Index of Elliptic Boundary Value Problems Associated with Isometric Group Actions
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Index of Elliptic Boundary Value Problems Associated with Isometric Group Actions A. V. Boltachev∗,1 and A. Yu. Savin∗,∗∗,2 ∗
RUDN University, Moscow, Russia Leibniz Universit¨ at Hannover, Germany, E-mail: 1 [email protected], 2 [email protected] ∗∗
Received May 17, 2020; Revised May 21, 2020; Accepted July 15, 2020
Abstract. Given a compact manifold with boundary, endowed with an isometric action of a discrete group of polynomial growth, we state an index theorem for elliptic elements in the algebra of nonlocal operators generated by the Boutet de Monvel algebra of pseudodifferential boundary value problems on the manifold and the shift operators associated with the group action. DOI 10.1134/S1061920820030048
1. INTRODUCTION Let Γ be a discrete group of diffeomorphisms of a smooth manifold M . We consider the class of operators generated by pseudodifferential operators on M and shift operators Tγ u(x) = u(γ −1 (x)) for all γ ∈ Γ and u ∈ C ∞ (M ). The Fredholm property for operators in this class is known in a quite general situation (see [1, 2]). However, the index problem was studied in the case of manifolds without boundary only (see [3, 4, 5, 6, 7] and the references cited there). In this paper, we consider a compact smooth manifold with boundary, endowed with an isometric action of a discrete group of polynomial growth in the sense of Gromov [8]. In this geometric situation, we state an index theorem for elliptic elements in the algebra generated by the Boutet de Monvel pseudodifferential boundary value problems [9] on the manifold and the shift operators associated with the group action. Our index formula gives, as special cases, the index formula for elliptic elements in the Boutet de Monvel algebra [10] (see also [11]) and the index formula for elliptic operators associated with isometric group actions on closed manifolds [4]. 2. BOUTET DE MONVEL ALGEBRA Let M be a compact smooth manifold with boundary denoted by X. Suppose that M is endowed with a Riemannian metric and consider the induced Riemannian metric on X. The local coordinates on M and X are denoted by x and x , respectively. In addition, in a neighborhood of the boundary, we use coordinates x = (x , xn ), xn 0, such that the boundary has the equation xn = 0, while xn is equal to the distance to the boundary. The dual coordinates in T ∗ M are denoted by ξ = (ξ , ξn ). We consider the Boutet de Monvel operators of order and type equal to zero. We refer the reader to [9, 12, 13, 14] for a complete exposition of the Boutet de Monvel algebra and recall here only several facts about this algebra, which are used below. The Boutet de Monvel operators of order and type equal to zero define continuous mappings of the form D=
L2 (M ) L2 (M ) A+G C : ⊕ −→ ⊕ B AX L2 (X) L2 (X)
(2.1)
Here A is a classical pseudodifferential operator of order zero on M , and the complete symbol of A satisfies the so-called transmission property; AX is a classical pseudodifferential operator of order zero on X; B, C, and G
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