A. Juhl's history of generalised Selberg zeta functions

This is an excerpt from section 8.2 ("Historical comments") of A. Juhl's book Cohomological Theory of Dynamical Zeta Functions (Birkhauser, 2001) 667-669.

"In the case of finite volume surfaces Selberg discovered the trace formula and the related zeta function now bearing his name in the seminal paper [261]. Gelfand, Graev and Piateski-Shapiro ([104]) discussed the trace formual for compact quotients of PSL(2,R) from th eright representation theoretical point of view via the spectral decomposition of the right regular representation of PSL(2,R) on the Hilbert space $L^{2}(\Gamma/\PSL(2,R))$.

The two classical volumes of Hejhal ([127], [128]) cover many aspects of the trace formula and the zeta function for finite volume surfaces. The methods used by Hejhal belong to the more classical framework, avoiding representation theory. The first volume of [275] is an introduction to the subject which covers its relations to a great variety of problems ranging from number theory to mathematical physics. For a role of the trace formula and the zeta function in quantum mechanics we refer to [150], [151] and the references there.

Although already [261] contains some remarks on the higher-dimensional real hyperbolic case, the Selberg zeta function ZS for compact space forms of rank on e symmetric spaces was introduced only much later in 1977 in [101]. Gangolli proved the existence of a meromorphic continuation of a certan integral power of ZS (but not for ZS itself) and characterized its zeros and poles. Gangolli and Warner extended these results in [103] also to the finite volume case.

In 1985 Wakayama [288] introduced even more general Selberg type zeta functions associated to a locally homogeneous vector bundle on X.

Also motivated by [261] Smale posed in section II.4 of his seminal paper [266] the problem of studying [a] zeta function for general classes of flows. In the case of the geodesic flow of a surface of constant negative curvature the latter zeta function coincides with Selberg's zeta function (as pointed out to Smale by Sinai and Langlands).

Then in his 1976 paper [244] Ruelle published the prototype of a new argument for the existence of meromorphic continuations of the zeta functions for hyperbolic flows...with real analytic Anosov foliations on compact manifolds of arbitrary dimension. The basic ingredients of Ruelle's proof are the Markov decomposition of the underlying phase space, the associated symbolic dynamics, the Bowen-Manning countng lemma and a trace formula for the transfer operators. Later Fried ([93]) extended Ruelle's method to cover all generalized Selberg zeta functions. In particular, Fried gave alternative definitions for the zeta functions introduced by Gangolli and Wakayama in terms of the dynamics of a corresponding flow. These definitions emphasize the dynamical nature of the zeta functions.

Now in 1988/89 we wrote a paper ([158]) in which we proved that the Selberg trace formula for a compact quotient...can be reformulated in the form which now is the explicit form of the dynamical Lefschetz formula (see Chapter 3). The method of [158] was to start with the explicit form of the trace formula proved in [80] and to express the identity term in the trace formula in terms of multiplicities...by using suitable index formulas for Dirac operators on Xn.

The main motivation for doing this was the paper [112] in which Guillemin gave a formal argument for regarding the Selberg trace formula for compact Riemann surfaces X as a Lefschetz fixed fpoint formula for the geometric quantization of the geodesic flow of X on SX. since Guillemin's argument rested on a much more general (but conjectural) Lefschetz trace formula for the geometric quantization of a Hamiltonian flow with an invariant real polarization, it was a natural problem to ask whether in higher dimensions the Selberg trace formula still fits into this framework."

[Juhl continues for several pages, covering more recent work.]



[80] J.J. Duistermaat and V. Guillemin, "The spectrum of positive elliptic operators and periodic bicharacteristics", Invent. Math. 29 (1979) 39-79.

[93] D. Fried, "The zeta functions of Ruelle and Selberg I", Ann. Sci. Ecole Norm. Sup. 19 (1986) 491-517.

[101] R. Gangolli, "Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one", Illinois J. Math 21 (1977) 1-21.

[103] R. Gangolli and G. Warner, "Zeta functions of Selberg's type for some non-compact quotients of symmetric spaces of rank one", Nagoya Math. J. 78 (1980) 1-44.

[104] I.M. Gelfand, M.I. Graev and I.I. Piatetskii-Shapiro, Representation Theory and Automorphic Functions. Generalized Functions 6 (Academic Press, 1990 - reprint of 1969 edition)

[112] V. Guillemin, "Lectures on the spectral theory of elliptic operators", Duke Math. J. 44 (1977) 485-517.

[127] D.A. Hejhal, The Selberg Trace Formula for PSL(2,R), volume 548 of Lecture Notes in Mathematics (Springer, 1976)

[128] D.A. Hejhal, The Selberg Trace Formula for PSL(2,R), volume 1001 of Lecture Notes in Mathematics (Springer, 1983)

[150] N. Hurt, Geometric Quantization in Action, Number 8 in Math. Appl.. (Reidel, Dordrecht, 1983)

[151] N. Hurt, "Zeta functions and periodic orbit theory: a review", Resultate Math. 23 (1993) 55-120.

[158] A. Juhl, "Quantized geodesic flow and the Selberg trace formula I", In Surveys on Global Analysis, number 117 in Teubner Texte Für Mathematik (Teubner, 1990) 138-197.

[244] D. Ruelle, "Zeta function for expanding maps and Anosov Flows", Invent. Math. 34 (1976) 231-242.

[261] A. Selberg, "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc. 20 (1956) 47-87.

[266] S. Smale, "Differentiable dynamical systems", Bull. Amer. math. Soc. 73 (1967) 747-817.

[275] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I,II (Springer, 1985)

[288] M. Wakayama, "Zeta functions of Selberg's type for non-compact quotient of SU(n,1) (n<2)", Hiroshima Math. J. 15 (1985) 235-295.


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