M. Scott Osborne

Multiplicities of the integrable discrete series: The case of a nonuniform lattice in an R-rank one semisimple group

Let G be a noncompact connected simple Lie group of split-rank 1. Assume that G possesses a compact Cartan subgroup so that the discrete series for G is not empty. Let Γ be a nonuniform lattice in G. In this paper, we give an explicit formula for the multiplicity with which an integrable discrete series representation of G occurs in the space of cusp forms in L2(GΓ).

The Selberg trace formula. V. Questions of trace class

The purpose of this paper is to develop criteria which will ensure that the K-finite elements of C^°(G) are represented on L¿ia(G/T) by trace class operators.

Topological Vector Spaces

A topological vector space X over \(\mathbb{R}\) or \(\mathbb{C}\) is a vector space, which is also a topological space, in which the vector space operations are continuous.

Duals of Fréchet Spaces

A good title for this chapter might have been “Weird Countability.” The point is that, while “countability” applies to a Frechet space X in basically one way (it is first countable), it affects X ∗ in some rather strange ways.

Colimits and Tor

Of the two functors defined in Chapter 3, Ext is the more “universal”; Section 6.6 describes how Tor can, in principle at least, be defined using Ext. On the other hand, Tor has more properties. Up to now, this has only been reflected in the fact that Tor mixes rightR -modules with left R -modules. But there’s more. Tor behaves well with respect to certain colimits, and that is the general subject of this chapter.

Change of Rings

In the discussions of the preceding chapters, the ring has stayed fixed. It has been arbitrary, but unvarying. That is about to change; we shall now be concerned with what happens when certain modifications are made to a ring. The three structural operations addressed later are the formation of matrix rings, polynomial rings, and localizations. There are some generalities to the theory, however, and they not only form a backdrop for change of rings, they also ease computations for specific rings.

Ext and Tor

As stated in the last chapter, homological algebra is primarily concerned with measuring how much modules depart from being projective, injective, or flat. The measure of this is contained in two new sequences of functors, Tor and Ext. While there are several ways of defining these functors (a fact that is the source of some theorems), the most straightforward way is via complexes.

Abstract Homological Algebra

A close look at much of the earlier material, especially in the last chapter, reveals the strong connection between projectives and injectives. The idea is this: Formulate your result purely in terms of arrows (morphisms), then reverse them. That is, work in the opposite category. Not everything can be done this way, but a surprising amount can.

The Selberg trace formula. VIII. Contribution from the continuous spectrum

The purpose of this paper is to isolate the contribution from the continuous spectrum to the Selberg trace formula.