UDC 513.88 : 513.83 + 517.948

MATHEMATICS

A. B. ANTONĔVICH

ON THE INDEX OF A PSEUDODIFFERENTIAL OPERATOR WITH A FINITE GROUP OF SHIFTS

(Presented by Academician I. G. Petrovskii, 20 VI 1969)

Let \(G\) be a finite group of order \(N\); let \(X\) be a connected \(G\)-manifold and let the action of the group \(G\) on \(X\) be effective. In the space \(C_\infty(X)\), by the formula

\[ T_g u(x)=u(g^{-1}x), \qquad g\in G,\qquad x\in X,\qquad u\in C_\infty(X), \]

we define a representation \(T_g\) of the group \(G\).

By a pseudodifferential operator of order \(r\) with a finite group of shifts we shall mean an operator \(A\) acting in \(C_\infty(X)\) by the formula

\[ Au=\sum_{g\in G} A_g T_g u, \]

where the \(A_g\) are pseudodifferential operators of order \(r\). The operator \(A\) belongs to the operators with a deviating argument. In the present paper conditions for the Noether property of the operator \(A\) are indicated and a formula for its index is given.

Let \(\widetilde A\) be the operator acting in the space \(C_\infty(X\times G)\) by the formula

\[ \widetilde A u(x,g')=\sum_{g\in G} T_{g'}A_gT_{g'}^{-1}u(x,g'g). \]

If functions from \(C_\infty(X\times G)\) are regarded as vector-functions on \(X\), then the operator \(\widetilde A\) is a \(G\)-invariant pseudodifferential operator \(\left({}^{1}\right)\).

Theorem 1. In order that the extension of the operator \(A\) be a Noether operator from \(H_l(X)\) into \(H_{l-r}(X)\), it is necessary and sufficient that the operator \(\widetilde A\) be elliptic.

Proof. In the space \(C_\infty(X\times G)\) define a representation \(\widetilde T_g\) of the group \(G\)

\[ \widetilde T_{g''}u(x,g')=u(g''^{-1}x,g''^{-1}g'). \]

The space \(C_\infty(X\times G)\) decomposes into a direct sum of a finite number of invariant subspaces \(M^i\) \((i=1,\ldots,k)\) such that in \(M^i\) there acts a representation which is a multiple of the irreducible representation \(D^i\) of the group \(G\). The operator \(\widetilde A\) commutes with all operators \(\widetilde T_{g''}\) and, consequently, the subspaces \(M^i\) are invariant with respect to the operator \(\widetilde A\), i.e. the operator \(\widetilde A\) decomposes into the direct sum of operators \(A_i\), \(i=1,\ldots,k\). Let \(M^1\) and \(A_1\) be the subspace and the operator corresponding to the identity representation of the group \(G\). The mapping \(p:M^1\to C_\infty(X)\), acting by the formula

\[ pu(x)=u(x,e), \]

is an isomorphism and carries the operator \(A_1\) into \(A\). The remaining operators \(A_i\) can also be realized in \(C_\infty^{n_i}(X)\) as pseudodifferential operators with shift \((n_i\) is the dimension of the representation \(D^i)\).

Let now \(\widetilde A\) be an elliptic operator. Then it can be extended to a Noetherian operator acting from \(H_l^N(X)\) to \(H_{l-r}^N(X)\), and consequently the operator \(A\) is also extended to a Noetherian operator from \(H_l(X)\) to \(H_{l-r}(X)\). Suppose that \(\widetilde A\) is not an elliptic operator. Then one can construct a function \(u_\lambda \in M^1\) such that, as \(\lambda \to \infty\), the a priori estimate is not satisfied for \(u_\lambda\), which contradicts the Noetherian property of the operator \(A\). The theorem is proved.

In work \({}^{1}\) the analytic index of a \(G\)-invariant operator is defined as an element of the ring \(R(G)\) of characters of representations of the group \(G\):

\[ \operatorname{index} B=[\operatorname{Ker} B]-[\operatorname{Coker} B]\in R(G), \]

where \([\operatorname{Ker} B]\), \([\operatorname{Coker} B]\) are the characters of the representations induced by the action of the group \(G\) in \(\operatorname{Ker} B\) and \(\operatorname{Coker} B\). The value of \(\operatorname{index} B\) at an element \(g\) is called the Lefschetz number and is denoted by \(L(g,B)\). In \({}^{1-3}\) a method is indicated for computing the index for pseudodifferential operators.

Theorem 2. The index of a pseudodifferential operator \(A\) with a finite group of shifts is expressed by the formula

\[ \operatorname{ind} A=\frac{1}{N}\sum_{g\in G} L(g,\widetilde A). \tag{1} \]

Proof. From the decomposition of the operator \(\widetilde A\) we obtain

\[ \operatorname{index} \widetilde A=\sum_{i=1}^{k}\operatorname{index} A_i, \tag{2} \]

and since in \(M^i\) there acts a representation that is a multiple of the irreducible representation \(D^i\), we have

\[ \operatorname{index} A_i=\frac{\operatorname{ind} A_i}{n_i}\chi_i, \tag{3} \]

where \(\chi_i\) is the character of the irreducible representation \(D^i\), \(\operatorname{ind} A_i=\dim \operatorname{Ker} A_i-\dim \operatorname{Coker} A_i\). By virtue of the orthogonality of the characters of irreducible representations, from formulas (2) and (3) we obtain

\[ \operatorname{ind} A_i=\frac{n_i}{N}\sum_{g\in G} L(g,\widetilde A)\overline{\chi_i}(g). \tag{4} \]

From (4), for \(i=1\), we obtain the assertion of the theorem.

Theorem 3. If every transformation \(g\in G\), \(g\ne e\), has a finite number of fixed points, then

\[ \operatorname{ind} A=\frac{1}{N}\operatorname{ind}\widetilde A. \tag{5} \]

Proof. If \(g\) has a finite number of fixed points, then in work \({}^{2}\) \(L(g,\widetilde A)\) is expressed in terms of the trace of the operator \(\widetilde T_g\) in the fiber over a fixed point. Since in the fiber over a fixed point \(\widetilde T_g\) acts as the regular representation, \(\operatorname{Tr} T_g=0\), \(L(g,\widetilde A)=0\), and formula (1) gives (5).

The results of the article are easily transferred to operators in sections of vector bundles. In the case of one-dimensional singular integral operators and the cyclic group \(G\), formula (5) was obtained by other methods in works \({}^{4,5}\).

Belorussian State University
named after V. I. Lenin
Minsk

Received
11 VI 1969

CITED LITERATURE

\({}^{1}\) M. F. Atiyah, I. M. Singer, UMN, 23, no. 5 (143), 99 (1968).
\({}^{2}\) M. F. Atiyah, G. B. Segal, UMN, 23, no. 6 (144), 135 (1968).
\({}^{3}\) M. F. Atiyah, I. M. Singer, UMN, 24, no. 1 (145), 127 (1969).
\({}^{4}\) G. S. Litvinchuk, Izv. AN SSSR, ser. matem., 31, 563 (1967).
\({}^{5}\) G. S. Litvinchuk, Izv. AN SSSR, ser. matem., 32, 1414 (1968).