Spectral theory for non-unitary twists.

*(English)*Zbl 1427.11045For unitary representations of locally compact groups, there is a well-developed general spectral theory which expresses such a representation as a direct integral or a direct sum of irreducibles. However, for non-unitary representations there is no spectral theory in general. The aim of this paper is to develop a spectral theory for certain non-unitary representations induced from cocompact lattices.

The contents of this interesting article may be summarised as follows. In Section 1, the basic definitions are recalled. For example, the definition of the space \(C_c^\infty(G)\) of test functions on a locally compact topological group \(G\): For a Lie group \(G\), this has the usual meaning; for a locally compact \(G\) with \(G/G^0\) compact (with \(G^0\) the identity component), \(C_c^\infty(G)\) is the inductive limit, in the category of locally convex spaces, of \(C_c^\infty(L)\) over all Lie quotients \(L\) of \(G\); for general locally compact \(G\), one chooses an open subgroup \(H\) with \(H/H^0\) compact and then \(C_c^\infty(G)\) is the sum of all \(C_c^\infty(gH)\) with \(g\) running over \(G\). Also, in this section, is given a counterexample to a question asked in [A. Deitmar and G. van Dijk, J. Lie Theory 26, No. 1, 269–291 (2016; Zbl 1342.22009)] as to whether a locally compact \(G\) admitting a cocompact lattice must be a trace class group.

In Section 2, the notions of complete \(L\)-filtration of a representation with respect to a “tower” \(L\) (a type of linearly ordered set), subquotients and discrete representations of a locally compact group \(G\) (on Banach spaces) are introduced. This filtration will take the place of direct sum or direct integral expression in cases where this is possible.

Section 3 deals with Lie groups. The notions of \(\pi\)-filtration for an irreducible representation \(\pi\) of \(G\), and of \(\Delta\)-admissibility of a representation (where \(\Delta\) is a group-Laplacian, which is the Laplace operator for a chosen invariant metric on the Lie group \(G\)) are introduced, and a spectral theorem is proved for such representations, in terms of complete filtrations.

In Section 4, the main spectral theorem for nonunitary representations of a Lie group, induced from a cocompact lattice, is proved. Also proved is a trace formula for locally compact groups, which we restate here in a rather short form:

Theorem 2 (Trace formula). Let \( G \) be a locally compact group and \(\Gamma\subset G\) be a cocompact lattice. Let \((\omega,V_\omega)\) be a finite-dimensional complex representation of \(\Gamma\) and let \(H= L^2(\Gamma\backslash G,\omega)\) denote the usual space of all measurable \(f:G\to V_\omega\) with \(f(\gamma x)=\omega(\gamma)f(x)\) (for \(\gamma\in\Gamma, x\in G\)) such that \(\int_F \langle f(x),f(x)\rangle dx<\infty\) where \(F\subset G\) is a fixed compact fundamental domain for \(\Gamma\backslash G\). For the right regular representation \( R \) of \( G \) on the Hilbert space \( H \), the operator \(R(f)\) is of trace class and its trace equals either side of the equation \[ \sum_{\pi \in \widetilde{G}}N_{\Gamma,\omega}(\pi)\, \mathrm{tr} (\pi (f))=\sum_{[\gamma]} \,\mathrm{vol} (\Gamma_\gamma\backslash G_\gamma)\,\mathcal{O}_\gamma(f)\,\mathrm{tr} (\omega(\gamma)). \] where \(N_{\Gamma,\omega}(\pi)\) denotes the maximal length of a \(\pi\)-filtration in \(H\), the sum on the right running over all conjugacy classes \([\gamma]\) in \(\Gamma\), the groups \(\Gamma_\gamma,G_\gamma\) are the centralizers of \(\gamma\) in \(\Gamma,G\) respectively and \(\mathcal{O}_\gamma(f)\) denotes the orbital integral \[ \mathcal{O}_\gamma(f)=\int_{G_\gamma\backslash G}f(x^{-1}\gamma x) dx. \] The other symbols have their usual meanings. The left hand side of the above trace formula is called the spectral side and the right hand side the geometric side.

In Section 5, a slightly stronger spectral theorem is proved for semisimple Lie groups, which says that the right regular representation of a semisimple Lie group \(G\) on \(L^2(\Gamma \backslash G\times V_\omega)\) is a direct sum of representations of finite length. Here \(\Gamma \backslash G\times V_\omega\) is the quotient space under the diagonal action of \(\Gamma\) on \(G\times V_\omega\), where \((\omega,V_\omega)\) is a finite-dimensional complex representation of \(\Gamma\).

Overall, this article is well-written, quite self-contained and the results proved are interesting.

The contents of this interesting article may be summarised as follows. In Section 1, the basic definitions are recalled. For example, the definition of the space \(C_c^\infty(G)\) of test functions on a locally compact topological group \(G\): For a Lie group \(G\), this has the usual meaning; for a locally compact \(G\) with \(G/G^0\) compact (with \(G^0\) the identity component), \(C_c^\infty(G)\) is the inductive limit, in the category of locally convex spaces, of \(C_c^\infty(L)\) over all Lie quotients \(L\) of \(G\); for general locally compact \(G\), one chooses an open subgroup \(H\) with \(H/H^0\) compact and then \(C_c^\infty(G)\) is the sum of all \(C_c^\infty(gH)\) with \(g\) running over \(G\). Also, in this section, is given a counterexample to a question asked in [A. Deitmar and G. van Dijk, J. Lie Theory 26, No. 1, 269–291 (2016; Zbl 1342.22009)] as to whether a locally compact \(G\) admitting a cocompact lattice must be a trace class group.

In Section 2, the notions of complete \(L\)-filtration of a representation with respect to a “tower” \(L\) (a type of linearly ordered set), subquotients and discrete representations of a locally compact group \(G\) (on Banach spaces) are introduced. This filtration will take the place of direct sum or direct integral expression in cases where this is possible.

Section 3 deals with Lie groups. The notions of \(\pi\)-filtration for an irreducible representation \(\pi\) of \(G\), and of \(\Delta\)-admissibility of a representation (where \(\Delta\) is a group-Laplacian, which is the Laplace operator for a chosen invariant metric on the Lie group \(G\)) are introduced, and a spectral theorem is proved for such representations, in terms of complete filtrations.

In Section 4, the main spectral theorem for nonunitary representations of a Lie group, induced from a cocompact lattice, is proved. Also proved is a trace formula for locally compact groups, which we restate here in a rather short form:

Theorem 2 (Trace formula). Let \( G \) be a locally compact group and \(\Gamma\subset G\) be a cocompact lattice. Let \((\omega,V_\omega)\) be a finite-dimensional complex representation of \(\Gamma\) and let \(H= L^2(\Gamma\backslash G,\omega)\) denote the usual space of all measurable \(f:G\to V_\omega\) with \(f(\gamma x)=\omega(\gamma)f(x)\) (for \(\gamma\in\Gamma, x\in G\)) such that \(\int_F \langle f(x),f(x)\rangle dx<\infty\) where \(F\subset G\) is a fixed compact fundamental domain for \(\Gamma\backslash G\). For the right regular representation \( R \) of \( G \) on the Hilbert space \( H \), the operator \(R(f)\) is of trace class and its trace equals either side of the equation \[ \sum_{\pi \in \widetilde{G}}N_{\Gamma,\omega}(\pi)\, \mathrm{tr} (\pi (f))=\sum_{[\gamma]} \,\mathrm{vol} (\Gamma_\gamma\backslash G_\gamma)\,\mathcal{O}_\gamma(f)\,\mathrm{tr} (\omega(\gamma)). \] where \(N_{\Gamma,\omega}(\pi)\) denotes the maximal length of a \(\pi\)-filtration in \(H\), the sum on the right running over all conjugacy classes \([\gamma]\) in \(\Gamma\), the groups \(\Gamma_\gamma,G_\gamma\) are the centralizers of \(\gamma\) in \(\Gamma,G\) respectively and \(\mathcal{O}_\gamma(f)\) denotes the orbital integral \[ \mathcal{O}_\gamma(f)=\int_{G_\gamma\backslash G}f(x^{-1}\gamma x) dx. \] The other symbols have their usual meanings. The left hand side of the above trace formula is called the spectral side and the right hand side the geometric side.

In Section 5, a slightly stronger spectral theorem is proved for semisimple Lie groups, which says that the right regular representation of a semisimple Lie group \(G\) on \(L^2(\Gamma \backslash G\times V_\omega)\) is a direct sum of representations of finite length. Here \(\Gamma \backslash G\times V_\omega\) is the quotient space under the diagonal action of \(\Gamma\) on \(G\times V_\omega\), where \((\omega,V_\omega)\) is a finite-dimensional complex representation of \(\Gamma\).

Overall, this article is well-written, quite self-contained and the results proved are interesting.

Reviewer: Ramdin Mawia (Kolkata)