UDC 519.41/47

MATHEMATICS

S. V. SMIRNOV

POSITIVE DEFINITE FUNCTIONS ON ALGEBRAIC NILPOTENT GROUPS OVER A DISCRETE FIELD

(Presented by Academician P. S. Aleksandrov, 23 XII 1965)

Let \(G\) be a discrete group. Denote by \(M(G)\) the set of positive definite functions \(\varphi\) on \(G\), constant on conjugacy classes and such that \(\varphi(e)=1\) (where \(e\) is the identity of \(G\)). It is known that there is a one-to-one correspondence between the extreme points of \(M(G)\) and factor representations of type \(I_n\) and \(II_1\).

The first complete description of the set \(M(G)\) for a group not belonging to type I was given by Thoma \((^{1,2})\) for the group of affine transformations of a one-dimensional space over the field of rational numbers and for the infinite symmetric group. In the work of A. A. Kirillov \((^3)\) this problem was solved for matrix groups of arbitrary order.

The present note is devoted to the case where \(G\) is a nilpotent linear algebraic group over a field \(k\) of characteristic zero.

Theorem. Let \(H\) be an algebraic normal divisor of \(G\); \(C\) the center of the factor group \(G/H\); \(p\) the natural projection of \(G\) onto \(G/H\). Denote by \(\pi\) a character of \(C\) which is not equal to \(1\) on any one-parameter subgroup of \(C\). Then the function \(\varphi(g)\), given by the formula

\[ \varphi(g)= \begin{cases} \pi(p(g)), & \text{if } p(g)\in C,\\ 0, & \text{if } p(g)\notin C, \end{cases} \]

is an extreme point of the set \(M(G)\). Conversely, every extreme point of \(M(G)\) can be obtained by such a construction.

Proof. First consider the group \(S_n\) corresponding to the Lie algebra \(\mathscr{S}_n\) with basis \(Y, X_1,\ldots,X_n\) and commutation relations
\[ [Y,X_i]=X_{i+1}\quad (i=1,\ldots,n-1);\qquad [Y,X_n]=0 \quad \text{and} \quad [X_i,X_j]=0. \]
Direct computations show that the center \(C\) of this group consists of elements of the form \(c=\exp tX_n\), where \(\exp\) is the canonical mapping of the algebra onto the group. The elements \(cg\) and \(g\), where \(c\in C,\ g\notin C\), are conjugate.

Lemma 1. If a function \(\varphi\) on \(S_n\) belongs to \(M(S_n)\), then for any \(g\notin C\) the function \(\varphi(c)-|\varphi(g)|^2\) is positive definite on \(C\).

The proof is obtained by considering the set \(c_1,\ldots,c_n\in C\) and \(g,\ldots,g\) (\(n\) times).

Let \(G\) be a nilpotent algebraic linear group over a field of characteristic \(0\), and let \(\mathscr{G}\) be the corresponding Lie algebra. Then we have
\[ \mathscr{G}=Y_n \supset Y_{n-1}\supset \cdots \supset Y_1=Z, \]
where the \(Y_i\) are ideals such that \([\mathscr{G},Y_i]\subset Y_{i-1}\), and \(Z\) is the center.

Lemma 2. For any \(y\in G\) \((y\notin Z)\) there exists an algebra isomorphic to \(\mathscr{S}_n\) which contains \(y\).

Proof. Let \(Y_m\) be an ideal such that \([y,Y_i]=0\) for \(i<m\), but \([y,Y_m]\ne 0\); then there exists an element \(t\in Y_m\) such that \([t,y]=y_1\ne 0\). Let \([t,y_1]=y_2\ne 0,\ldots,[t,y_{l-1}]=y_l\ne 0\), but \([t,y_l]=0\). We shall prove that \([y_i,y_j]=0\) for all \(i\) and \(j\). Indeed, \([y_i,y_j]=\)

\[ = [[t, y_{i-1}]y_j] = -[[y_{i-1}, y_j], t] - [[y_j, t], y_{i-1}] = -[[y_{i-1}, y_j], t] - [y_{i-1}, y_{i+1}], \]
and by induction on \(i\) we obtain the assertion of the lemma.

Now take an arbitrary extreme function \(\varphi\) from \(M(G)\). From the work of Thoma \((^4)\) it follows that the restriction of \(\varphi\) to the center is a character of the center.

Lemma 3. Suppose that the restriction of \(\varphi\) to the center does not degenerate on any one-parameter subgroup of the center (i.e., does not become \(1\) on it). Then \(\varphi\) is concentrated on the center.

Proof. We shall show by induction that the restriction of \(\varphi\) to \(\exp Y_k\) is concentrated on \(\exp Z\). Suppose that we have already proved this for \(Y_{m-1}\), and prove it for \(Y_m\). Take an arbitrary element \(y \in Y_m\) and consider \(\varphi\) on \(\exp \mathscr{S}_n\) (\(\mathscr{S}_n\) is constructed by Lemma 2). If \(y_l \in Z\), then the restriction of \(\varphi\) to the center \(C\) of the group \(S_n\) will be a nondegenerate character; if \(y_l \notin Z\), then this restriction will be a function concentrated at \(e\). In both of these cases the function \(\varphi(c)-d\) (\(d\) is a positive constant) on \(C\) cannot be positive definite. Using Lemma 1, we obtain
\[ \varphi(\exp y)=0. \]

Conversely, it is not difficult to check that a nondegenerate character of the center, extended trivially to the whole group, is an extreme function. Indeed, if \(\varphi(g)=\frac12(\varphi_1(g)+\varphi_2(g))\), then on the center \(\varphi_1=\varphi_2=\varphi\). But then, by Lemma 3, \(\varphi_1\) and \(\varphi_2\) are concentrated on the center, i.e. \(\varphi\) is extreme.

Now take an arbitrary extreme function \(\varphi\). Let \(J\) be a maximal ideal for which the restriction of \(\varphi\) to \(\exp J\) becomes \(1\). Straightforward computations show that the function \(\varphi\), considered as a function on \(G/\exp J\), no longer degenerates on any one-parameter subgroup of the center. Therefore, by Lemma 3, \(\varphi\) has the required form.

This work was carried out under the supervision of A. A. Kirillov, to whom the author expresses his gratitude.

Moscow State University
named after M. V. Lomonosov

Received
3 XII 1965

REFERENCES

1. E. Thoma, Math. Zs., 84, No. 4, 389 (1964).
2. E. Thoma, Math. Zs., 85, No. 2, 40 (1964).
3. A. A. Kirillov, DAN, 162, No. 3 (1965).
4. E. Thoma, Math. Ann., 153, No. 2 (1964).