The trace formula for reductive groups.

*(English)*Zbl 0533.10029
Journées automorphes, Dijon 1981, Publ. Math. Univ. Paris VII, No. 15, 1-41 (1983).

[For the entire collection see Zbl 0516.00011.]

This paper is a report on the author’s work on the Selberg trace formula for a general reductive group, and describes the present state of the formula.

If one attempts for a general group to do what is now reasonably familiar for GL(2), say, it is possible to get a sort of trace formula, but one which is not as useful as one would like. A major difficulty is that the formula is expressed in terms of non-invariant distributions. The first major improvement the author has made was to develop a method for expressing the formula in terms of invariant distributions using an inductive process on subgroups. This is explained in considerable detail. Unfortunately the expressions one gets are still not readily computed, and additional techniques are required. These are also described quite clearly.

This article should be helpful to anybody who might be interested in applying the trace formula, or even simply in getting some understanding of it, without necessarily proving all the details. Especially interesting are discussions of some shortcomings or limitations of the formula as it currently stands, including the author’s assessment of how serious they are.

This paper is a report on the author’s work on the Selberg trace formula for a general reductive group, and describes the present state of the formula.

If one attempts for a general group to do what is now reasonably familiar for GL(2), say, it is possible to get a sort of trace formula, but one which is not as useful as one would like. A major difficulty is that the formula is expressed in terms of non-invariant distributions. The first major improvement the author has made was to develop a method for expressing the formula in terms of invariant distributions using an inductive process on subgroups. This is explained in considerable detail. Unfortunately the expressions one gets are still not readily computed, and additional techniques are required. These are also described quite clearly.

This article should be helpful to anybody who might be interested in applying the trace formula, or even simply in getting some understanding of it, without necessarily proving all the details. Especially interesting are discussions of some shortcomings or limitations of the formula as it currently stands, including the author’s assessment of how serious they are.

Reviewer: J.Repka

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |