^{(3)}\) decomposes into two three-dimensional components with highest weights \(M_3'\) and \(M_3''\), while all of \(R^{(3)}\) decomposes into components with highest weights
\(k\Lambda_3,\ k\Lambda_3+M_3',\ k\Lambda_3+M_3''\).

The case \(n=2\) could have been investigated similarly, but it is simpler to use the fact that every field tangent to \(S_2\) is representable in the form
\(\operatorname{grad}_{S_2} f+J\operatorname{grad}_{S_2} g\), where \(J\) is the operator of vector multiplication by the outward normal, and \(f\) and \(g\) are functions on the sphere. It follows at once that all irreducible subspaces of \(R^{(2)}\) have the form
\(\{\operatorname{grad}_{S_2} f_i\}\) or \(\{J\operatorname{grad}_{S_2} f_i\}\), where \(f_i\) are elements of an irreducible subspace of spherical functions on \(S_2\).

In conclusion we give the schemes of the representations realized in spherical vector fields. (Following E. B. Dynkin\(^{1}\), we specify the scheme of a representation by writing above a simple root \(\alpha\) the number \(\Lambda_\alpha=\dfrac{2(\Lambda,\alpha)}{(\alpha,\alpha)}\), where \(\Lambda\) is the highest weight of the representation. Zero values of \(\Lambda_\alpha\) are omitted.)

I express my gratitude to E. B. Dynkin for posing the problem and for his guidance.

Moscow State University
named after M. V. Lomonosov

Received
29 III 1957

REFERENCES

\(^{1}\) E. B. Dynkin, Tr. Mosk. matem. obshch. 1 (1952).
\(^{2}\) E. Cartan, Rend. d. Circolo mat. di Palermo, 53 (1929).