ANISOTROPIC HARMONIC ANALYSIS ON HOMOGENEOUS TREES 3
principal series of representations of SL(2, R). Through the mediation of the
Poisson Kernel, P, and its powers,
Pz,
the principal series representations can
also be realized on
L2(dD,d6),
where dO is Lebesgue measure. The represen-
tations na of G can be realized in an analogous way on the boundary of the
tree.
If in place of 5L(2,R) we consider the group PGL(2, Qg), where q is prime
and Qq is the field of g-adic numbers, then this analogy becomes stronger. (See
[Cartl] and [Cart3] for more details.) PGL(2, Qq) acts naturally on the unla-
belled tree G. The principal series of PGL(2, Qq) can be obtained by decompos-
ing the regular representation of PGL(2, Qq) on
£2(G)
with respect to a certain
"Laplace-Beltrami" or Hecke operator on £2(G). This Hecke operator is equal to
the R obtained from the isotropic random walk on G. (The walk is isotropic if
Pi = ''' = Pq+i = 1/(Z + !)•) Moreover, if the labelling of G is chosen properly
with respect to the action of PGL(2, Q9), then G is a subgroup of PGL(2, Qq)
[FT-Pi2]. Consequently, the representations 7ra of G associated to the isotropic
random walk are simply the restrictions from PGL(2, Qq) of principal series
representations. The book [FT-Pil] is devoted to the study of the isotropic
case when G is replaced by its close relative, the free group. Finally, note that
the quotient PGL(2, Qq)/G is compact, which is to say that G is a cocompact
subgroup of PGL(2, Qq).
There are two general programs which the present work can be considered
part of. First of all, one can replace the nearest neighbor random walk, which we
have been discussing, with a general symmetric, left invariant random walk on G.
Then
£2(G)
will again decompose into generalized eigenspaces of the transition
matrix, P, and again G will have an induced action on (almost) each of these
eigenspaces. Now one asks: How does the random walk behave?
For instance, what is the asymptotic behavior of the probability of return-
ing to the starting point after n steps? Can the representations of G obtained
this way be realized on the boundary of the tree? Are these representations
irreducible? What features do these representations share (independently of the
random walk)? For a random walk defined by a finitely supported measure on G,
some of these questions are answered in [Saw-St], [Angel] and [Lalley]. Further-
more, this program can be extended to other lattice subgroups of PGL(2, Qq)
and 5L(2, R), and also to lattice subgroups of other semisimple Lie groups. The
objective is to find properties of the random walk and/or its associated repre-
sentations which depend only on properties of the Lie group in question. The
geometry of the Lie group is inherited (in some ways) by its lattice subgroups,
and this geometry in turn should influence the random walk and the associated
representations. This general program was initiated by Furstenberg's brilliant
results on the Poisson boundaries of random walks defined on lattice subgroups
of semisimple Lie groups [Furst].
The second (related) program which the present investigation might be consid-
ered part of is as follows. Suppose that G is any lattice subgroup of 5L(2, R) or
PGL(2, Qq). Consider the restriction to G of an irreducible representation of the
larger group. Is this restriction still irreducible? As mentioned above, the ran-
dom walk representations associated to the isotropic random walk on Z2 * Z2
Previous Page Next Page