4 ALESSANDRO FIGA-TALAMANCA AND TIM STEGER

are such restrictions, with respect to a particular embedding of Z2 • • • Z2 as a

lattice subgroup of PGL(2,Qq). This example is not atypical. [Cow-St] deals

with a general lattice subgroup inside a general connected semisimple Lie group.

Assuming the lattice is irreducible, the result is that all irreducible unitary rep-

resentations of the Lie group except for discrete series representations remain

irreducible when restricted to the lattice. The general method of this paper is

also applicable to matrix groups over Qq.

In addition to being an early step in these two large projects, this work is

an intermediate step in a more limited investigation. Nearest neighbor random

walks on trees, (those that proceed along the edges,) have an important geo-

metrical property: any walk which starts at x and ends at y must pass through

each of the vertices intermediate between x and y. This geometrical fact has im-

portant consequences for the analytical study of the random walk (see [Cart2].)

Many of the pleasant simplifications arising in the present work stem from this

geometrical phenomenon. Suppose we replace G with an arbitrary subgroup of

the automorphism group of the unlabelled tree. Also, suppose we replace our

random walk with an arbitrary symmetric, group invariant, nearest neighbor

random walk on the tree. Then we can again find a series of representations

of the new group, and again profit from the simplifications mentioned above.

This line of investigation might be called harmonic analysis on the tree, since

the tree is taken as primary and the group as secondary. This way of proceeding

is suggested by some of the results in [Aomoto].

We now give a brief outline of the paper, mentioning first that [Aomoto], which

we consulted, contains most of the results of Chapter I, many of the results of

Chapter II and a few of the results of Chapter III, at least for the very closely

related case of the free group.

In Chapter I we first describe in more detail the basic objects of our study.

Then we consider an arbitrary G-invariant random walk on G, constrained to

skip only finitely many edges at each step. In this situation we show that the

matrix elements of (7 —

R)~l

are algebraic functions of 7. We also derive the

special formula for (7 —

R)~l

in the case when the random walk proceeds along

the edges of G. The succeeding chapters are restricted to this basic special case.

The main object of Chapter II is the study of the transition matrix R, con-

sidered as an operator. We show how to find the spectrum of R as an operator

on

£r(G)

for 1 r 2. We provide complete descriptions for the spectrum

of R on

£2

and for the real part of the spectrum of R on

i1.

Finally, we use

standard spectral theory to decompose

£2(G)

as a direct integral of generalized

eigenspaces of R.

Chapter III has two main purposes. First, we realize each representation 7ra of

G (originally defined on the generalized 7-eigenspaces

of#)onL

2

(ft,zv). Here ft

is the boundary of the tree and va is an appropriate positive measure. Secondly,

we analytically continue the representations 7ra and obtain complementary series

representations of G, denoted n^ and 7r7. Chapter III also contains the proof of

a limited form of the Kunze-Stein phenomenon for G.

A useful tool in achieving these results is the Helgason-type theorem: any

7-eigenfunction of R, (even an unbounded eigenfunction), can be written as

Jn

P1{X,UJ)

dv(uj) where P1 is any of several appropriate Poisson kernels and v