On tempered representations

David Kazhdan; Alexander Yom Din

doi:10.1515/crelle-2022-0019

Article

On tempered representations

David Kazhdan and Alexander Yom Din

Published/Copyright: April 28, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 788

Abstract

Let G be a unimodular locally compact group. We define a property of irreducible unitary G-representations V which we call c-temperedness, and which for the trivial V boils down to Følner’s condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness.

We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V’s, as well as for all tempered V’s in the cases of G:=SL2⁢(ℝ) and of G=PGL2⁢(Ω) for a non-Archimedean local field Ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K-finite vectors.

In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V, generalising a formula of Harish-Chandra from the case when V is square-integrable.

Funding statement: Alexander Yom Din’s research was supported by the Israel Science Foundation (grant No. 1071/20). David Kazhdan’s research was partially supported by ERC grant No. 669655.

A Auxiliary claims regarding polynomial growth of exponential integrals and sums

A.1 Some notation

We denote [n]:={1,2,…,n} and

ℂ≤0:={z∈ℂ:Re⁢(z)≤0},D:={z∈ℂ:|z|≤1}.

Given x=(x1,…,xn)∈ℝ≥0n and m=(m1,…,mn)∈ℤ≥0n, we write xm:=x1m1⁢⋯⁢xnmn. Given λ∈ℂ≤0n, we denote

Jλ:={1≤j≤n:Re⁢(λj)=0}.

Given (λ,m)∈ℂ≤0n×ℤ≥0n, we denote d⁢(λ,m):=∑j∈Jλ(1+mj). Given J⊂[n] and some set X, let us denote by resJ:Xn→XJ the natural restriction and by extJ:XJ→Xn the natural extension by zero.

We fix a finite set ℐ⊂ℝ≥0n with the property that given j∈[n] there exists v∈ℐ such that 〈v,ej〉≠0, where ej is the j-th standard basis vector. We denote

P<r:={x∈ℝ≥0n:〈v,x〉<r⁢ for all ⁢v∈ℐ}.

Given J⊂[n], we denote by PJ⊂ℝ≥0J the convex pre-compact subset

{y∈ℝ≥0J:extJ⁢(y)∈P<1}.

In Section A.3 we will also use the following notations. We consider a compact space B equipped with a nowhere vanishing Radon measure db. Let us say that a function

ϕ:B×ℝ≥0n→ℂ

is nice if it is expressible as

B×ℝ≥0n→idB×eiB×Dn→ϕ∘ℂ,

where ei⁢(x1,…,xn):=(e-x1,…,e-xn) and ϕ∘ is continuous and holomorphic in the second variable (in the sense that when we fix the variable in B it is the restriction of a holomorphic function on a neighbourhood of Dn). Given J⊂[n], we denote by resJ⁢ϕ:B×ℝ≥0J→ℂ the function given by

resJ⁢ϕ⁢(b,y):=ϕ∘⁢(b,extJ⁢(ei⁢(y))).

We also write ϕ⁢(b,+∞) for ϕ∘⁢(b,0), etc.

A.2 Growth – The case of summation over a lattice

Lemma A.1.

Let λ:=(λ1,…,λn)∈C≤0n, let m:=(m1,…,mn)∈Z≥0, and let K be a compact subset of R≥0n. Assume that Re⁢(λ)=0 and λ∉(2⁢π⁢i)⁢Zn. We have

supQ⊂K⁡|1rn⁢∑x∈1r⁢ℤ≥0n∩Qxm⁢er⁢〈λ,x〉|=O⁢(r-1)

as r→+∞, where Q denote convex subsets.

Proof.

Let us re-order the variables, assuming that λ1∉2⁢π⁢i⁢ℤ. Let us write x=(x1,x′), where x′=(x2,…,xn) and analogously write m′, etc. Given a convex subset Q⊂K and x′∈ℝ≥0n-1, let us denote by Qx′⊂ℝ≥0 the subset consisting of x1 for which (x1,x′)∈Q (it is an interval). Let us enlarge K for convenience, writing it in the form K=K1×K′, where K1⊂ℝ≥0 is a closed interval and K′⊂ℝ≥0n-1 is the product of closed intervals.

We have

∑x∈1r⁢ℤ≥0n∩Qxm⁢er⁢〈λ,x〉=∑x′∈1r⁢ℤ≥0n-1∩K′(x′)m′⁢er⁢〈λ′,x′〉⁢(∑x1∈1r⁢ℤ≥0∩Qx′x1m1⁢er⁢λ1⁢x1).

We have Qx′⊂K′ and it is elementary to see that

supR⊂K′⁡|∑x1∈1r⁢ℤ≥0∩Rx1m1⁢er⁢λ1⁢x1|=O⁢(1)

as r→+∞, where R denote intervals. Therefore we obtain, for some C>0 (not depending on Q) and all r≥1,

|1rn⁢∑x∈1r⁢ℤ≥0n∩Qxm⁢er⁢〈λ,x〉|≤C⁢(1rn-1⁢∑x′∈1r⁢ℤ≥0n-1∩K′(x′)m′)⁢r-1.

Since the expression in brackets is clearly bounded independently of r, we are done. ∎

Lemma A.2.

Let (λ,m)∈C≤0n×Z≥0n. Then the limit

limr→+∞⁡1rd⁢(λ,m)⁢∑x∈ℤ≥0n∩P<rxm⁢e〈λ,x〉⁢d⁢x

exists, and it is equal to 0 if resJλ⁢(λ)∉2⁢π⁢i⋅ZJλ and otherwise equal to

(∫PJλyresJλ⁢(m)⁢𝑑y)⁢(∑z∈ℤ≥0JλczresJλc⁢(m)⁢e〈resJλc⁢(λ),z〉)

(the sum converging absolutely).

Proof.

Let us abbreviate J:=Jλ. Let us denote λ′:=resJ⁢(λ) and λ′′:=resJc⁢(λ), and similarly for m. Given x′′∈ℤ≥0Jc, let us denote by P(<r)x′′⊂ℝ≥0J the subset consisting of y′ for which extJ⁢(r⁢y′)+extJc⁢(x′′)∈P<r.

We have

∑x∈ℤ≥0n∩P<rxm⁢e〈λ,x〉=rd⁢(λ,m)-|J|⁢∑x′′∈ℤ≥0Jc(x′′)m′′⁢e〈λ′′,x′′〉⁢∑y′∈1r⁢ℤ≥0J∩P(<r)x′′(y′)m′⁢er⁢〈λ′,y′〉

=:△.

Let us assume first that λ′∉2⁢π⁢i⋅ℤJ. Then by Lemma A.1 there exists C>0 such that for all convex subsets Q⊂PJ and all r≥1 we have

|1r|J|⁢∑y′∈1r⁢ℤ≥0J∩Q(y′)m′⁢er⁢〈λ′,y′〉|≤C⋅r-1.

Therefore

|△|≤C⁢rd⁢(λ,m)-1⁢∑x′′∈ℤ≥0Jc(x′′)m′′⁢e〈Re⁢(λ′′),x′′〉,

giving the desired.

Now we assume that λ′∈2⁢π⁢i⋅ℤJ. It is not hard to see that

limr→+∞⁡1r|J|⁢∑y′∈1r⁢ℤ≥0J∩P(<r)x′′(y′)m′=∫PJ(y′)m′⁢𝑑y′.

Hence we have (by dominated convergence)

limr→+∞⁡1rd⁢(λ,m)⁢△=∑x′′∈ℤ≥0Jc(x′′)m′′⁢e〈λ′′,x′′〉⁢∫PJ(y′)m′⁢𝑑y′.∎

Claim A.3.

Let p≥1, let {(λ(ℓ),m(ℓ))}ℓ∈[p]⊂C≤0n×Z≥0n be a collection of pairwise different couples and let {c(ℓ)}ℓ∈[p]⊂C∖{0} be a collection of non-zero scalars. Denote

d:=maxℓ∈[p]⁡d⁢(2⁢R⁢e⁢(λ(ℓ)),2⁢m(ℓ)).

The limit

limr→+∞⁡1rd⁢∑x∈ℤ≥0n∩P<r|∑ℓ∈[p]c(ℓ)⁢xm(ℓ)⁢e〈λ(ℓ),x〉|2

exists and is strictly positive.

Proof.

Let us break the integrand into a sum following

|∑ℓ∈[p]Aℓ|2=∑ℓ1,ℓ2∈[p]Aℓ1⁢Aℓ2¯.

Using Lemma A.2, we see the that resulting limit breaks down as a sum, over (ℓ1,ℓ2)∈[p]2, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the (ℓ1,ℓ2) place is zero unless d⁢(λ(ℓ1),m(ℓ1))=d, d⁢(λ(ℓ2),m(ℓ2))=d, Jλ(ℓ1)=Jλ(ℓ2) and resJ(ℓ1)⁢(λ(ℓ2))-resJ(ℓ1)⁢(λ(ℓ1))∈2⁢π⁢i⋅ℤJ. We thus can reduce to the case when, for a given J⊂[n], we have Jλ(ℓ)=J for all ℓ∈[p], we have d⁢(λ(ℓ),m(ℓ))=d for all ℓ∈[p], and we have resJ⁢(λ(ℓ2))-resJ⁢(λ(ℓ1))∈2⁢π⁢i⋅ℤJ for all ℓ1,ℓ2∈[p]. We then obtain, using Lemma A.2, that our overall limit equals

∑z∈ℤ≥0Jc∫PJ|∑ℓ∈[p]c(ℓ)⁢yresJ⁢(m(ℓ))⁢zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉|2⁢𝑑y.

It is therefore enough to check that

∑ℓ∈[p]c(ℓ)⁢yresJ⁢(m(ℓ))⁢zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉,

a function in (z,y)∈ℤ≥0Jc×PJ, is not identically zero. By the local linear independence of powers of y, we can further assume that resJ⁢(m(ℓ)) is independent of ℓ∈[p], and want to check that

∑ℓ∈[p]c(ℓ)⁢zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉,

a function in z∈ℤ≥0Jc, is not identically zero. Notice that, by our assumptions, the elements in the collection {(resJc⁢(λ(ℓ)),resJc⁢(m(ℓ)))}ℓ∈[p] are pairwise different. Thus the non-vanishing of our sum is clear (by linear algebra of generalized eigenvectors of shift operators on ℤJc). ∎

A.3 Growth – The case of an integral

Lemma A.4.

Let λ:=(λ1,…,λn)∈C≤0n, let m:=(m1,…,mn)∈Z≥0, and let K be a compact subset of R≥0n. Assume that Re⁢(λ)=0 and λ≠0. We have

supQ⊂K⁡|∫Qxm⁢er⁢〈λ,x〉⁢𝑑x|=O⁢(r-1)

as r→+∞, where Q denote convex subsets.

Proof.

Let us re-order the variables, assuming that λ1≠0. Let us write x=(x1,x′), where x′=(x2,…,xn) and analogously write m′, etc. Given a convex subset Q⊂K and x′∈ℝ≥0n-1, let us denote by Qx′⊂ℝ≥0 the subset consisting of x1 for which (x1,x′)∈Q (it is an interval). Let us enlarge K for convenience, writing it in the form K=K1×K′, where K1⊂ℝ≥0 is a closed interval and K′⊂ℝ≥0n-1 is the product of closed intervals.

Using Fubini’s theorem,

∫Qxm⁢er⁢〈λ,x〉⁢𝑑x=∫K′(x′)m′⁢er⁢〈λ′,x′〉⁢(∫Qx′x1m1⁢er⁢λ1⁢x1⁢𝑑x1)⁢𝑑x′.

We have Qx′⊂K′ and it is elementary to see that

supR⊂K′⁡|∫Rx1m1⁢er⁢λ1⁢x1⁢𝑑x1|=O⁢(r-1)

as r→+∞, where R denote intervals. Therefore we obtain, for some C>0 and all r≥1,

|∫Qxm⁢er⁢〈λ,x〉⁢𝑑x|≤C⁢(∫K′(x′)m′⁢𝑑x′)⁢r-1,

as desired. ∎

Lemma A.5.

Let (λ,m)∈C≤0n×Z≥0n and let ϕ:B×R≥0n→C be a nice function. Then the limit

limr→+∞⁡1rd⁢(λ,m)⁢∫B∫P<rxm⁢e〈λ,x〉⁢ϕ⁢(b,x)⁢𝑑x⁢𝑑b

exists, and it is equal to 0 if resJλ⁢λ≠0 and otherwise equal to

(∫PJλyresJλ⁢(m)⁢𝑑y)⁢(∫B∫ℝ≥0JλczresJλc⁢(m)⁢e〈resJλc⁢(λ),z〉⁢resJλc⁢ϕ⁢(b,z)⁢𝑑z⁢𝑑b)

(the double integral converging absolutely).

Proof.

Let us re-order the variables, assuming that J:=Jλ=[k]. Write x=(x′,x′′), where x′ consists of the first k components and x′′ consists of the last k components. Let us write analogously m′,λ′, etc.

First, let us notice that if k≠0, we can write

ϕ⁢(b,x)=e-x1⁢ϕ0⁢(b,x)+ϕ1⁢(b,x),

where ϕ0,ϕ1:B×ℝ≥0n→ℂ are nice functions and ϕ1 does not depend on x1. Dealing with e-x1⁢ϕ0⁢(b,x) instead of ϕ⁢(b,x) makes us consider λ with smaller set Jλ and thus (λ,m) with a smaller d⁢(λ,m) and from this, reasoning inductively, we see that we can assume that ϕ only depends on (b,x′′). Let us write ϕ′′:=resJc⁢ϕ.

Let us perform a change of variables y′:=1r⁢x′. Let P(<r)⊂ℝ≥0n denote the transform of P<r under this changes of variables (i.e. (x′,x′′)∈P<r if and only if (y′,x′′)∈P(<r)). We obtain

∫B∫P<rxm⁢e〈λ,x〉⁢ϕ⁢(b,x)⁢𝑑x⁢𝑑b

=rd∫B∫P(<r)(y′)m′er⁢〈λ′,y′〉(x′′)m′′e〈λ′′,x′′〉ϕ′′(b,x′′)dy′dx′′db=:△.

Given x′′∈ℝ≥0Jc, let us denote by P(<r)x′′⊂ℝ≥0J the set

{y′∈ℝ≥0J:(y′,x′′)∈P(<r)}.

Notice that P(<r1)x′′⊂P(<r2)x′′ for r1<r2 and ⋃rP(<r)x′′=PJ. Using Fubini’s theorem,

△=rd⁢∫B∫ℝ≥0Jc(x′′)m′′⁢e〈λ′′,x′′〉⁢ϕ′′⁢(b,x′′)⁢(∫P(<r)x′′(y′)m′⁢er⁢〈λ′,y′〉⁢𝑑y′)⁢𝑑x′′⁢𝑑b.

If λ′≠0, by Lemma A.4 there exists C>0 such that for all convex subsets Q⊂PJ and all r≥1 we have

|∫Q(y′)m′⁢er⁢〈λ′,y′〉⁢𝑑y′|≤C⋅r-1.

We have therefore

|△|≤C⋅rd-1⋅∫B∫ℝ≥0Jc(x′′)m′′⁢e〈Re⁢(λ′′),x′′〉⁢|ϕ′′⁢(b,x′′)|⁢𝑑x′′⁢𝑑b

and thus indeed the desired limit is equal to 0.

Now we assume λ′=0. Using Lebesgue’s dominated convergence theorem, we have

limr→+∞⁡1rd⁢△=limr→+∞⁡∫B∫ℝ≥0Jc(x′′)m′′⁢e〈λ′′,x′′〉⁢ϕ′′⁢(b,x′′)⁢(∫P(<r)x′′(y′)m′⁢𝑑y′)⁢𝑑x′′⁢𝑑b

=∫B∫ℝ≥0Jc(x′′)m′′⁢e〈λ′′,x′′〉⁢ϕ′′⁢(b,x′′)⁢(∫PJ(y′)m′⁢𝑑y′)⁢𝑑x′′⁢𝑑b,

as desired. ∎

Claim A.6.

Let {(λ(ℓ),m(ℓ))}ℓ∈[p]⊂C≤0n×Z≥0n be a collection of pairwise different couples. Let {ϕ(ℓ)}ℓ∈[p] be a collection of nice functions B×R≥0n→C such that for every ℓ∈[p] the function b↦ϕ(ℓ)⁢(b,+∞) on B is not identically zero. Denote

d:=maxℓ∈[p]⁡d⁢(2⁢R⁢e⁢(λ(ℓ)),2⁢m(ℓ)).

The following limit exists and is strictly positive:

limr→+∞⁡1rd⁢∫B∫P<r|∑ℓ∈[p]xm(ℓ)⁢e〈λ(ℓ),x〉⁢ϕ(ℓ)⁢(b,x)|2⁢𝑑x⁢𝑑b.

Proof.

Let us break the integrand into a sum following

|∑ℓ∈[p]Aℓ|2=∑ℓ1,ℓ2∈[p]Aℓ1⁢Aℓ2¯.

Using Lemma A.5, we see the that resulting limit breaks down as a sum, over (ℓ1,ℓ2)∈[p]2, of limits which exist, so the only thing to check is that the resulting limit is non-zero. It is easily seen that the limit at the (ℓ1,ℓ2) place is zero unless d⁢(λ(ℓ1),m(ℓ1))=d, d⁢(λ(ℓ2),m(ℓ2))=d, Jλ(ℓ1)=Jλ(ℓ2) and resJ(ℓ1)⁢(λ(ℓ1))=resJ(ℓ1)⁢(λ(ℓ2)). We thus can reduce to the case when, for a given J⊂[n], we have Jλ(ℓ)=J for all ℓ∈[p], we have d⁢(λ(ℓ),m(ℓ))=d for all ℓ∈[p], and we have resJ⁢(λ(ℓ1))=resJ⁢(λ(ℓ2)) for all ℓ1,ℓ2∈[p]. We then obtain, using Lemma A.5, that our overall limit equals

∫B∫ℝ≥0Jc∫PJ|∑ℓ∈[p]yresJ⁢(m(ℓ))⁢zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉⁢resJc⁢ϕ(ℓ)⁢(b,z)|2⁢𝑑y⁢𝑑z⁢𝑑b.

It is therefore enough to check that

∑ℓ∈[p]yresJ⁢(m(ℓ))⁢zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉⁢resJc⁢ϕ(ℓ)⁢(b,z),

a function in (b,z,y)∈B×ℝ≥0Jc×PJ, is not identically zero. By the local linear independence of powers of y, we can further assume that resJ⁢(m(ℓ)) is independent of ℓ∈[p], and want to check that

∑ℓ∈[p]zresJc⁢(m(ℓ))⁢e〈resJc⁢(λ(ℓ)),z〉⁢resJc⁢ϕ(ℓ)⁢(b,z),

a function in (b,z)∈B×ℝ≥0Jc, is not identically zero. Notice that, by our assumptions, the elements in the collection {(resJc⁢(λ(ℓ)),resJc⁢(m(ℓ)))}ℓ∈[p] are pairwise different and for every ℓ∈[p], the function b↦ϕ(ℓ)⁢(b,extJc⁢(+∞)) on B is not identically zero. Considering the partial order on ℂJc given by μ1≤μ2 if μ2-μ1∈ℤ≥0Jc, we can pick ℓ∈[p] for which resJc⁢(λ(ℓ)) is maximal among the {resJc⁢(λ(ℓ′))}ℓ′∈[p]. We can then pick b∈B such that ϕ(ℓ)⁢(b,extJc⁢(+∞))≠0. We then boil down to Lemma A.7 that follows. ∎

In the end of the proof of Claim A.6 we have used the following:

Lemma A.7.

Let {(λ(ℓ),m(ℓ))}ℓ∈[p]⊂Cn×Z≥0n be a collection of pairwise different couples. Let {ϕ(ℓ)}ℓ∈[p] be a collection of nice functions R≥0n→C (so here B={1}). Suppose that ϕ(ℓ)⁢(+∞)≠0 for some ℓ∈[p] for which λ(ℓ) is maximal among the {λ(ℓ′)}ℓ′∈[p] with respect to the partial order λ1≤λ2 if λ2-λ1∈Z≥0n. Then the function

x↦∑ℓ∈[p]xm(ℓ)⁢e〈λ(ℓ),x〉⁢ϕ(ℓ)⁢(x)

on R≥0n is not identically zero.

Proof.

We omit the proof of the lemma – one develops the ϕ(ℓ) into power series in e-x1,…,e-xn and uses separation by generalized eigenvalues of the partial differentiation operators ∂x1,…,∂xn. ∎

Acknowledgements

We would like to thank Pavel Etingof for great help with the proof of Claim 7.3 in the case G=SL2⁢(ℝ), which was present in a prior draft of the paper, before we encountered the work [6]. We would like to thank Vincent Lafforgue for a very useful discussion. We thank Erez Lapid for useful correspondence.

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Received: 2022-02-11

Published Online: 2022-04-28

Published in Print: 2022-07-01

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Articles in the same Issue

https://doi.org/10.1515/crelle-2022-0019