Uniform pointwise bounds for Matrix coefficients of unitary representations on semidirect products
Abstract.
Let be a local field of characteristic , and let be a connected semisimple almost algebraic group. Suppose rank and is an excellent representation of on a finite dimensional vector space . We construct uniform pointwise bounds for the finite matrix coefficients restricted on of all unitary representations of the semidirect product without nontrivial fixed vectors. These bounds turn out to be sharper than the bounds obtained from
itself for some cases. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of the pair .
Keywords: Matrix coefficient, unitary representation, Fourier transform, projectionvalued measure, Mackey machine, Kazhdan constant.
1. Introduction and main results
Let be a local field of char . We say that is a (connected) almost algebraic group if is a (connected) Lie group with finite center for isomorphic to or is the group of rational points of a (connected) linear algebraic group over for nonarchimedean or isomorphic to . Unless stated otherwise, denotes a connected semisimple almost algebraic group with rank and denotes its underlying algebraic group; that is, for nonarchimedean or isomorphic to .
1.1. Finitedimensional representations of
For a finite dimensional vector space over , a representation is called normal if is continuous for isomorphic to ; or if is a rational map for nonarchimedean or isomorphic to .
There is a decomposition (resp. ) where (resp. ) is the product of compact factors (resp. anisotropic factors) of (resp. ) and (resp. ) is the product of noncompact factors (resp. isotropic factors) of (resp. ) when is isomorphic to (resp. nonarchimedean or isomorphic to ).
Denote by (resp. ), the noncompact factors (resp. isotropic factors) of (resp. ). Also set and for nonarchimedean or isomorphic to . We call these the noncompact almost simple factors of .
Definition 1.1.
A normal representation of on is called good if the fixed points in are ; and is called excellent if fixed points in are for each noncompact almost simple factor of .
In this paper we present an “upper bound function” for matrix coefficients for all unitary representations of without nontrivial fixed vectors if is an excellent representation of on . Special cases of and are considered in [16] and [20] for a local field . For these cases the following conditions are satisfied: every orbit is locally closed (intersection of an open and a closed set) in the dual group ; and for each the stabilizer is amenable. The first one allows us to use the “Mackey machine” and the latter one implies that the matrix coefficients are bounded by the HarishChandra functions. Margulis also used this criterion in [29, Theorem 2] to prove Kazhdan’s property of the pair . In fact, if is a connected almost algebraic group and is a normal representation, then “Mackey machine” applies to the semidirect product and hence we have complete descriptions of the dual groups of (see [40, Theorem 7.3.1] or [38, Chapter 5.4]). Therefore any irreducible representation of without nontrivial fixed vectors is induced from the ones on the stabilizers , . However, for general cases, the complexity of these stabilizers would require heavy analysis calculations.
Our work is an extension of the ideas of R. Howe and E. C. Tan [20, Chap V, Theorem 3.3.1]. For , they considered the system of imprimitivity based on instead of “Mackey machine” to calculate upper bounds of finite matrix coefficients restricted on . More precisely, the deformation of orbits under the natural dual action of the Cartan subgroup on gives enough information to get upper bounds of finite matrix coefficients on . In their proof the commutativity of , a maximal compact subgroup in , is essential. However, for general examples, the complexity of orbits and noncommutativity of do seem to require some new method for handling it; the same is true in an attempt at extending the results to nonarchimedean fields.
1.2. Main results
Let be a connected semisimple almost algebraic group with rank, a maximal split torus, a minimal parabolic subgroup containing , the closed positive Weyl chamber of given by the choice of , a good maximal compact subgroup such that the Cartan decomposition holds where is a finite subset of the centralizer of (Section 2.2).
Denote by be the set of nonzero roots of relative to and by the set of positive roots. Let be a finite dimensional vector space over . Fix a normal representation of on . Since char , then by full reducibility of semisimple groups there is a natural decomposition of under : such that for each , is an irreducible representation of . Denote by the set of weights of on relative to and by and the highest weight and lowest weight respectively, compatible with the ordering of . Set , where . Moreover, If , .
Definition 1.2.
Set
where is the set of simple roots of and is the modular function of .
For a unitary representation of , a vector in is called finite if the subspace spanned by is finitedimensional. We use the term finite matrix coefficients of (on ) to refer to its matrix coefficients with respect to finite vectors (restricted on ).
Definition 1.3.
Let be a positive function on , invariant under left and right translations by , and such that , for any . A unitary representation of is said to be bounded on where , if for any finite unit vectors and , one has the estimate
Here denotes the subspace spanned by and similarly for .
Note that if is excellent. By the following theorem, determines a uniform pointwise bound for all finite matrix coefficients of restricted on .
Theorem 1.4.
Fix an excellent representation of on . Let be an integer with . Then for any unitary representation of without nontrivial fixed vectors, is bounded on . Here is the HarishChandra function of (see Section 4.1).
Theorem 1.4 does not provide the sharpest uniform pointwise bounds. The novelty of the above theorem lies in the simplicity of our method giving an upper bound for finite matrix coefficients.
Proposition 7.5 shows that if each almost simple factor of has Kazhdan’s property then the upper bounds of finite matrix coefficients on can be obtained from semisimple part itself by using property or higher rank trick. Theorem 1.4 provides upper bounds of finite matrix coefficients on even if the above condition fails on . Moreover, using the orbitdeformation method (in proving Theorem 1.4) one may obtain sharper upper bounds for finite matrix coefficients on than those offered by semisimple part itself (see various examples in Section 8). For instance, for , and the standard representation of on , the best possible decay rate from itself for finite matrix coefficients is at most one half of the decay rate (provided by the combination of orbitdeformation method and higher rank trick) (see Remark 8.3).
We also have the following interesting result, in which the group is defined by the symmetric or Hermitian form on :
Proposition 1.5.
Let , and be the standard representation of on . Then for any unitary representation of without nontrivial fixed vectors is bounded on for , is bounded on for and is bounded on for .
It turns out that the uniform pointwise bounds for finite matrix coefficients in above proposition are in fact much sharper than the bounds obtained from itself: and don’t have property ; for such that it is not bounded on for any (see Remark 8.3). , while there exists an irreducible representation of
It is proved in [35, Proposition 2.3] and [34, Proposition p.22] that the pair has Kazhdan’s property if is good. Then Theorem 1.4 gives, in the case of local fields with character , the quantitative results. The pointwise bound provides us with a simple and general method of calculating Kazhdan constants (see Section 9 for definition) for various compact subsets of semisimple , in particular for any compact subset properly containing .
It is well known that any almost simple algebraic group with rank does not have property if is nonarchimedean or if Lie algebra of is isomorphic to or for archimedean. We have the following example including the case of not having property :
Proposition 1.6.
Suppose rank and is irreducible and good on . Denote by is the noncompact simple factor of and by a good maximal compact subgroup in . Let be the smallest integer such that where is defined in (8.11). For any such that , is a Kazhdan constant for .
Let be a real semisimple connected Lie group of rank without compact factors and with finite center and a cocompact torsion free lattice in , which admits a representation without nontrivial invariant subspace with eigenvalue . By Margulis’ superrigidity theorem [30], the representation of extends to a homomorphism (otherwise we consider a finite cover of ). Combined with some results of A. Katok and R. Spatzier in [21] and [22], Theorem 1.4 in the case of archimedean fields yields an application in obtaining tame estimates of cocycle equations on order Sobolev space [37].
Acknowledgements. I would like to thank Professor Roger Howe for valuable suggestions and for his encouragement.
2. Preliminaries on almost algebraic groups
We denote by the Lie algebra of (resp. ) when is archimedean (resp. nonarchimedean). Let be the Lie algebra. Throughout the paper will denote a finitedimensional vector space over .
2.1. Unipotent, nilpotent elements and exponential map
Let be a connected linear algebraic group. Denote by the set of unipotent elements of and the subspace of nilpotent elements of where is the Lie algebra of . Then implies that
(2.1) 
belongs to . Conversely, if , then the logarithm
(2.2) 
belongs to . The set and are subvarieties in and respectively. The maps and are inverses of each other, biregular and defined over (see [30, Chapter 0.20]).
Remark 2.1.
The condition char is necessary in guaranteeing the existence of exponential map (resp. logarithm map) of nilpotent elements (resp. unipotent elements) in (resp. group ).
Under an algebraic group morphism the image of any unipotent element is unipotent and every unipotent algebraic group over a field of characteristic is connected. Moreover, we have:
Proposition 2.2.
(see [30]) If is a group morphism, then for each we have , where is the differential of and is Lie algebra of .
2.2. Cartan decomposition
Let (resp. ) be a maximal split Cartan subgroup (resp. split torus) in (resp. ) and (resp. ) a minimal parabolic subgroup of (resp. ) containing (resp. ) when is archimedean (resp. nonarchimedean). For the nonarchimedean case, set and . Let (resp. ) denote the set of characters of (resp. characters of over ) whose ordering is induced from (resp. ). Denote by the set of positive characters in (resp. ) with respect to that ordering.
When is archimedean, we set
When is nonarchimedean, we fix a uniformizer of such that is the cardinality of the residue field of , and set
We set
Equivalently for each . We call a positive Weyl chamber of .
Let (resp. ) denote the centralizer of (resp. ) in (resp. ) for archimedean (resp. nonarchimedean) and set for nonarchimedean. Since (resp. ) can be considered as a subset of (resp. ) in a natural way, it has an induced ordering from this inclusion. Define
For any subgroup of , denotes the normalizer of , denotes the centralizer of and denotes the center of .
Lemma 2.3.
There exists a maximal compact subgroup of such that

,

the Cartan decomposition holds, in the sense that for any , there are elements (unique up to mod ) such that .
See [17] for archimedean case and see [6] and [33] for nonarchimedean case. In general, the positive Weyl chamber has finite index in . Hence for some finite subset , , i.e., for any , there exist unique elements and such that .
A maximal compact subgroup is called a good maximal compact subgroup of if it satisfies the properties listed in the above lemma.
Remark 2.4.
We have if is archimedean or; is split over or; if is quasisplit or split over an unramified extension over a nonarchimedean local field .
2.3. Roots and weights relative to a split torus
Let be a normal representation of on . A character (resp. ) is said to be a weight of in the representation for archimedean (resp. nonarchimedean) if there exists a nonzero vector such that
(2.3) 
and the corresponding weight space of is described as all vectors in satisfying (2.3).
Since char, then by full reducibility of semisimple groups there is a decomposition of under : where is the weight space of and is the set of weights of . Notice that if char then full reducibility fails.
Nontrivial characters of (resp. ) in the adjoint representation of are said to be the roots of for archimedean (resp. nonarchimedean). Denote the set of roots by . For each let be the corresponding root space, i.e.,
Lemma 2.5.
Let be a normal representation of , then the following hold:

If then every root of is a rational combination of weights of ;

the sum of all weights of is trivial.
Proof.
(1) Let be the set of roots. By full reducibility of , there is a decomposition of under : where is the weight space of and is the set of weights.
For each choose and let denote the oneparameter subgroup , . Since is infinite (2.1) and (2.2) yield that the subgroup is infinite. The assumption implies that there exists such that is nontrivial on and maps into [19]. Then it follows that
Then we finish the proof for (1).
(2) Let (resp. ), then is a root system of and the factor group (resp. ) coincides with the weyl group of the root system for archimedean (resp. nonarchimedean) (see [17] and [30]). Moreover, there exists be a complete set of representatives for nonarchimedean [30, Chapter 0.27].
It is clear that the sum of all weights in is invariant under the action of the Weyl group. To prove (2), it is sufficient to prove the following statement: the only element in invariant under the Weyl group is . Let be the subset of containing all vectors invariant under the Weyl group. It is obvious that is a subspace. Suppose . Note that there exists a positive definite inner product on invariant under the Weyl group [30, Chapter 0.26]. Denote by the orthogonal complement of under the inner product. Then is also invariant under the Weyl group. For any , we have a unique decomposition where and . There exists in the Weyl group such that , then we have
By uniqueness of the decomposition, it follows that , that is, . Then immediately we find that , which implies that . Then which contradicts the assumption. Hence our claim is proved. ∎
Remark 2.6.
The condition is weaker than the condition is excellent. If is irreducible, then the two conditions are equivalent.
Next, we will give a detailed description of good maximal compact subgroups for different .
3. Good maximal compact subgroups in
3.1. Maximal compact subgroups in when
For each there exists such that the following subspace
(3.1) 
is a compact subalgebra and the Lie group in with Lie algebra is a maximal compact subgroup in [17, Chapter III]. Let
(3.2) 
3.2. Maximal compact subgroups in when
In this part, we follow the notations and quote the conclusions from [17, Chapter VI] with minor modifications. If is a Cartan decomposition of and is the set of fixed points of the corresponding Cartan involution, then the Lie subgroup in with Lie algebra is a maximal compact subgroup of . Let be the complexification of , put and let and denote the conjugations of with respect to and . The automorphism of of will be denoted by .
Let denote any maximal abelian subspace of and let be any maximal abelian subalgebra of containing . Obviously . We put . Let denote the subspace of generated by . Then is a Cartan subalgebra of . Let . We denote by the set of nontrivial roots of and denote by the corresponding root space of in for each . Since each root is real valued on we get in this way an ordering of . Let denote the set of positive roots. Now for each the linear function , , and defined by
for any are again members of . The root is trivial on if and only if . We divide the positive roots in two classes as follows:
Define
For let be the real vector space spanned by
It is clear that if . Furthermore, if and if .
The following follows from (the proof) of Theorem in [17, Chapter VI]:
Proposition 3.1.
Let , , then


For any , if ; and if .

on ; and for any .
Select a basis (resp. ) of for each (resp. ). Set (resp. ) if (resp. ). Let
(3.3) 
Here we use the unified notations compatible with (3.1).
Let be the connected Lie group in with Lie algebra . Denote by the restriction of to for and let be the set of exponentials of nontrivial restricted roots, i.e.,
where for any . The ordering in induces a compatible ordering in .
Remark 3.2.
Set for each and call a positive Weyl chamber of .
3.3. Maximal compact subgroups in if is nonarchimedean
Let be a good maximal compact subgroup in and let be a metric on induced from an absolute value on . Since is compact and open (see [32, Appendix]), from (2.1) and (2.2) we see that
Lemma 3.4.
There exists such that for any and any with , if then .
For each , choose a basis of where . Let
(3.4) 
Remark 3.5.
As an immediate consequence of Lemma 3.4 we have the following statement: there exists such that for all with and .
4. Preliminaries on matrix coefficients
In this section we list some notations and wellknown properties about matrix coefficients which will be used in this paper.
Definition 4.1.
For a locally compact group , a (continuous) unitary representation of is said to be strongly if there is a dense subset in the Hilbert space attached to such that for any and in , the matrix coefficient lies in . We say is strongly if it is strongly for any .
Since the matrix coefficients of a unitary representation with respect to unit vectors are bounded by , a strongly representation is also strongly for any .
4.1. The HarishChandra function
We denote by the modular function of ; in particular for ,
(4.1) 
where denotes the multiplicity of . The HarishChandra function is defined by
As is well known, is the diagonal matrix coefficient where is the representation which is unitarily induced from the trivial representation and is its unique (up to scalar) invariant unit vector. In fact, is given by
We list some wellknown properties of (see [15], [33] and [38]):
Proposition 4.2.

is a continuous biinvariant function of with values in .

For any , there exist constants and such that
Then the formal sum determines the decay rate of .

is integrable for any .
For archimedean we can write the Haar measure of in terms of the Cartan decomposition : where is a positive function on satisfying
for all and for some constants and if (see [18] and [23, Proposition 5.2.8]).
For nonarchimedean and for any biinvariant function of , we have
Moreover, there exist positive constants and such that for all (see [33, Lemma 4.1.1]). Then we have the following result (detailed proof can be found in [16, Lemma 7.3]):
Lemma 4.3.
Let be a biinvariant continuous function on . If for some then .
4.2. Useful results about matrix coefficients
The following follows from (the proof) of [9, Corollary in pg. 108]:
Theorem 4.4.
For a connected semisimple almost algebraic group and its unitary representation , if is strongly for some positive integer , then is bounded on .
Even though it is assumed that is an semisimple algebraic group in [9, Theorem 2], the proof works for any semisimple almost algebraic group case as well without any change.
Definition 4.5.
Let be a locally compact group and let and be unitary representations of . We say that is weakly contained in if and only if each matrix coefficient of is the limit, uniformly on compacta, of sums of matrix coefficients of
subject to the condition that (see [9] and [11]) and an equivalent definition given by diagonal matrix coefficient is in [12]. Loosely speaking, an irreducible representation is weakly contained in the regular representation if it appears in the Plancherel formula.
In particular, to an arbitrary representation of , we may assign a set supp in the unitary dual of consisting of all which are weakly contained in . If is of type I, as we will assume, then supp is exactly the support of the projectionvalued measure on defining (up to unitary equivalence). The following establishes equivalent definitions of a bounded representation [18, Lemma 6.2].
Proposition 4.6.
Let be a locally compact group with compact subgroup and be a subgroup of . Then a representation of is bounded on if and only if all are bounded on .
We shall also make use of the following lemma which is an obvious consequence of [9], Theorem 4.4 and above proposition (since a connected semisimple almost algebraic group is known to be of Type I (see [4] and [38])).
Lemma 4.7.
Let be a positive integer. Suppose of is a direct integral of unitary representations. Then it is strongly if and only if almost all the integrands are.
5. The Fourier transform and projectionvalued measure
5.1. The Fourier transform
Let be a locally compact abelian group with a Haar measure and denote by its dual group. The Fourier transform of is obtained by restriction:
the bar denoting complex conjugation. In particular, belongs to for all , where is the space of complexvalued continuous functions vanishing at infinity [13, pg. 93]. The space of functions , known as the SchwartzBruhat space of (rapidly decreasing functions on ), is defined such that it has the property: the Fourier transform induces a topological isomorphism (see [5] and [31]). The definition by François Bruhat is a direct limit of spaces of infinitely differentiable, rapidly decreasing functions on quotient spaces of Lie type of and it generalizes the familiar Schwartz space of (see [14]). In particular, if is isomorphic to a finite dimensional vector space over a local field , in the archimedean case, is the space of Schwartz functions; for nonarchimedean case is the space of locally constant functions with compact support.
Theorem 5.1.
For a suitable normalization of the dual Haar measure on , we have:

The Fourier transform from to extends to an isometry from onto .

If and , then for almost every ,

Every defines a unitary character on