Abstract Generated abstract
The paper studies pseudodifferential operators with a finite group of shifts on a connected manifold with an effective finite group action. By associating such an operator to a group invariant pseudodifferential operator on the product of the manifold with the group, it gives a necessary and sufficient Fredholm, or Noether, condition in Sobolev spaces: ellipticity of the associated invariant operator. It then derives an index formula expressing the index of the original shifted operator as the average of the Lefschetz numbers of the associated operator over the group. When all nonidentity group elements have only finitely many fixed points, the formula reduces to one N-th of the index of the associated invariant operator, with the results extending to operators on vector bundle sections.
Full Text
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).