On Uniform Admissibility of Unitary and Smooth Representations
Uriya A. First Thomas Rüd
math.RT math.GR 22D10, 22D12


Revisiting D_s0^*(2317) as a 0^+ tetraquark state from QCD sum rules
    Jian-Rong Zhang
    
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Let G be a  reductive algebraic group over a non-Archimedean local field F,
or more precisely, its group of F-points.
The study of complex representations of G often
divides into investigating  two classes of representations:  
unitary representations, i.e. representations
on Hilbert spaces via unitary operators, and smooth representations, i.e.representation on an abstract -vector space such that the stabilizers of all
vectors are   open. 
Historically, the unitary representations served as the original object of interest,
and the smooth representations were introduced in  as a tool to study them.
Interplay between the  two types of representations is abundant 
and is now part of the standard theory.  

      

Building on the work of Harish-Chandra <cit.>,
Bernstein <cit.> proved that 
the irreducible unitary representations of G
are uniformly admissible.
That is to say, for any compact open subgroup  K≤ G, there exists an integer
n=n(K,G) such that the dimension of the K-fixed vectors in any irreducible
unitary 
representation of G is at most n.
Here, “irreducible” should be interpreted in the   category of unitary
representations,
namely, the irreducible unitary  representations 
are those representations admitting no proper closed  subrepresentations other than 0.
Bernstein later showed   that the irreducible smooth
representations of G are  uniformaly admissible as well;
here, “irreducible” means irreducible as a complex G-module.
A standard source for the latter result  is Section II.2.2 
in Bernstein's lecture notes
<cit.>.
Both  results are fundamental in the theory of reductive p-adic Lie groups.

It was noted by  
Bernstein and Zelevinsky in <cit.>

that by combining <cit.>
with a modification of results by Godement
<cit.>, one can formally
derive uniform admissiblity of the irreducible
unitary representations of G from uniform admissiblity of 
its irreducible smooth representations. 
The purpose of this note is to extend this to arbitrary ℓ-groups, 
provide a converse, and give self-contained proofs in the process.
We show:






































	Let G be a  totally disconnected locally compact group such
	that |G/K|<2^ℵ_0 for some compact open subgroup K≤ G. Then
	the following conditions are equivalent:
	
		
  (a) The irreducible unitary representations of G
		are uniformly admissible.
		
  (b) The irreducible smooth representations of G
		are uniformly admissible.
	
	In fact, (b) implies (a) without the assumption |G/K|<2^ℵ_0.


The assumption   |G/K|<2^ℵ_0 is necessary in general; see Remark <ref>.
The proof makes use of polynomial identities, a technique which
goes back at least as far as Kaplansky <cit.>.

Theorem <ref> was applied in <cit.> to show that
irreducible unitary representations of  quasi-reductive groups
over a non-Archimedean local field are uniformly admissible.








































 



























In addition to Theorem <ref>, we   show, separately, that uniform admissibility of the  
irreducible smooth representations is  invariant  under passing to a subgroup or
an overgroup of finite index.


	Let G be a  totally disconnected locally compact group
	and let H≤ G be an open subgroup of finite index. Then
	the following conditions are equivalent:
	
		
  (a) The irreducible smooth representations of G
		are uniformly admissible.
		
  (b) The irreducible smooth representations of H
		are uniformly admissible.
	
	This statement also holds upon replacing “uniformly admissible”
	with “admissible”. 
 

We do not know whether Theorem <ref> holds under the milder
assumption that H is a group such that there exists
a continuous group homomorphism H→ G with finite kernel and cokernel.















	The paper is organized in three sections, the first of which   is preliminary, whereas
	the second and the third are concerned with proving Theorems <ref> and <ref>,
	respectively.
	


	We thank Maarten Solleveld for  encouraging us to publish this note.





§ PRELIMINARIES


We  begin with recalling some necessary facts and setting notation for the sequel.
Throughout, all vector spaces are over , and an algebra
means a unital associative -algebra.

 



 


 §.§ ℓ-Groups
 
 
 Throughout this paper, G denotes a  locally compact totally disconnected group, or an
 ℓ-group for short,
 and μ
 is a fixed left-invariant  measure on G. 
 The modular character of G is denoted
 δ:G→_>0; it 
 is determined by the identity μ(Sg)=μ(S)δ(g), holding for any compact
 S⊆ G and g∈ G. 
 
 
 
 We write K G to denote that K is a compact open subgroup of G. 
 It is well known that such subgroups
 form a basis of neighborhoods to the identity element 1_G; see, for instance, <cit.>.
 
 


	As usual,
	a unitary representation of G
	consists of a complex Hilbert space V endowed with a continuous
	G-action G× V→ V such that ⟨ gu,gv⟩ = ⟨ u,v⟩
	for all u,v∈ V, g∈ G. We say that V is topologically irreducible,
	or just irreducible, if V has no proper nonzero closed
	G-submodules.
	
	A smooth representation of G
	is an abstract complex vector space V together with a
	G-action G× V→ V such that the G-stabilizer of every
	v∈ V is open, or equivalently, V=⋃_K GV^K,
	where V^K:={v∈ V : gv=v for all g∈ K}.
	We say that V is algebraically irreducible, or just irreducible,
	if V is contains no proper nonzero G-modules.
	



	



 §.§ Unital *-Algebras


	By a (unital) *-algebra, we mean a  unital associative  -algebra
	A together with an involution *:A→ A, i.e., 
	an additive map satisfying
	a^**=a, (ab)^*=b^*a^*
	and   
	(α a)^*=α a^* for all a∈ A, α∈.
	
	A unitary representation of a *-algebra A
	is a Hilbert space V endowed with an A-module
	structure such that ⟨ au,v⟩=⟨ u,a^*v⟩
	for all a∈ A, u,v∈ V. A unitary representation 
	is topologically irreducible, or just irreducible, if
	it has no proper nonzero closed A-submodules.
	
	
	


	We shall need the following versions of Schur's lemma.
	
	
		Let A be a *-algebra with _ A<2^ℵ_0
		and let M be a simple left A-module.
		Then _A(M)=.
		
	
	
		Fix some 0≠ m∈ M.
		Then Am=M and hence any element
		f∈_A(M) is uniquely determined by f(m).
		As a result, __A(M)≤_ M≤_A<2^ℵ_0.
		Thus, _A(M) is a division algebra over  of -dimension <||.
		It is well known that such algebras are algebraic over ,
		and hence coincide with . We conclude that _A(M)=.
		
		
	
	
		Let A be a *-algebra  
		and let V be an irreducible unitary
		representation.
		Then all continuous A-endomorphisms of V
		are    scalar.
	
	
	
		See, for instance, <cit.>
	
	


 §.§ The Relative Hecke Algebra


	
	The most important example of a *-algebra that we shall consider
	is   the relative Hecke algebra (G,K)
	associated to a compact open subgroup K G.
	Recall that the underlying vector space
	of (G,K) consists of
	the bi-K-invariant compactly supported functions
	f:G→
	and that its multiplication is given by convolution
	
    (f⋆ g)(y):= ∫_x∈ Gf(x)g(x^-1y) dμ .
 
	
	The unity of ℋ(G,K)
	is e_K:=μ(K)^-11_K, where 1_K:G→
	denotes the characteristic function of K.
	We make (G,K) into a *-algebra
	by setting
	
    f^*(g)=δ(g)f(g^-1)

	
	for all f∈(G,K) and g∈ G (recall that δ is the modular
	character of G).



	Given a unitary or smooth representation V
	of G, the space V^K carries
	a left (G,K)-module structure
	given by
	
    f· v:=∫_g∈ Gf(g)· gv dμ    ∀  f∈(G,K), v∈ V^K

	
	Note that in the smooth case, if we write
	f=∑_iα_iμ(K)^-1g_iK, then the integral is 
	just the finite sum ∑_iα_ig_iv.
	
	When V is a unitary representation, 
	we further have
	⟨ fu,v⟩ = ⟨ u,f^*v⟩
	for all u,v∈ V and f∈(G,K), and so V^K
	becomes  a unitary representation of ℋ(G,K).
	We remark that the equality
	⟨ fu,v⟩ = ⟨ u,f^*v⟩ follows
	easily from the definitions and the identity
	∫_x∈ Gf(x^-1)δ(x)dμ =∫_x∈ Gf(x)dμ,
	which holds whenever the integrals exist.
	
	






























	We shall need the following two easy propositions.













 
Let V be an irreducible smooth representation of G and let K G.
Then either V^K=0, or V^K is a simple ℋ(G,K)-module.



We need to show that ℋ(G,K)· v= V^K for all 0≠ v∈ V^K.
Let u∈ V^K. Since V is irreducible,
there are g_1,…,g_n∈ G and α_1,…,α_n∈
such that 
∑_iα_ig_iv=u.
Let f=e_K⋆ (∑_i α_iμ(K)^-1g_iK)∈ℋ(G,K).
Then

    f· v=e_K· (∑_iα_iμ(K)^-11_g_iK)· v=e_K·∑_iα_i g_iv=e_Ku=u

and we conclude that u∈ℋ(G,K)· v.


	































































































































































 
Let V be an irreducible unitary representation of
G and let K G. Then either V^K=0,
or V^K is an irreducible unitary representation of ℋ(G,K).








Similarly to the proof of Proposition <ref>, we need
to show that V^K is the closure of (G,K)· v for
all 0≠ v∈ V^K.
Since V is topologically irreducible, gv |  g∈ G is dense in V.
For all g∈ G, we have μ^-1(K)gK· v=gv and thus,
gK· v | g∈ G is dense in V.
This in turn implies that
e_K·gK· v | g∈ G is dense in e_KV=V^K.
Since e_K⋆gK∈(G,K), it follows
that (G,K)· v is dense in V.







 §.§ Miscellaneous Results





We finish this   section with recalling the Amistur–Levitzky theorem
and two density theorems due to Jacobson and von Neumann. These will be needed
in proving Theorem <ref>.





To state the Amitsur–Leviski theorem, let ⟨ x_1,…,x_n⟩
denote the ring of polynomials in n non-commuting variables
over . The n-th standard polynomial is 
define by

    S_n(x_1,…,x_n)=∑_σsgn(σ)x_σ(1)… x_σ(n)

where σ runs over all permutations on {1,…,n}.
Recall that
a polynomial f∈⟨ x_1,…,x_n⟩ is called an identity
of a ring R if f vanishes on all inputs from R.
 





Let V be a -vector space and let n∈.
The polynomial S_n(x_1,…,x_n) is a polynomial identity of _(V) if and only
if n≥ 2  V.














	Let R be a ring, let M be a simple left R-module, and let D be the
	R-endomorphism ring of M. 
	Give M the discrete topology and _D(M) the topology induced
	by the product topology on M^M (the point-wise convergence topology).
	Then the image of R in _D(M) under the map sending r to (m↦ rm)
	is dense in _D(M).
 














See, for instance, <cit.>. 



















Given a Hilbert space V, let (V) denote
the algebra of bounded linear automorphisms of V.
For a subset S⊆(V),
we let S' denote the commutant of S, namely, the set of elements
of (V) that commute with all elements in S.  














 
Let V be a Hilbert space and let A⊂(V) be a unital subalgebra closed under taking adjoints. 
Then A is dense in A” in the strong  operator topology (and hence also in the weak operator topology). 



See, for instance, <cit.>.





§ PROOF OF THEOREM <REF>


















 
We begin with several results about -algebras and *-algebras.
The Jacobson radical of a ring R is denoted (R).








	Let A be a unital -algebra. Consider the conditions:
	

  (a) Every simple A-module has -dimension at most n.

  (b) A/(A) has S_2n 
as a polynomial identity. 
	
	Then (a) implies (b), and the converse holds provided _ A<2^ℵ_0.






	Suppose (a) holds.
	Let 𝒮 be a set of representative for the isomorphism
classes of simple left A-modules.
	By the definition of (A),  the map

    A/(A) 
    →∏_M∈𝒮_(M)

sending a+(A)∈ A/(A) to (x↦ ax)∈_(M) in the M-component, is injective.
Condition (a) and Theorem <ref> imply that S_2n
is a polynomial identity of _(M) for all S∈𝒮,
and hence also of A/(A). 

	Suppose now that (b) holds and _ A<2^ℵ_0.
	Let M be a simple A-module.
	Consider M as a discrete topological space and make
	_(M) into a topological ring by giving it the topology 
	induced from the product topology on M^M.
	By Proposition <ref>, _A(M)=,
	and hence, by  Theorem <ref>,
	the image of A in _(M) is dense.
	Since the map A→_(M) factors through A/(A),
	condition (b) implies that S_2n vanishes on all inputs from dense subset
	of _(M). Since the topology on _(M) is Hausdorff,
	it follows that S_2n is a polynomial identity of _(M).
	Thus, M≤ n by Theorem <ref>.



	Let A be a unitary *-algebra and let V
	be a unitary representation of A.
	Then (A)· V=0.



	Let a∈(A). Suppose first that a^*=a.
	Then the operator T_a:V→ V given by T_a(v)=av
	is self-adjoint. Since a∈(A), the element
	λ-a is invertible in A for all 0≠λ∈.
	As a result,  Spec(T_a)={0}, and 
	since T is self-adjoint, this forces T=0 and a· V=0.
	
	The general case follows by noting that any
	a∈(A) satisfies
	a=a+a^*/2+i(a-a^*/2i) with   a+a^*/2 and
	a-a^*/2i
	being elements of (A) fixed by *. 



	Let A be a unital *-algebra. Consider the   conditions:
	

  (a') Every irreducible unitary representation of A  has -dimension at most n.

  (b) A/(A) has S_2n 
as a polynomial identity.
	 
	Then (b) implies (a'), and the converse holds provided A/(A) admits a faithful unitary
	representation.






	Suppose (b) holds and let V be an irreducible unitary
	representation of A. 
	We may
	replace A with its image in (V)  and hence assume
	that (A)=0, by Lemma <ref>.
	Now,    by Proposition <ref>, the commutant
	A' is , and hence A”=(V).
	By Theorem <ref>, A is dense in (V)
	in the strong operator topology. Condition (b) therefore implies
	that S_2n is a polynomial identity of (V).
	It is easy to check that this is impossible if V is infinite-dimensional,
	whereas in the finite-dimensional case, Theorem <ref>
	implies that V≤ n, as required.
	
	Suppose now that (a') holds and A/(A) admits a faithful unitary
	representation  V. 
	
	By Lemma <ref>, we may replace A with its image in 
	(V) and assume that (A)=0 hereafter.
	Let 0≠ a∈ A. The the operator 
	v↦ av:V→ V is nonzero, and hence, by
	<cit.>, there exists an irreducible
	unitary representation V_a of A such that the image of
	a in (V_a) is nonzero.
	Consequently, the ring homomorphism
	
    A  ∏_a≠ Aℬ(V_a)

	is injective. By condition (a') and Theorem <ref>, 
	S_2n is a polynomial identity of each of the factors (V_a),
	and a fortiori of A as well.

 













































































We finally prove Theorem <ref> by establishing the following
slightly stronger version.

Notice that if   |G/K|<2^ℵ_0 holds
for some K G, then it holds for all K G.



Let G be an ℓ-group, let K be a compact open subgroup with |G/K|<2^ℵ_0,
and let n∈. 
Then the following conditions are equivalent:


  (a) All irreducible unitary representations V of G satisfy V^K≤ n.

  (b) All irreducible smooth representations V of G satisfy V^K≤ n.

In fact, the implication (b)⟹(a) holds without assuming |G/K|<2^ℵ_0.


 
Let A=(G,K).
By Propositions <ref> and <ref>,
it is enough to show that every simple A-module is of dimension ≤ n
if and only if every irreducible unitary representation of A is of dimension ≤ n.
This would follow from Propositions <ref> and <ref>
if we check that _A<2^ℵ_0 and A admits a faithful unitary
representation.
The former condition holds since |G/K|<2^ℵ_0,
whereas for the second condition one can consider the left action
of A=(G,K) on L^2(K∖G) by convolution.
Indeed, if a⋆ f=0 for all f∈L^2(K∖G),
then a=a⋆ e_K=0.
















	(i) The assumption that   |G/K|<2^ℵ_0 in Theorems <ref>
	and <ref> is necessary in general.
	For instance, consider the function field
	(t) and let G be the multiplicative group (t)^× with the discrete
	topology.
	Since G is commutative and locally compact, 
	
	any unitary
	representation of G is 1-dimensional.
	However, (t) with the G-action induced by the product
	in (t) is an infinite-dimensional
	irreducible smooth representation of G.
	
	(ii) The assumption that _ A<2^ℵ_0
	in Proposition <ref> cannot be dropped
	in general. 
	For example, take A=(t) and note that
	S_2(x_1,x_2)=x_1x_2-x_2x_1 is a polynomial identity
	of A, while A   itself is a simple
	A-module of infinite -dimension. 
	
	(iii) The assumption that A/(A) admits a faithful unitary
	representation in Proposition <ref> is necessary
	in general.
	A counterexample can be constructed as follows: Let * denote
	the unique involution of (t) extending the complex
	conjugation and fixing t.
	Let A=_2((t))
	and let *:A→ A be the involution given by
	(f_ij)↦ (f^*_ji).
	Then S_2(x_1,x_2)=x_1x_2-x_2x_1  is not a polynomial
	identity of A, but all irreducible unitary
	representations of A have dimension ≤ 1.
	In fact, the latter   holds vacuously, since A has no
	nonzero unitary representations. To see this, note that
	if V is a Hilbert space and A→(V) is a *-homomorphism,
	then the image of [[ t 0; 0 t ]]
	in (V) is a self-adjoint operator with an empty spectrum, forcing V=0.



 

§ PROOF OF THEOREM <REF>






	Theorem <ref>  follows readily from the following theorem.


	Let H be an open  subgroup of G of finite index  
	and let K be a compact open subgroup of H. Then there is n∈ 
	and a compact open subgroup L≤ K such
	that:
	
		
  (i) For any irreducible smooth
		representation V of H, there are   irreducible 
		smooth representations U_1,…,U_m (m≤ n) of G
		such that V^K≤∑_i U_i^L.
		
  (ii) For any irreducible smooth
		representation U of G, there are   irreducible 
		smooth representations V_1,…,V_m (m≤ n) of H
		such that U^K≤∑_i V_i^L.
	

 












	Since [G:H] is finite, H has only finitely
	many conjugates. Their intersection, N, is an open  normal subgroup
	of finite index. We may replace K with K∩ N.
	The theorem would now follow if we establish it for
	the  pairs (G,N) and (H,N). It is therefore enough to prove
	the theorem when H is normal in G.
	Under this extra assumption, we  shall see that L=K  
	and n:=[G:H] will suffice.





	(i) Let g_1,…,g_n be representatives
	for the H-cosets in G.
	Let V_i be the irreducible smooth representation of H obtained by giving
	V  the H-action (h,v)↦ (g_ihg_i^-1)v (note that
	g_ihg_i^-1∈ H since H G). 
	
	Consider the  (non-normalized) induction 
	
    Ind_H^G(V) := {f:G→ V | f(hg)=h· f(g) for all h∈ H, g∈ G} .

	As usual, the group G acts on Ind_H^G(V) by translations on the right.
	Let V'_i denote the functions in Ind_H^G(V)
	supported on g_iH=Hg_i. It is easy to see that V'_i
	is an H-subrepresention of Ind_H^G(V),
	that Ind_H^G(V)=⊕_i=1^n V'_i
	and that f↦ f(g_i):V'_i→ V_i
	defines an isomorphism of H-modules.
	Thus, the length of Ind_H^G(V) as a complex
	H-module is   n, and so the length of Ind_H^G(V) as a complex
	G-module is at most n. 
	
	Let U_1,…,U_m denote the composition
	factors of Ind_H^G(V), regarded as a G-module.
	Then m≤ n, and ∑_i U_i^K=Ind_H^G(V)^K	≥ V^K,
	so we have proved (i).
		























	(ii) Let U be an irreducible smooth representation of G, and let
	g_1,…,g_n be as in (i). Since U is a simple   G-module,
	it is finitely generated. As [G:H]<∞, it follows
	that U is also finitely generated as a complex H-module.
	Thus, there exists a maximal complex H-submodule M_0≤ U.
	Since H is normal in G, the space gM_0 is  a maximal H-submodule for
	all g∈ G. Let ={gM_0 | g∈ G}. Then  consists of 
	at most [G:H]=n elements, call them M_1,…,M_m.
	Write V_i=U/M_i and set V=⊕_i=1^m U/M_i=⊕_M∈ U/M.
	Then action of H on V extends to a G-action
	by setting g·⊕_M∈(v_M+M)=⊕_M∈(gv_g^-1M+M)
	and the map u↦⊕_M∈ (u+M):U→ V defines
	a nontrivial G-homomorphism, which must be injective since V is irreducible.
	It follows that U^K≤ V^K=∑_i=1^m V_i^K, 
	and the proof is complete.












 


	The proof of part (ii) also shows that any irreducible complex G-module
	has   finite length as a complex H-module.


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